CBSE Toll-free counselling helpline to handle exam stress

With board examination just a month away, the Central Board of Secondary Education (CBSE) is all set to start its psychological counselling helpline from February 9 for students grappling with pre-exam stress.

The board on Tuesday announced that this service will be provided for the 20th consecutive year to both

"The pre-examination counselling for students and parents will be on from February 9 till April 29. It is an outreach programme which caters to the heterogeneous student population and vast geographical network of schools," reads an official circular that was sent to more than 100 CBSE-affiliated schools in Bhopal.

The board has asked schools to inform students about the helpline and what help principals and trained counsellors from CBSE-affiliated schools located in and outside India will offer.

This year, 90 principals, trained counsellors from CBSE-affiliated government and private schools, a few psychologists and special educators will participate in tele-counselling to address exam-related issues of students and parents.

CBSE Helpline toll free number 1800 11 8004

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Olympiad: Homi Bhabha Centre For Science Education

About:

The international Olympiad movement is aimed at bringing the most gifted secondary and higher secondary students of the world together in a friendly competition of the highest level. The Olympiads do not lead directly to any career benefits; rather, they provide a stimulus to begin a career in science or mathematics, to undertake a lifelong journey into the realms of exciting intellectual challenges. The Olympiads are not merely competitions, they are the meeting places of the brightest young minds of the world, and many friendships forged at the Olympiads form the seeds of scientific collaboration later in life. Much like the Olympics in sports, the Olympiads are a celebration of the very best in school level science and mathematics.

A major national Olympiad programme in basic sciences and mathematics which connects to the international Olympiads is in operation in India. The Homi Bhabha Centre for Science Education is the nodal centre of the country for this programme. The programme aims at promoting excellence in science and mathematics among pre-university students.

Among the sciences, the Olympiad programme in Astronomy (junior and senior level), Biology, Chemistry, Junior Science and Physics is a five stage process for each subject separately. The first stage for each subject is organized by the Indian Association of Physics Teachers (IAPT) in collaboration with teacher associations in other subjects. All the remaining stages are organized by Homi Bhabha Centre for Science Education (HBCSE). Read More....

Important Links:

• Olympiad Math

• Olympiad Physics

• Olympiad chemistry

• Olympiad Biology

Contact Details:

Homi Bhabha Centre for Science Education(TIFR)
V. N. Purav Marg,
Next to Anushakti Nagar bus depot,
Mankhurd, Mumbai - 400 088.
2507 2322 (Science Olympiads)
2507 2207 (Mathematical Olympiad)
Website: www.olympiads.hbcse.tifr.res.in

Courtesy: Olympiad

CBSE Papers

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• Download Model Papers

NCERT Mathematics Question Paper (Class - 12)

:: Chapter 1 - Number System ::

Q1. Determine whether each of the following relations are reflexive, symmetric and transitive:

(ii) Relation R in the set N of natural numbers defined as
R = {(x, y) : y = x + 5 and x < 4}

(iii) Relation R in the set A = {1, 2, 3, 4, 5, 6} as
R = {(x, y) : y is divisible by x}

(iv) Relation R in the set Z of all integers defined as
R = {(x, y) : x – y is an integer}

Q 1. Determine whether each of the following relations are reflexive, symmetric and transitive:

(i) Relation R in the set A = {1, 2, 3, ..., 13, 14} defined as
R = {(x, y) : 3x – y = 0}

Q 1.Determine whether each of the following relations are reflexive, symmetric and transitive:

(v) Relation R in the set A of human beings in a town at a particular time given by

(a) R = {(x, y) : x and y work at the same place}
(b) R = {(x, y) : x and y live in the same locality}
(c) R = {(x, y) : x is exactly 7 cm taller than y}
(d) R = {(x, y) : x is wife of y}
(e) R = {(x, y) : x is father of y}

2. Show that the relation R in the set R of real numbers, defined as
R = {(a, b) : a ? b2} is neither reflexive nor symmetric nor transitive.

3. Check whether the relation R defined in the set {1, 2, 3, 4, 5, 6} as
R = {(a, b) : b = a + 1} is reflexive, symmetric or transitive.

4. Show that the relation R in R defined as R = {(a, b) : a ? b}, is reflexive and transitive but not symmetric.

5. Check whether the relation R in R defined by R = {(a, b) : a ? b3} is reflexive, symmetric or transitive.

6. Show that the relation R in the set {1, 2, 3} given by R = {(1, 2), (2, 1)} is symmetric but neither reflexive nor transitive.

7. Show that the relation R in the set A of all the books in a library of a college, given by R = {(x, y) : x and y have same number of pages} is an equivalence relation.

8. Show that the relation R in the set A = {1, 2, 3, 4, 5} given by
R = {(a, b) : |a – b| is even}, is an equivalence relation. Show that all the elements of {1, 3, 5} are related to each other and all the elements of {2, 4} are related to each other. But no element of {1, 3, 5} is related to any element of {2, 4}.

9. Show that each of the relation R in the set A = {x ? Z : 0 ? x ? 12}, given by
(i) R = {(a, b) : |a – b| is a multiple of 4}
(ii) R = {(a, b) : a = b}

:: Chapter 2 - Inverse Trigonometric Functions ::

EXERCISE

Question 1. Find the principal values of the following:

Question 6. Show that the relation R in the set {1, 2, 3} given by R = {(1, 2), (2, 1)} is symmetric but neither reflexive nor transitive.

Question 7. Show that the relation R in the set A of all the books in a library of a college, given by R = {(x, y) : x and y have same number of pages} is an equivalence relation.

Question 8. Show that the relation R in the set A = {1, 2, 3, 4, 5} given by R = {(a, b) :
|a – b| is even}, is an equivalence relation. Show that all the elements of {1, 3, 5} are related to each other and all the elements of {2, 4} are related to each other. But no element of {1, 3, 5} is related to any element of {2, 4}.

Question 9. Show that each of the relation R in the set A = {x ∈ Z : 0 ≤ x ≤ 12}, given by (i) R = {(a, b) : |a – b| is a multiple of 4} (ii) R = {(a, b) : a = b} is an equivalence relation. Find the set of all elements related to 1 in each case.

Question 10. Give an example of a relation. Which is
(i) Symmetric but neither reflexive nor transitive.
(ii) Transitive but neither reflexive nor symmetric.
(iii) Reflexive and symmetric but not transitive.
(iv) Reflexive and transitive but not symmetric.
(v) Symmetric and transitive but not reflexive.

Question 11. Show that the relation R in the set A of points in a plane given by R = {(P, Q) : distance of the point P from the origin is same as the distance of the point Q from the origin}, is an equivalence relation. Further, show that the set of all points related to a point P ≠ (0, 0) is the circle passing through P with origin as centre.

Question 12. Show that the relation R defined in the set A of all triangles as R = {(T1, T2) : T1 is similar to T2}, is equivalence relation. Consider three right angle triangles T1 with sides 3, 4, 5, T2 with sides 5, 12, 13 and T3 with sides 6, 8, 10. Which triangles among T1, T2 and T3 are related?

Question 13. Show that the relation R defined in the set A of all polygons as R = {(P1, P2) : P1 and P2 have same number of sides}, is an equivalence relation. What is the set of all elements in A related to the right angle triangle T with sides 3, 4 and 5?

Question 14. Let L be the set of all lines in XY plane and R be the relation in L defined as R = {(L1, L2) : L1 is parallel to L2}. Show that R is an equivalence relation. Find the set of all lines related to the line y = 2x + 4.

Question 15. Let R be the relation in the set {1, 2, 3, 4} given by R = {(1, 2), (2, 2), (1, 1), (4,4), (1, 3), (3, 3), (3, 2)}. Choose the correct answer.
(A) R is reflexive and symmetric but not transitive.
(B) R is reflexive and transitive but not symmetric.
(C) R is symmetric and transitive but not reflexive.
(D) R is an equivalence relation.

Question 16. Let R be the relation in the set N given by R = {(a, b) : a = b – 2, b > 6}. Choose the correct answer.
(A) (2, 4) ∈ R
(B) (3, 8) ∈ R
(C) (6, 8) ∈ R
(D) (8, 7) ∈ R

EXERCISE

Question 1. Show that the function f : R → R defined by f (x) = 1 x is one-one and onto, where R is the set of all non-zero real numbers. Is the result true, if the domain R is replaced by N with co-domain being same as R?

Question 2. Check the injectivity and surjectivity of the following functions:

(i) f : N → N given by f (x) = x2
(ii) f : Z → Z given by f (x) = x2
(iii) f : R → R given by f (x) = x2
(iv) f : N → N given by f (x) = x3
(v) f : Z → Z given by f (x) = x3


Question 3. Prove that the Greatest Integer Function f :
R→R, given by f (x) = [x], is neither one-one nor onto, where [x] denotes the greatest integer less than or equal to x.

Question 4. Show that the Modulus Function f : R→R, given by f (x) = | x |, is neither oneone nor onto, where | x | is x, if x is positive or 0 and | x | is – x, if x is negative. 5. Show that the Signum Function f : R→R, given by

is neither one-one nor onto.

State whether the function f is bijective. Justify your answer.

Question 10. Let A = R – {3} and B = R – {1}. Consider the function f : A → B defined by

Question 11. Let f : R → R be defined as f(x) = x4. Choose the correct answer.
(A) f is one-one onto (B) f is many-one onto
(C) f is one-one but not onto
(D) f is neither one-one nor onto.

Question 12. Let f : R → R be defined as f (x) = 3x. Choose the correct answer.
(A) f is one-one onto
(B) f is many-one onto
(C) f is one-one but not onto
(D) f is neither one-one nor onto.
 

EXERCISE

EXERCISE

Show that is commutative and associative. Find the identity element for on A, if any.

Question 12. State whether the following statements are true or false. Justify.

(i) For an arbitrary binary operation on a set N, a a = a ∀ a ∈ N.
(ii) If is a commutative binary operation on N, then a (b c) = (c b) a

Question 13. Consider a binary operation on N defined as a b = a3 + b3. Choose the correct answer.

(A) Is both associative and commutative?
(B) Is commutative but not associative?
(C) Is associative but not commutative?
(D) Is neither commutative nor associative?

Define the relation R in P(X) as follows: For subsets A, B in P(X), ARB if and only if A ⊂ B. Is R an equivalence relation on P(X)? Justify your answer.

Question 9. Given a non-empty set X, consider the binary operation :

P(X) × P(X) → P(X) given by A B = A ∩ B ∀ A, B in P(X), where P(X) is the power set of X. Show that X is the identity element for this operation and X is the only invertible element in P(X) with respect to the operation

Question 10. Find the number of all onto functions from the set {1, 2, 3, ... , n} to itself.

Question 11. Let S = {a, b, c} and T = {1, 2, 3}. Find F–1 of the following functions F from S to T, if it exists.
(i) F = {(a, 3), (b, 2), (c, 1)}
(ii) F = {(a, 2), (b, 1), (c, 1)}

Question 12. Consider the binary operations : R × R → R and o : R × R → R defined as a b = |a – b| and a o b = a, ∀ a, b ∈ R. Show that is commutative but not associative, o is associative but not commutative. Further, show that ∀ a, b, c ∈ R, a (b o c) = (a b) o (a b). [If it is so, we say that the operation distributes over the operation o]. Does o distribute over ? Justify your answer.

Question 13. Given a non-empty set X, let : P(X) × P(X) → P(X) be defined as A * B = (A – B) ∪ (B – A), ∀ A, B ∈ P(X). Show that the empty set φ is the identity for the operation and all the elements A of P(X) are invertible with A–1 = A. (Hint : (A – φ) ∪ (φ – A) = A and (A – A) ∪ (A – A) = A A = φ).

Question 14. Define a binary operation on the set {0, 1, 2, 3, 4, 5} as

:: Chapter 3 - Matrix ::

EXERCISE

Question 1. In the matrix , write

(i) The order of the matrix, (ii) The number of elements,
(iii) Write the elements a13, a21, a33, a24, a23.

Question 2. If a matrix has 24 elements, what are the possible orders it can have? What, if it has 13 elements?

Question 3. If a matrix has 18 elements, what are the possible orders it can have? What, if it has 5 elements ?

Question 4. Construct a 2 × 2 matrix, A = [aij], whose elements are given by:
(i)aij=(i + j)2/2

Question 4. Construct a 2 × 2 matrix, A = [aij], whose elements are given by:
(ii)aij=i/j

Question 4. Construct a 2 × 2 matrix, A = [aij], whose elements are given by:
(iii)aij=(i + 2j)2/2

Question 5. Construct a 3 × 4 matrix, whose elements are given by:
(i)aij=1/2| -3i + j |

Question 5. Construct a 3 × 4 matrix, whose elements are given by:
(ii)aij=2i - j

Question 6. Find the values of x, y and z from the following equations:

(i)

Question 6. Find the values of x, y and z from the following equations:

(ii)

Question 6. Find the values of x, y and z from the following equations:

(iii)

Question 7. Find the value of a, b, c and d from the equation

Question 8 .is a square matrix, if
(A) m < n (B) m > n (C) m = n (D) None of these

Question 9.Which of the given values of x and y make the following pair of matrices equal

Question 10. The number of all possible matrices of order 3 × 3 with each entry 0 or 1 is:

EXERCISE

=Question 1. Let Find each of the following:
(i) A + B (ii) A – B (iii) 3A – C

=Question 1. Let Find each of the following:
(iv) AB (v) BA

=Question 2. Compute the following:

=Question 3. Compute the indicated products.

=Question 4. If , then compute (A+B) and (B – C). Also, verify that A + (B – C) = (A + B) – C.

=Question 5. If , then compute 3A – 5B.

=Question 6. Simplify

:: Chapter 4 - Determinants ::

Exercise

Question 1. Evaluate the determinants in Exercises 1 and 2.

Question 2. Evaluate the determinants in Exercises 1 and 2.

Question 3. If , then show that | 2A | = 4 | A |

Question 4. If , then show that | 3 A | = 27 | A |

Question 5. Evaluate the determinants

Question 6. If , find | A |

Question 7. Find values of x, if 

Question 8. If , then x is equal to

Exercise

Using the property of determinants and without expanding in Exercises 1 to 7, prove That 


Using the property of determinants and without expanding in Exercises 1 to 7, prove That


Using the property of determinants and without expanding in Exercises 1 to 7, prove That


Using the property of determinants and without expanding in Exercises 1 to 7, prove That


Using the property of determinants and without expanding in Exercises 1 to 7, prove That 

By using properties of determinants, in Exercises 8 to 14, show that:

 

Question 15.Let A be a square matrix of order 3 × 3, then | kA| is equal to

(A) k| A|
(B) k2 | A|
(C) k3 | A|
(D) 3k | A |

Question 16. Which of the following is correct?

(A) Determinant is a square matrix.
(B) Determinant is a number associated to a matrix.
(C) Determinant is a number associated to a square matrix.
(D) None of these

Exercise

Question 1. Find area of the triangle with vertices at the point given in each of the following :

(i) (1, 0), (6, 0), (4, 3)
(ii) (2, 7), (1, 1), (10, 8)
(iii) (–2, –3), (3, 2), (–1, –8)

Question 2. Show that points A (a, b + c), B (b, c + a), C (c, a + b) are collinear.

Question 3. Find values of k if area of triangle is 4 sq. units and vertices are
(i) (k, 0), (4, 0), (0, 2)
(ii) (–2, 0), (0, 4), (0, k)

Question 4.
(i) Find equation of line joining (1, 2) and (3, 6) using determinants.
(ii) Find equation of line joining (3, 1) and (9, 3) using determinants.

Question 5. If area of triangle is 35 sq units with vertices (2, – 6), (5, 4) and (k, 4). Then k is

(A) 12 (B) –2 (C) –12, –2 (D) 12, –2

Exercise

Write Minors and Cofactors of the elements of following determinants:

Exercise

=Find adjoint of each of the matrices in Exercises 1 and 2.

=Verify A (adj A) = (adj A) A = |A| I in Exercises 3 and 4

=Find the inverse of each of the matrices (if it exists) given in Exercises 5 to 11.

17. Let A be a nonsingular square matrix of order 3 × 3. Then |adj A| is equal to

(A) |A|
(B) |A|2
(C) |A|3
(D) 3|A|

18. If A is an invertible matrix of order 2, then det (A–1) is equal to

(A) det
(A) (B)1/det (A)
(C) 1
(D) 0

Exercise

Question 1. x + 2y = 2 and 2x + 3y = 3

Question 2. 2x – y = 5 and x + y = 4

Question 3. x + 3y = 5 and 2x + 6y = 8

Question 4. x + y + z = 1 , 2x + 3y + 2z = 2 and ax + ay + 2az = 4

Question 3x–y – 2z = 2, 2y – z =-1 and –3x – 5y = 3

Question 6. 5x – y + 4z = 5,2x + 3y + 5z = 2 and 5x – 2y + 6z = –1

Solve system of linear equations, using matrix method, in Exercises 7 to 14.

Question 7. 5x + 2y = 4 and 7x + 3y = 5

Question 8. 2x – y = –2 and 3x + 4y = 3

Question 9. 4x – 3y = 3 and 3x – 5y = 7

Question 10. 5x + 2y = 3 and 3x + 2y = 5

Question 11. 2x + y + z = 1, x – 2y – z =3/2 and 3y – 5z = 9

Question 12. x – y + z = 4, 2x + y – 3z = 0 and x + y + z = 2

Question 13. 2x + 3y +3 z = 5, x – 2y + z = – 4 and 3x – y – 2z = 3

Question 14. x – y + 2z = 7,3x + 4y – 5z = – 5 and 2x – y + 3z = 12

Question 15. If , find A–1. Using A–1 solve the system of equations 2x – 3y + 5z = 11 3x + 2y – 4z = – 5 x + y – 2z = – 3

Question 16. The cost of 4 kg onion, 3 kg wheat and 2 kg rice is Rs 60. The cost of 2 kg onion, 4 kg wheat and 6 kg rice is Rs 90. The cost of 6 kg onion 2 kg wheat and 3 kg rice is Rs 70. Find cost of each item per kg by matrix method.

:: Chapter 6 - Application of Derivatives ::

EXERCISE

Question 1. Find the rate of change of the area of a circle with respect to its radius r when (a) r = 3 cm (b) r = 4 cm

Question 2. The volume of a cube is increasing at the rate of 8 cm3/s. How fast is the surface area increasing when the length of an edge is 12 cm?

Question 3. The radius of a circle is increasing uniformly at the rate of 3 cm/s. Find the rate at which the area of the circle is increasing when the radius is 10 cm.

Question 4. An edge of a variable cube is increasing at the rate of 3 cm/s. How fast is the volume of the cube increasing when the edge is 10 cm long?

Question 5. A stone is dropped into a quiet lake and waves move in circles at the speed of 5 cm/s. At the instant when the radius of the circular wave is 8 cm, how fast is the enclosed area increasing? 6. The radius of a circle is increasing at the rate of 0.7 cm/s. What is the rate of increase of its circumference?

Question 7. The length x of a rectangle is decreasing at the rate of 5 cm/minute and the width y is increasing at the rate of 4 cm/minute. When x = 8cm and y = 6cm, find the rates of change of (a) the perimeter, and (b) the area of the rectangle.

Question 8. A balloon, which always remains spherical on inflation, is being inflated by pumping in 900 cubic centimetres of gas per second. Find the rate at which the radius of the balloon increases when the radius is 15 cm.

Question 9. A balloon, which always remains spherical has a variable radius. Find the rate at which its volume is increasing with the radius when the later is 10 cm. 10. A ladder 5 m long is leaning against a wall. The bottom of the ladder is pulled along the ground, away from the wall, at the rate of 2cm/s. How fast is its height on the wall decreasing when the foot of the ladder is 4 m away from the wall

Question 11. A particle moves along the curve 6y = x3 +2. Find the points on the curve at which the y-coordinate is changing 8 times as fast as the x-coordinate.

Question 12. The radius of an air bubble is increasing at the rate of 1 2 cm/s. At what rate is the volume of the bubble increasing when the radius is 1 cm?

Question 13. A balloon, which always remains spherical, has a variable diameter 3 (2 1) 2 x + . Find the rate of change of its volume with respect to x.

Question 14. Sand is pouring from a pipe at the rate of 12 cm3/s. The falling sand forms a cone on the ground in such a way that the height of the cone is always one-sixth of the radius of the base. How fast is the height of the sand cone increasing when the height is 4 cm?

Question 15. The total cost C(x) in Rupees associated with the production of x units of an item is given by C(x) = 0.007x3 – 0.003x2 + 15x + 4000. Find the marginal cost when 17 units are produced.

Question 16. The total revenue in Rupees received from the sale of x units of a product is given by R(x) = 13x2 + 26x + 15. Find the marginal revenue when x = 7.

Question 17. The rate of change of the area of a circle with respect to its radius r at r = 6 cm is

(A) 10π
(B) 12π
(C) 8π
(D) 11π

Question 18. The total revenue in Rupees received from the sale of x units of a product is given by R(x) = 3x2 + 36x + 5. The marginal revenue, when x = 15 is

(A) 116
(B) 96
(C) 90
(D) 126

EXERCISE

Question 1. Show that the function given by f (x) = 3x + 17 is strictly increasing on R.

Question 2. Show that the function given by f (x) = e2x is strictly increasing on R.

Question 3. Show that the function given by f (x) = sin x is (a) strictly increasing in 0, 2 (b) strictly decreasing in , 2 (c) neither increasing nor decreasing in (0, π)

Question 4. Find the intervals in which the function f given by f (x) = 2x2 – 3x is (a) strictly increasing (b) strictly decreasing

Question 5. Find the intervals in which the function f given by f (x) = 2x3 – 3x2 – 36x + 7 is (a) strictly increasing (b) strictly decreasing

Question 6. Find the intervals in which the following functions are strictly increasing or decreasing:

(a) x2 + 2x – 5
(b) 10 – 6x – 2x2
(c) –2x3 – 9x2 – 12x + 1
(d) 6 – 9x – x2
(e) (x + 1)3 (x – 3)3

Question 7. Show that log(1 ) 2 2 y x x x = + −+ , x > – 1, is an increasing function of x throughout its domain.

Question 8. Find the values of x for which y = [x(x – 2)]2 is an increasing function.

Question 9. Prove that 4sin (2 cos ) y θ = −θ + θ is an increasing function of θ in 0, 2

Question 10. Prove that the logarithmic function is strictly increasing on (0, ∞).

Question 11. Prove that the function f given by f (x) = x2 – x + 1 is neither strictly increasing nor strictly decreasing on (– 1, 1)

Question 12. Which of the following functions are strictly decreasing on 0, 2?

(A) cos x
(B) cos 2x
(C) cos 3x
(D) tan x

Question 13. On which of the following intervals is the function f given by f (x) = x100 + sin x –1 strictly decreasing ? (A) (0,1) (B) , 2 (D) None of these

Question 14. Find the least value of a such that the function f given by f (x) = x2 + ax + 1 is strictly increasing on (1, 2).

Question 15. Let I be any interval disjoint from (–1, 1). Prove that the function f given by f (x) x 1 x = + is strictly increasing on I.

Question 16. Prove that the function f given by f (x) = log sin x is strictly increasing on 0, 2and strictly decreasing on .

Question 17. Prove that the function f given by f (x) = log cos x is strictly decreasing on 0, 2 ⎠ and strictly increasing on , 2 .

Question 18. Prove that the function given by f (x) = x3 – 3x2 + 3x – 100 is increasing in R.

Question 19. The interval in which y = x2 e–x is increasing is

(A) (– ∞, ∞)
(B) (– 2, 0)
(C) (2, ∞)
(D) (0, 2)

EXERCISE

Question 1. Find the slope of the tangent to the curve y = 3x4 – 4x at x = 4.

Question 2. Find the slope of the tangent to the curve 1, 2 2 y x x x − = ≠ − at x = 10.

Question 3. Find the slope of the tangent to curve y = x3 – x + 1 at the point whose x-coordinate is 2.

Question 4. Find the slope of the tangent to the curve y = x3 –3x + 2 at the point whose x-coordinate is 3.

Question 5. Find the slope of the normal to the curve x = acos3 θ, y = asin3 θ at . 4 π θ =

Question 6. Find the slope of the normal to the curve x = 1− asinθ, y = bcos2 θ at . 2 π θ =

Question 7. Find points at which the tangent to the curve y = x3 – 3x2 – 9x + 7 is parallel to the x-axis.

Question 8. Find a point on the curve y = (x – 2)2 at which the tangent is parallel to the chord joining the points (2, 0) and (4, 4).

Question 9. Find the point on the curve y = x3 – 11x + 5 at which the tangent is y = x – 11.

Question 10. Find the equation of all lines having slope – 1 that are tangents to the curve 1 1 y x = − , x ≠ 1.

Question 11. Find the equation of all lines having slope 2 which are tangents to the curve 1 3 y x = − , x ≠ 3.

Question 12. Find the equations of all lines having slope 0 which are tangent to the curve 2 1 . 2 3 y x x = − +

Question 13. Find points on the curve 2 2 1 9 16 x + y = at which the tangents are (i) parallel to x-axis (ii) parallel to y-axis.

Question 14. Find the equations of the tangent and normal to the given curves at the indicated points:

(i) y = x4 – 6x3 + 13x2 – 10x + 5 at (0, 5)
(ii) y = x4 – 6x3 + 13x2 – 10x + 5 at (1, 3)
(iii) y = x3 at (1, 1)
(iv) y = x2 at (0, 0)
(v) x = cos t, y = sin t at 4 t π =1

Question 15. Find the equation of the tangent line to the curve y = x2 – 2x +7 which is (a) parallel to the line 2x – y + 9 = 0 (b) perpendicular to the line 5y – 15x = 13.

Question 16. Show that the tangents to the curve y = 7x3 + 11 at the points where x = 2 and x = – 2 are parallel.

Question 17. Find the points on the curve y = x3 at which the slope of the tangent is equal to the y-coordinate of the point.

Question 18. For the curve y = 4x3 – 2x5, find all the points at which the tangent passes through the origin.

Question 19. Find the points on the curve x2 + y2 – 2x – 3 = 0 at which the tangents are parallel to the x-axis.

Question 20. Find the equation of the normal at the point (am2,am3) for the curve ay2 = x3.

Question 21. Find the equation of the normals to the curve y = x3 + 2x + 6 which are parallel to the line x + 14y + 4 = 0.

Question 22. Find the equations of the tangent and normal to the parabola y2 = 4ax at the point (at2, 2at).

Question 23. Prove that the curves x = y2 and xy = k cut at right angles* if 8k2 = 1.

Question 24. Find the equations of the tangent and normal to the hyperbola 2 2 2 2 1 x y a b − = at the point (x0, y0).

Question 25. Find the equation of the tangent to the curve y = 3x − 2 which is parallel to the line 4x − 2y + 5 = 0 . Choose the correct answer in Exercises 26 and 27.

Question 26. The slope of the normal to the curve y = 2x2 + 3 sin x at x = 0 is

(A) 3
(B) 1 3
(C) –3
(D) 1 3 −

Question 27. The line y = x + 1 is a tangent to the curve y2 = 4x at the point

(A) (1, 2)
(B) (2, 1)
(C) (1, – 2)
(D) (– 1, 2)

EXERCISE

Question 1. Using differentials, find the approximate value of each of the following up to 3 places of decimal.

(i) 25.3
(ii) 49.5
(iii) 0.6
(iv) 1 (0.009)3
(v) 1 (0.999)10
(vi) 1 (15)4
(vii) 1 (26)3
(viii) 1 (255)4
(ix) 1 (82)4
(x) 1 (401)2
(xi) 1 (0.0037)2
(xii) 1 (26.57)3
(xiii) 1 (81.5)4
(xiv) 3 (3.968)2
(xv) 1 (32.15)5

Question 2. Find the approximate value of f (2.01), where f (x) = 4x2 + 5x + 2.

Question 3. Find the approximate value of f (5.001), where f (x) = x3 – 7x2 + 15.

Question 4. Find the approximate change in the volume V of a cube of side x metres caused by increasing the side by 1%.

Question 5. Find the approximate change in the surface area of a cube of side x metres caused by decreasing the side by 1%.

Question 6. If the radius of a sphere is measured as 7 m with an error of 0.02 m, then find the approximate error in calculating its volume.

Question 7. If the radius of a sphere is measured as 9 m with an error of 0.03 m, then find the approximate error in calculating its surface area.

Question 8. If f(x) = 3x2 + 15x + 5, then the approximate value of f (3.02) is (A) 47.66 (B) 57.66 (C) 67.66 (D) 77.66 9. The approximate change in the volume of a cube of side x metres caused by increasing the side by 3% is

(A) 0.06 x3 m3
(B) 0.6 x3 m3
(C) 0.09 x3 m3
(D) 0.9 x3 m3

EXERCISE

Question 1. Find the maximum and minimum values, if any, of the following functions given by

(i) f
(x) = (2x – 1)2 + 3
(ii) f (x) = 9x2 + 12x + 2
(iii) f
(x) = –
(x – 1)2 + 10
(iv) g
(x) = x3 +

Question 1. It is given that at x = 1, the function x4 – 62x2 + ax + 9 attains its maximum value, on the interval [0, 2]. Find the value of a.

Question 2. Find the maximum and minimum values of x + sin 2x on [0, 2π].

Question 3. Find two numbers whose sum is 24 and whose product is as large as possible.

Question 4. Find two positive numbers x and y such that x + y = 60 and xy3 is maximum.

Question 5. Find two positive numbers x and y such that their sum is 35 and the product x2 y5 is a maximum.

Question 6. Find two positive numbers whose sum is 16 and the sum of whose cubes is minimum.

Question 7. A square piece of tin of side 18 cm is to be made into a box without top, by cutting a square from each corner and folding up the flaps to form the box. What should be the side of the square to be cut off so that the volume of the box is the maximum possible.

Question 8. A rectangular sheet of tin 45 cm by 24 cm is to be made into a box without top, by cutting off square from each corner and folding up the flaps. What should be the side of the square to be cut off so that the volume of the box is maximum ?

Question 9. Show that of all the rectangles inscribed in a given fixed circle, the square has the maximum area.

Question 10. Show that the right circular cylinder of given surface and maximum volume is such that its height is equal to the diameter of the base.

Question 11. Of all the closed cylindrical cans (right circular), of a given volume of 100 cubic centimetres, find the dimensions of the can which has the minimum surface area?

Question 12. A wire of length 28 m is to be cut into two pieces. One of the pieces is to be made into a square and the other into a circle. What should be the length of the two pieces so that the combined area of the square and the circle is minimum?

Question 13. Prove that the volume of the largest cone that can be inscribed in a sphere of radius R is 8 27 of the volume of the sphere.

Question 14. Show that the right circular cone of least curved surface and given volume has an altitude equal to 2 time the radius of the base.

Question 15. Show that the semi-vertical angle of the cone of the maximum volume and of given slant height is tan−1 2 .

Question 16. Show that semi-vertical angle of right circular cone of given surface area and maximum volume is sin 1 1

Miscellaneous Exercise on Chapter 6

Question 1. Using differentials, find the approximate value of each of the following: (a) 1 17 4 81 (b) ( ) 1 33 5 −

Question 2. Show that the function given by f (x) log x x = has maximum at x = e.

Question 3. The two equal sides of an isosceles triangle with fixed base b are decreasing at the rate of 3 cm per second. How fast is the area decreasing when the two equal sides are equal to the base

Question 4. Find the equation of the normal to curve x2 = 4y which passes through the point (1, 2).

Question 5. Show that the normal at any point θ to the curve x = a cosθ + a θ sin θ, y = a sinθ – aθ cosθ is at a constant distance from the origin.

Question 6. Find the intervals in which the function f given by ( ) 4sin 2 cos 2 cos f x x x x x x − − = + is

(i) increasing
(ii) decreasing.

Question 7. Find the intervals in which the function f given by 3 3 f (x) x 1 , x 0 x = + ≠ is

(i) increasing
(ii) decreasing.

Question 8. Find the maximum area of an isosceles triangle inscribed in the ellipse 2 2 2 2 1 x y a b + = with its vertex at one end of the major axis.

Question 9. A tank with rectangular base and rectangular sides, open at the top is to be constructed so that its depth is 2 m and volume is 8 m3. If building of tank costs Rs 70 per sq metres for the base and Rs 45 per square metre for sides. What is the cost of least expensive tank?

Question 10. The sum of the perimeter of a circle and square is k, where k is some constant. Prove that the sum of their areas is least when the side of square is double the radius of the circle.

Question 11. A window is in the form of a rectangle surmounted by a semicircular opening. The total perimeter of the window is

Question 12. m. Find the dimensions of the window to admit maximum light through the whole opening.

Question 13. A point on the hypotenuse of a triangle is at distance a and b from the sides of the triangle. Show that the maximum length of the hypotenuse is 2 2 3 (a3 + b3 )2 .

Question 14. Find the points at which the function f given by f (x) = (x – 2)4 (x + 1)3 has

(i) local maxima
(ii) local minima
(iii) point of inflexion

14. Find the absolute maximum and minimum values of the function f given by f (x) = cos2 x + sin x, x ∈ [0, π]

Question 15. Show that the altitude of the right circular cone of maximum volume that can be inscribed in a sphere of radius r is 4 3 r .

Question 16. Let f be a function defined on [a, b] such that f ′(x) > 0, for all x ∈ (a, b). Then prove that f is an increasing function on (a, b).

Question 17. Show that the height of the cylinder of maximum volume that can be inscribed in a sphere of radius R is 2R 3 . Also find the maximum volume.

Question 18. Show that height of the cylinder of greatest volume which can be inscribed in a right circular cone of height h and semi vertical angle α is one-third that of the cone and the greatest volume of cylinder is 4 3 tan2 27 πh α . Choose the correct answer in the Exercises from 19 to 24.

Question 19. A cylindrical tank of radius 10 m is being filled with wheat at the rate of 314 cubic metre per hour. Then the depth of the wheat is increasing at the rate of

(A) 1 m3/h
(B) 0.1 m3/h
(C) 1.1 m3/h
(D) 0.5 m3/h

Question 20. The slope of the tangent to the curve x = t2 + 3t – 8, y = 2t2 – 2t – 5 at the point (2,– 1) is

(A) 22 7
(B) 6 7
(C) 7 6
(D) 6 7

Question 21. The line y = mx + 1 is a tangent to the curve y2 = 4x if the value of m is

(A) 1
(B) 2
(C) 3
(D) 1 2

Question 22. The normal at the point (1,1) on the curve 2y + x2 = 3 is

(A) x + y = 0
(B) x – y = 0
(C) x + y +1 = 0
(D) x – y = 0

Question 23. The normal to the curve x2 = 4y passing (1,2) is

(A) x + y = 3
(B) x – y = 3
(C) x + y = 1
( D) x – y = 1

Question 24. The points on the curve 9y2 = x3, where the normal to the curve makes equal intercepts with the axes are

(A) 4, 8 3
(B) 4, 8 3
(C) 4, 3 8
(D) 4, 3 8

:: Chapter 7 - Integral ::

EXERCISE

Question 1. Find the area of the region bounded by the curve y2 = x and the lines x = 1, x = 4 and the x-axis.

Question 2. Find the area of the region bounded by y2 = 9x, x = 2, x = 4 and the x-axis in the first quadrant. Fig

Question 3. Find the area of the region bounded by x2 = 4y, y = 2, y = 4 and the y-axis in the first quadrant.

Question 4. Find the area of the region bounded by the ellipse 2 2 1 16 9 x y + = .

Question 5. Find the area of the region bounded by the ellipse 2 2 1 4 9 x y + = .

Question 6. Find the area of the region in the first quadrant enclosed by x-axis, line x = 3 y and the circle x2 + y2 = 4.

Question 7. Find the area of the smaller part of the circle x2 + y2 = a2 cut off by the line 2 x= a .

Question 8. The area between x = y2 and x = 4 is divided into two equal parts by the line x = a, find the value of a.

Question 9. Find the area of the region bounded by the parabola y = x2 and y = x .

Question 10. Find the area bounded by the curve x2 = 4y and the line x = 4y – 2.

Question 11. Find the area of the region bounded by the curve y2 = 4x and the line x = 3. Choose the correct answer in the following Exercises 12 and 13.

Question 12. Area lying in the first quadrant and bounded by the circle x2 + y2 = 4 and the lines x = 0 and x = 2 is

(A) π
(B) 2 π
(C) 3 π
(D) 4 π

Question 13. Area of the region bounded by the curve y2 = 4x, y-axis and the line y = 3 is

(A) 2
(B) 9 4
(C) 9 3
(D) 9 2

EXERCISE

Question 1. Find the area of the circle 4x2 + 4y2 = 9 which is interior to the parabola x2 = 4y.

Question 2. Find the area bounded by curves (x – 1)2 + y2 = 1 and x2 + y2 = 1.

Question 3. Find the area of the region bounded by the curves y = x2 + 2, y = x, x = 0 and x = 3.

Question 4. Using integration find the area of region bounded by the triangle whose vertices are (– 1, 0), (1, 3) and (3, 2).

Question 5. Using integration find the area of the triangular region whose sides have the equations y = 2x + 1, y = 3x + 1 and x =

Question 4.Choose the correct answer in the following exercises 6 and 7.

Question 6. Smaller area enclosed by the circle x2 + y2 = 4 and the line x + y = 2 is

(A) 2 (π – 2)
(B) π – 2
(C) 2π – 1
(D) 2 (π + 2)

Question 7. Area lying between the curves y2 = 4x and y = 2x is

(A) 2 3
(B) 1 3
(C) 1 4
(D) 3 4

Miscellaneous Exercise on Chapter

Question 1. Find the area under the given curves and given lines:

(i) y = x2, x = 1, x = 2 and x-axis
(ii) y = x4, x = 1, x = 5 and x-axis

Question 2. Find the area between the curves y = x and y = x2.

Question 3. Find the area of the region lying in the first quadrant and bounded by y = 4x2, x = 0, y = 1 and y = 4.

Question 4. Sketch the graph of y = x + 3 and evaluate 0 6 3 − ∫ x + dx .

Question 5. Find the area bounded by the curve y = sin x between x = 0 and x = 2π.

Question 6. Find the area enclosed between the parabola y2 = 4ax and the line y = mx.

Question 7. Find the area enclosed by the parabola 4y = 3x2 and the line 2y = 3x + 12.

Question 8. Find the area of the smaller region bounded by the ellipse 2 2 1 9 4 x + y = and the line 1 3 2 x y + = .

Question 9. Find the area of the smaller region bounded by the ellipse 2 2 2 2 x y 1 a b + = and the line 1 x y a b + = .

Question 10. Find the area of the region enclosed by the parabola x2 = y, the line y = x + 2 and the x-axis.

Question 11. Using the method of integration find the area bounded by the curve x + y = 1 . [Hint: The required region is bounded by lines x + y = 1, x– y = 1, – x + y = 1 and – x – y = 1].

Question 12. Find the area bounded by curves {(x, y) : y ≥ x2 and y = | x |}.

Question 13. Using the method of integration find the area of the triangle ABC, coordinates of whose vertices are A(2, 0), B (4, 5) and C (6, 3).

Question 14. Using the method of integration find the area of the region bounded by lines: 2x + y = 4, 3x – 2y = 6 and x – 3y + 5 = 0

Question 15. Find the area of the region {(x, y) : y2 ≤ 4x, 4x2 + 4y2 ≤ 9} Choose the correct answer in the following Exercises from 16 to 20.

Question 16. Area bounded by the curve y = x3, the x-axis and the ordinates x = – 2 and x = 1 is

(A) – 9
(B) 15 4 −
(C) 15 4
(D) 17 4

Question 17. The area bounded by the curve y = x | x | , x-axis and the ordinates x = – 1 and x = 1 is given by

(A) 0
(B) 1 3
(C) 2 3
(D) 4 3 [Hint : y = x2 if x > 0 and y = – x2 if x < 0].

Question 18. The area of the circle x2 + y2 = 16 exterior to the parabola y2 = 6x is

(A) 4 (4 3) 3 π −
(B) 4 (4 3) 3 π +
(C) 4 (8 3) 3 π −
(D) 4 (8 3) 3 π +

Question 19. The area bounded by the y-axis, y = cos x and y = sin x when 0 2 x π ≤ ≤ is

(A) 2 ( 2 −1)
(B) 2 −1
(C) 2 +1
(D) 2

:: Chapter 8 - Application Of Integrals ::

EXERCISE

1. Find the area of the region bounded by the curve y^2 =x and the lines x = 1 , x = 4 and the x axis

2.Find the area of the region bounded by y^2 = 9x, x=2, x =4 and the x axis in the first quadrant.

3. Find the area of the region bounded by x^2 = 4y, y = 2, y = 4 and the y-axis in the first quadrant.

Find the equation of the region bounded by the ellipse x^2/16 + y^2/9 =1

Find the equation of the region bounded by the ellipse x^2/4 + y^2/9 =1

6. find the area of the region in the first quadrant enclosed by x axis, line x =root 3 y and the circle x^2 + y^2 = 4

7.Area between x=y2 and x=4 is divided in two equal parts by the line x = a, find the value of a

8. The area between x^2 = y and x = 4 is divided into two equal parts by the line x = a, find the value of a.

9. Find the area of the region bounded by the parabola y = x^2 and y= |x|\

10. Find the area bounded by the curve x^2 =4y and the line x = 4y- 2

11. Find the area of the region bounded by the curve y2 = 4x and the line x = 3.

12. Area lying in the first quadrant and bounded by the circle x2 + y = 4 and the lines x = 0 and x = 2 is 2

13. Area of the region bounded by the curve y^2 =4x , y axis and the line y=3 is

EXERCISE

1. Find the area of the circle 4x^2 + 4y^2 = 9 which is interior to the parabola x^2 =4y

2. Find the area bounded by curves (x – 1)^2 + y^2 = 1 and x^2 + y^2 = 1

3.Find the area of the region bounded by the curves y= x^2 + 2 , y=x , x =0 and x = 3

4.Using integration find the area of region bounded by the triangle whose vertices are (– 1, 0), (1, 3) and (3, 2).

5. Using integration find the area of the triangular region whose sides have the equations y = 2x + 1, y = 3x + 1 and x = 4.

6. Smaller area enclosed by the circle x^2 +y^2 = 4 and the lines x + y = 2 is (A) 2 (π – 2) (B) π – 2 (C) 2π – 1 (D) 2 (π + 2)

7. Area lying between the curves y^2 = 4x and y = 2x is

Miscellaneous Solutions

1. Find the area under the given curves and given lines (i) y = x^2 , x=1 , x= 2 and x axis (ii) y = x^4 , x=1 , x= 5 and x axis

2. Find the area between the curves y = x and y = x^2

3.Find the area of the region lying in the first quadrant and bounded by y = 4x^2, x=0, y=1 and y= 4

4. Sketch the graph of y = x + 3 and evaluate integration limits 6 to 0 of x + 3 dx

5. Find the area bounded by the curve y = sin x between x = 0 and x = 2π.

6. Find the area enclosed between the parabola y^2 = 4ax and the line y =mx

7. Find the area enclosed by the parabola 4y = 3x^2 and the line 2y = 3x + 12 using integration to find area,

8. Find the area of the smaller region bounded by the ellipse x^2/9 + y^2/4 = 1 and the line x/3 + y/2 =1

:: Chapter 9 - Differential Equations ::

EXERCISE 9.2

In each of the Exercises 1 to 10 verify that the given functions (explicit or implicit) is a solution of the corresponding differential equation:

Question 1. y = ex + 1 : y″ – y′ = 0

Question 2. y = x² + 2x + C : y′ – 2x – 2 = 0

Question 3. y = cos x + C : y′ + sin x = 0

Question 4. y = 1+ x² : y′ = 1 2 xy + x

Question 5. y = Ax : xy′ = y (x ≠ 0)

Question 6. y = x sin x : xy′ = y + x x²− y² (x ≠ 0 and x > y or x < – y)

Question 7. xy = log y + C : y′ = 2 1 y − xy (xy ≠ 1)

Question 8. y – cos y = x : (y sin y + cos y + x) y′ = y

Question 9. x + y = tan–1y : y² y′ + y² + 1 = 0

Question 10. y = a²

 − x2 x ∈ (–a, a) : x + y dy dx = 0 (y ≠ 0)

Question 11. The number of arbitrary constants in the general solution of a differential equation of fourth order are:

(A) 0
(B) 2
(C) 3
(D) 4

Question 12. The number of arbitrary constants in the particular solution of a differential equation of third order are:

(A) 3
(B) 2
(C) 1
(D) 0

EXERCISE 9.3

In each of the Exercises 1 to 5, form a differential equation representing the given family of curves by eliminating arbitrary constants a and b.

Question 1. x y 1 a b + =

Question 2. y² = a (b²– x²)

Question 3. y = a e3x + b e– 2x

Question 4. y = e²x (a + bx)

Question 5. y = ex (a cos x + b sin x)

Question 6. Form the differential equation of the family of circles touching the y-axis at origin.

Question 7. Form the differential equation of the family of parabolas having vertex at origin and axis along positive y-axis.

Question 8. Form the differential equation of the family of ellipses having foci on y-axis and centre at origin.

Question 9. Form the differential equation of the family of hyperbolas having foci on x-axis and centre at origin.

Question 10. Form the differential equation of the family of circles having centre on y-axis and radius 3 units.

Question 11. Which of the following differential equations has y = c1 ex + c2 e–x as the general solution?
(A) 2 2 d y y 0 dx + =
(B) 2 2 d y y 0 dx − =
(C) 2 2 d y 1 0 dx + =
(D) 2 2 d y 1 0 dx − =

Question 12. Which of the following differential equations has y = x as one of its particular solution?
(A) 2 2 2 d y x dy xy x dx dx − + =
(B) 2 2 d y x dy xy x dx dx + + =
(C) 2 2 2 d y x dy xy 0 dx dx − + =
(D) 2 2 d y x dy xy 0 dx dx |
 

EXERCISE 9.4

EXERCISE 9.3

Question 17. Find the equation of a curve passing through the point (0, –2) given that at any point (x, y) on the curve, the product of the slope of its tangent and y coordinate of the point is equal to the x coordinate of the point.

Question 18. At any point (x, y) of a curve, the slope of the tangent is twice the slope of the line segment joining the point of contact to the point (– 4, –3). Find the equation of the curve given that it passes through (–2, 1).

Question 19. The volume of spherical balloon being inflated changes at a constant rate. If initially its radius is 3 units and after 3 seconds it is 6 units. Find the radius of balloon after t seconds.

Question 20. In a bank, principal increases continuously at the rate of r% per year. Find the value of r if Rs 100 double itself in 10 years (loge2 = 0.6931).

Question 21. In a bank, principal increases continuously at the rate of 5% per year. An amount of Rs 1000 is deposited with this bank, how much will it worth after 10 years (e0.5 = 1.648).

Question 22. In a culture, the bacteria count is 1,00,000. The number is increased by 10% in 2 hours. In how many hours will the count reach 2,00,000, if the rate of growth of bacteria is proportional to the number present?

Question 23. The general solution of the differential equation dy ex y dx = + is (A) ex + e–y = C (B) ex + ey = C (C) e–x + ey = C (D) e–x + e–y = C

Question 16. A homogeneous differential equation of the from dx h x dy y = can be solved by making the substitution.
(A) y = vx
(B) v = yx
(C) x = vy
(D) x = v

Question 17. Which of the following is a homogeneous differential equation?
(A) (4x + 6y + 5) dy – (3y + 2x + 4) dx = 0
(B) (xy) dx – (x3 + y3) dy = 0
(C) (x3 + 2y2) dx + 2xy dy = 0
(D) y2 dx + (x2 – xy – y2) dy = 0

Miscellaneous Exercise on Chapter 9

Question 1. For each of the differential equations given below, indicate its order and degree (if defined).
(i) 2 2 2 d y 5x dy 6y log x dx dx +
(ii) 3 2 dy 4 dy 7 y sin x dx dx
(iii) 4 3 4 3 d y sin d y 0 dx dx

Question 2. For each of the exercises given below, verify that the given function (implicit or explicit) is a solution of the corresponding differential equation.
(i) y = a ex + b e–x + x2 : 2 2 2 x d y 2 dy xy x 2 0 dx dx + − + − =
(ii) y = ex (a cos x + b sin x) : 2 2 d y 2 dy 2y 0 dx dx − + =
(iii) y = x sin 3x : 2 2 d y 9y 6cos3x 0 dx + − =
(iv) x2 = 2y2 log y : (x2 y2 ) dy xy 0 dx + − =

Question 3. Form the differential equation representing the family of curves given by (x – a)2 + 2y2 = a2, where a is an arbitrary constant.

Question 4. Prove that x2 – y2 = c (x2 + y2)2 is the general solution of differential equation (x3 – 3x y2) dx = (y3 – 3x2y) dy, where c is a parameter.

Question 5. Form the differential equation of the family of circles in the first quadrant which touch the coordinate axes.

Question 6. Find the general solution of the differential equation 2 2 1 0 1 dy y dx x − + = − .

Question 7. Show that the general solution of the differential equation 2 2 1 0 1 dy y y dx x x + + + = + + is given by (x + y + 1) = A (1 – x – y – 2xy), where A is parameter.

Question 8. Find the equation of the curve passing through the point 0, 4 whose differential equation is sin x cos y dx + cos x sin y dy = 0.

Question 9. Find the particular solution of the differential equation (1 + e2x) dy + (1 + y2) ex dx = 0, given that y = 1 when x = 0.

Question 10. Solve the differential equation 2 ( 0) x x y e ydx x e y y dy y ≠ .

Question 11. Find a particular solution of the differential equation (x – y) (dx + dy) = dx – dy, given that y = –1, when x = 0. (Hint: put x – y = t)

Question 12. Solve the differential equation 2 1( 0) x e y dxx x x dy.

Question 13. Find a particular solution of the differential equation cot dy y x dx + = 4x cosec x (x ≠ 0), given that y = 0 when 2 x π = .

Question 14. Find a particular solution of the differential equation (x + 1) dy dx = 2 e–y – 1, given that y = 0 when x = 0.

Question 15. The population of a village increases continuously at the rate proportional to the number of its inhabitants present at any time. If the population of the village was 20, 000 in 1999 and 25000 in the year 2004, what will be the population of the village in 2009?

Question 16. The general solution of the differential equation y dx x dy 0 y − = is
(A) xy = C
(B) x = Cy2
(C) y = Cx
(D) y = Cx2

Question 17. The general solution of a differential equation of the type P1 Q1 dx x dy + = is
(A) P1 ( P1 ) Q1 C dy dy y e∫ = ∫ e∫ dy +
(B) P1 ( P1 ) . Q1 C dx dx y e∫ = ∫ e∫ dx +
(C) P1 ( P1 ) Q1 C dy dy x e∫ = ∫ e∫ dy +
(D) P1 ( P1 ) Q1 C dx dx x e∫ = ∫ e∫ dx +

Question 18. The general solution of the differential equation ex dy + (y ex + 2x) dx = 0 is
(A) x ey + x2 = C
(B) x ey + y2 = C
(C) y ex + x2 = C
(D) y ey + x2 = C

:: Chapter 10 - Vector Algebra ::

Question 1. Represent graphically a displacement of 40 km, 30° east of north.

Question 2. Classify the following measures as scalars and vectors.
(i) 10 kg
(ii) 2 meters north-west
(iii) 40° (iv) 40 watt
(v) 10–19 coulomb
(vi) 20 m/s2

Question 3. Classify the following as scalar and vector quantities.
(i) time period
(ii) distance
(iii) force
(iv) velocity
(v) work done

Question 4. In Fig 10.6 (a square), identify the following vectors.
(i) Coinitial
(ii) Equal
(iii) Collinear but not equal

Question 5. Answer the following as true or false.
(i) a and −a are collin ear.
(ii) Two collinear vectors are always equal in magnitude.
(iii) Two vectors having same magnitude are collinear.
(iv) Two collinear vectors having the same magnitude are equal.

Question 2. Write two different vectors having same magnitude.

Question 3. Write two different vectors having same direction.

Question 4. Find the values of x and y so that the vectors 2iˆ + 3 ˆj and xiˆ + yˆj are equal.

Question 5. Find the scalar and vector components of the vector with initial point (2, 1) and terminal point (– 5, 7).

Question 6. Find the sum of the vectors a = iˆ − 2 ˆj + kˆ, b = −2iˆ + 4 ˆj + 5kˆ and c� = iˆ − 6 ˆj – 7kˆ .

Question 7. Find the unit vector in the direction of the vector a = iˆ + ˆj + 2kˆ .

Question 8. Find the unit vector in the direction of vector PQ, where P and Q are the points (1, 2, 3) and (4, 5, 6), respectively.

Question 9. For given vectors, a = 2iˆ − ˆj + 2kˆ and b = −iˆ + ˆj − kˆ , find the unit vector in the direction of the vector a + b .

Question 10. Find a vector in the direction of vector 5iˆ − ˆj + 2kˆ which has magnitude 8 units.

Question 11. Show that the vectors 2iˆ − 3 ˆj + 4kˆ and − 4iˆ + 6 ˆj − 8kˆ are collinear. 12. Find the direction cosines of the vector iˆ + 2 ˆj + 3kˆ .

Question 13. Find the direction cosines of the vector joining the points A(1, 2, –3) and B(–1, –2, 1), directed from A to B .

Question 14. Show that the vector iˆ + ˆj + kˆ is equally inclined to the axes OX, OY and OZ.

Question 15. Find the position vector of a point R which divides the line joining two points P and Q whose position vectors are iˆ + 2 ˆj − kˆ and – iˆ + ˆj + kˆ respectively, in the ratio 2 : 1
(i) internally
(ii) externally

:: Chapter 11 - Three Dimensional Geometry ::

EXERCISE 11.1

Question 1. If a line makes angles 90°, 135°, 45° with the x, y and z-axes respectively, find its direction cosines.

Question 2. Find the direction cosines of a line which makes equal angles with the coordinate axes.

Question 3. If a line has the direction ratios –18, 12, – 4, then what are its direction cosines ?

Question 4. Show that the points (2, 3, 4), (– 1, – 2, 1), (5, 8, 7) are collinear.

Question 5. Find the direction cosines of the sides of the triangle whose vertices are (3, 5, – 4), (– 1, 1, 2) and (– 5, – 5, – 2).

Question 2. Show that the line through the points (1, – 1, 2), (3, 4, – 2) is perpendicular to the line through the points (0, 3, 2) and (3, 5, 6).

Question 3. Show that the line through the points (4, 7, 8), (2, 3, 4) is parallel to the line through the points (– 1, – 2, 1), (1, 2, 5).

Question 4. Find the equation of the line which passes through the point (1, 2, 3) and is parallel to the vector 3 iˆ + 2 ˆj −2 kˆ .

Question 5. Find the equation of the line in vector and in cartesian form that passes through the point with position vector 2 iˆ− j + 4 kˆ and is in the direction iˆ + 2 ˆj − kˆ .

Question 9. Find the vector and the cartesian equations of the line that passes through the points (3, – 2, – 5), (3, – 2, 6).

Question 10. Find the angle between the following pairs of lines:

EXERCISE 11.3

Question 1. In each of the following cases, determine the direction cosines of the normal to the plane and the distance from the origin.
(a) z = 2
(b) x + y + z = 1
(c) 2x + 3y – z = 5
(d) 5y + 8 = 0

Question 2. Find the vector equation of a plane which is at a distance of 7 units from the origin and normal to the vector 3 iˆ + 5 ˆj − 6 kˆ.

Question 3. Find the Cartesian equation of the following planes:
(a) r (iˆ + ˆj − kˆ) = 2
(b) r (2iˆ +3 ˆj − 4kˆ) = 1
(c) r [(s − 2t) iˆ + (3 − t) ˆj +(2 s +t ) kˆ] = 15

Question 4. In the following cases, find the coordinates of the foot of the perpendicular drawn from the origin.
(a) 2x + 3y + 4z – 12 = 0 (b) 3y + 4z – 6 = 0
(c) x + y + z = 1 (d) 5y + 8 = 0

Question 5. Find the vector and cartesian equations of the planes
(a) that passes through the point (1, 0, – 2) and the normal to the plane is iˆ + ˆj − kˆ.
(b) that passes through the point (1,4, 6) and the normal vector to the plane is iˆ−2 ˆj + kˆ.

Question 6. Find the equations of the planes that passes through three points.
(a) (1, 1, – 1), (6, 4, – 5), (– 4, – 2, 3)
(b) (1, 1, 0), (1, 2, 1), (– 2, 2, – 1)

Question 7. Find the intercepts cut off by the plane 2x + y – z = 5.

Question 8. Find the equation of the plane with intercept 3 on the y-axis and parallel to ZOX plane.

Question 9. Find the equation of the plane through the intersection of the planes 3x – y + 2z – 4 = 0 and x + y + z – 2 = 0 and the point (2, 2, 1).

Question 10. Find the vector equation of the plane passing through the intersection of the planes r .(2 iˆ + 2 ˆj − 3 kˆ ) = 7 , r .(2 iˆ + 5 ˆj + 3 kˆ ) = 9 and through the point (2, 1, 3).

Question 11. Find the equation of the plane through the line of intersection of the planes x + y + z = 1 and 2x + 3y + 4z = 5 which is perpendicular to the plane x – y + z = 0.

Question 12. Find the angle between the planes whose vector equations are r (2 iˆ + 2 ˆj − 3 kˆ) = 5 and r (3 iˆ − 3 ˆj + 5 kˆ) = 3 .

Question 13. In the following cases, determine whether the given planes are parallel or perpendicular, and in case they are neither, find the angles between them.
(a) 7x + 5y + 6z + 30 = 0 and 3x – y – 10z + 4 = 0
(b) 2x + y + 3z – 2 = 0 and x – 2y + 5 = 0
(c) 2x – 2y + 4z + 5 = 0 and 3x – 3y + 6z – 1 = 0
(d) 2x – y + 3z – 1 = 0 and 2x – y + 3z + 3 = 0
(e) 4x + 8y + z – 8 = 0 and y + z – 4 = 0

Question 14. In the following cases, find the distance of each of the given points from the corresponding given plane.

Miscellaneous Exercise on Chapter 11

Question 1. Show that the line joining the origin to the point (2, 1, 1) is perpendicular to the line determined by the points (3, 5, – 1), (4, 3, – 1).

Question 2. If l1, m1, n1 and l2, m2, n2 are the direction cosines of two mutually perpendicular lines, show that the direction cosines of the line perpendicular to both of these are 1 2 2 1 1 2 2 1 1 2 2 1 m n − m n , n l − n l , l m − l m

Question 3. Find the angle between the lines whose direction ratios are a, b, c and b – c, c – a, a – b.

Question 4. Find the equation of a line parallel to x-axis and passing through the origin.

Question 5. If the coordinates of the points A, B, C, D be (1, 2, 3), (4, 5, 7), (– 4, 3, – 6) and (2, 9, 2) respectively, then find the angle between the lines AB and CD.

Question 10. Find the coordinates of the point where the line through (5, 1, 6) and (3, 4,1) crosses the YZ-plane.

Question 11. Find the coordinates of the point where the line through (5, 1, 6) and (3, 4, 1) crosses the ZX-plane.

Question 12. Find the coordinates of the point where the line through (3, – 4, – 5) and (2, – 3, 1) crosses the plane 2x + y + z = 7.

Question 13. Find the equation of the plane passing through the point (– 1, 3, 2) and perpendicular to each of the planes x + 2y + 3z = 5 and 3x + 3y + z = 0.

Question 14. If the points (1, 1, p) and (– 3, 0, 1) be equidistant from the plane (3 ˆ + 4 ˆ −12 ˆ) +13 = 0, r i j k then find the value of p.

Question 15. Find the equation of the plane passing through the line of intersection of the planes r (iˆ + ˆj + kˆ) =1 and r (2 iˆ + 3 ˆj − kˆ) + 4 = 0 and parallel to x-axis.

Question 16. If O be the origin and the coordinates of P be (1, 2, – 3), then find the equation of the plane passing through P and perpendicular to OP.

Question 17. Find the equation of the plane which contains the line of intersection of the planes r (iˆ + 2 ˆj + 3 kˆ) − 4 = 0 , r (2 iˆ + ˆj − kˆ) + 5 = 0 and which is perpendicular to the plane r (5 iˆ + 3 ˆj − 6kˆ) + 8 = 0 ] .

Question 18. Find the distance of the point (– 1, – 5, – 10) from the point of intersection of the line r = 2 iˆ − ˆj + 2 kˆ + λ (3 iˆ + 4 ˆj + 2 kˆ) and the plane r (iˆ − ˆj + kˆ) = 5 .

Question 19. Find the vector equation of the line passing through (1, 2, 3) and parallel to the planes r (iˆ − ˆj + 2kˆ) = 5 and r (3 iˆ + ˆj + kˆ) = 6 .

Question 20. Find the vector equation of the line passing through the point (1, 2, – 4) and perpendicular to the two lines:

:: Chapter 12 - Linear Programming ::

EXERCISE 12.1

Solve the following Linear Programming Problems graphically:

Question 1. Maximise Z = 3x + 4y subject to the constraints : x + y ≤ 4, x ≥ 0, y ≥ 0.

Question 2. Minimise Z = – 3x + 4 y subject to x + 2y ≤ 8, 3x + 2y ≤ 12, x ≥ 0, y ≥ 0.

Question 3. Maximise Z = 5x + 3y subject to 3x + 5y ≤ 15, 5x + 2y ≤ 10, x ≥ 0, y ≥ 0.

Question 4. Minimise Z = 3x + 5y such that x + 3y ≥ 3, x + y ≥ 2, x, y ≥ 0.

Question 5. Maximise Z = 3x + 2y subject to x + 2y ≤ 10, 3x + y ≤ 15, x, y ≥ 0.

Question 6. Minimise Z = x + 2y subject to 2x + y ≥ 3, x + 2y ≥ 6, x, y ≥ 0. Show that the minimum of Z occurs at more than two points.

Question 7. Minimise and Maximise Z = 5x + 10 y subject to x + 2y ≤ 120, x + y ≥ 60, x – 2y ≥ 0, x, y ≥ 0.

Question 8. Minimise and Maximise Z = x + 2y subject to x + 2y ≥ 100, 2x – y ≤ 0, 2x + y ≤ 200; x, y ≥ 0.

Question 9. Maximise Z = – x + 2y, subject to the constraints: x ≥ 3, x + y ≥ 5, x + 2y ≥ 6, y ≥ 0.

Question 10. Maximise Z = x + y, subject to x – y ≤ –1, –x + y ≤ 0, x, y ≥ 0.

EXERCISE 12.2

Question 1. Reshma wishes to mix two types of food P and Q in such a way that the vitamin contents of the mixture contain at least 8 units of vitamin A and 11 units of vitamin B. Food P costs Rs 60/kg and Food Q costs Rs 80/kg. Food P contains 3 units/kg of Vitamin A and 5 units / kg of Vitamin B while food Q contains 4 units/kg of Vitamin A and 2 units/kg of vitamin B. Determine the minimum cost of the mixture.

Question 2. One kind of cake requires 200g of flour and 25g of fat, and another kind of cake requires 100g of flour and 50g of fat. Find the maximum number of cakes which can be made from 5kg of flour and 1 kg of fat assuming that there is no shortage of the other ingredients used in making the cakes.

Question 3. A factory makes tennis rackets and cricket bats. A tennis racket takes 1.5 hours of machine time and 3 hours of craftman’s time in its making while a cricket bat takes 3 hour of machine time and 1 hour of craftman’s time. In a day, the factory has the availability of not more than 42 hours of machine time and 24 hours of craftsman’s time.
(i) What number of rackets and bats must be made if the factory is to work at full capacity?
(ii) If the profit on a racket and on a bat is Rs 20 and Rs 10 respectively, find the maximum profit of the factory when it works at full capacity.

Question 4. A manufacturer produces nuts and bolts. It takes 1 hour of work on machine A and 3 hours on machine B to produce a package of nuts. It takes 3 hours on machine A and 1 hour on machine B to produce a package of bolts. He earns a profit of Rs17.50 per package on nuts and Rs 7.00 per package on bolts. How many packages of each should be produced each day so as to maximise his profit, if he operates his machines for at the most 12 hours a day?

Question 5. A factory manufactures two types of screws, A and B. Each type of screw requires the use of two machines, an automatic and a hand operated. It takes 4 minutes on the automatic and 6 minutes on hand operated machines to manufacture a package of screws A, while it takes 6 minutes on automatic and 3 minutes on the hand operated machines to manufacture a package of screws B. Each machine is available for at the most 4 hours on any day. The manufacturer can sell a package of screws A at a profit of Rs 7 and screws B at a profit of Rs 10. Assuming that he can sell all the screws he manufactures, how many packages of each type should the factory owner produce in a day in order to maximise his profit? Determine the maximum profit.

Question 6. A cottage industry manufactures pedestal lamps and wooden shades, each requiring the use of a grinding/cutting machine and a sprayer. It takes 2 hours on grinding/cutting machine and 3 hours on the sprayer to manufacture a pedestal lamp. It takes 1 hour on the grinding/cutting machine and 2 hours on the sprayer to manufacture a shade. On any day, the sprayer is available for at the most 20 hours and the grinding/cutting machine for at the most 12 hours. The profit from the sale of a lamp is Rs 5 and that from a shade is Rs 3. Assuming that the manufacturer can sell all the lamps and shades that he produces, how should he schedule his daily production in order to maximise his profit?

Question 7. A company manufactures two types of novelty souvenirs made of plywood. Souvenirs of type A require 5 minutes each for cutting and 10 minutes each for assembling. Souvenirs of type B require 8 minutes each for cutting and 8 minutes each for assembling. There are 3 hours 20 minutes available for cutting and 4 hours for assembling. The profit is Rs 5 each for type A and Rs 6 each for type B souvenirs. How many souvenirs of each type should the company manufacture in order to maximise the profit?

Question 8. A merchant plans to sell two types of personal computers – a desktop model and a portable model that will cost Rs 25000 and Rs 40000 respectively. He estimates that the total monthly demand of computers will not exceed 250 units. Determine the number of units of each type of computers which the merchant should stock to get maximum profit if he does not want to invest more than Rs 70 lakhs and if his profit on the desktop model is Rs 4500 and on portable model is Rs 5000.

Question 9. A diet is to contain at least 80 units of vitamin A and 100 units of minerals. Two foods F1 and F2 are available. Food F1 costs Rs 4 per unit food and F2 costs Rs 6 per unit. One unit of food F1 contains 3 units of vitamin A and 4 units of minerals. One unit of food F2 contains 6 units of vitamin A and 3 units of minerals. Formulate this as a linear programming problem. Find the minimum cost for diet that consists of mixture of these two foods and also meets the minimal nutritional requirements.

Question 10. There are two types of fertilisers F1 and F2. F1 consists of 10% nitrogen and 6% phosphoric acid and F2 consists of 5% nitrogen and 10% phosphoric acid. After testing the soil conditions, a farmer finds that she needs atleast 14 kg of nitrogen and 14 kg of phosphoric acid for her crop. If F1 costs Rs 6/kg and F2 costs Rs 5/kg, determine how much of each type of fertiliser should be used so that nutrient requirements are met at a minimum cost. What is the minimum cost?

Question 11. The corner points of the feasible region determined by the following system of linear inequalities: 2x + y ≤ 10, x + 3y ≤ 15, x, y ≥ 0 are (0, 0), (5, 0), (3, 4) and (0, 5). Let Z = px + qy, where p, q > 0. Condition on p and q so that the maximum of Z occurs at both (3, 4) and (0, 5) is
(A) p = q
(B) p = 2q
(C) p = 3q
(D) q = 3p

Miscellaneous Exercise

Question 1. Refer to Example 9. How many packets of each food should be used to maximise the amount of vitamin A in the diet? What is the maximum amount of vitamin A

Question 2. A farmer mixes two brands P and Q of cattle feed. Brand P, costing Rs 250 per bag, contains 3 units of nutritional element A, 2.5 units of element B and 2 units of element C. Brand Q costing Rs 200 per bag contains 1.5 units of nutritional element A, 11.25 units of element B, and 3 units of element C. The minimum requirements of nutrients A, B and C are 18 units, 45 units and 24 units respectively. Determine the number of bags of each brand which should be mixed in order to produce a mixture having a minimum cost per bag? What is the minimum cost of the mixture per bag?

Question 3. A dietician wishes to mix together two kinds of food X and Y in such a way that the mixture contains at least 10 units of vitamin A, 12 units of vitamin B and 8 units of vitamin C. The vitamin contents of one kg food is given below: in the diet?

One kg of food X costs Rs 16 and one kg of food Y costs Rs

Question 20. Find the least cost of the mixture which will produce the required diet? 4. A manufacturer makes two types of toys A and B. Three machines are needed for this purpose and the time (in minutes) required for each toy on the machines is given below:

Each machine is available for a maximum of 6 hours per day. If the profit on each toy of type A is Rs 7.50 and that on each toy of type B is Rs 5, show that 15 toys of type A and 30 of type B should be manufactured in a day to get maximum profit.

Question 5. An aeroplane can carry a maximum of 200 passengers. A profit of Rs 1000 is made on each executive class ticket and a profit of Rs 600 is made on each economy class ticket. The airline reserves at least 20 seats for executive class. However, at least 4 times as many passengers prefer to travel by economy class than by the executive class. Determine how many tickets of each type must be sold in order to maximise the profit for the airline. What is the maximum profit?

Question 9. Refer to Question 8. If the grower wants to maximise the amount of nitrogen added to the garden, how many bags of each brand should be added? What is the maximum amount of nitrogen added?

Question 10. A toy company manufactures two types of dolls, A and B. Market tests and available resources have indicated that the combined production level should not exceed 1200 dolls per week and the demand for dolls of type B is at most half of that for dolls of type A. Further, the production level of dolls of type A can exceed three times the production of dolls of other type by at most 600 units. If the company makes profit of Rs 12 and Rs 16 per doll respectively on dolls A and B, how many of each should be produced weekly in order to maximise the profit?

:: Chapter 13 - Probability ::

Question 7. Two coins are tossed once, where (i) E : tail appears on one coin, F : one coin shows head (ii) E : no tail appears, F : no head appears

Question 8. A die is thrown three times, E : 4 appears on the third toss, F : 6 and 5 appears respectively on first two tosses

Question 9. Mother, father and son line up at random for a family picture E : son on one end, F : father in middle

Question 10. A black and a red dice are rolled.
(a) Find the conditional probability of obtaining a sum greater than 9, given that the black die resulted in a 5 .
(b) Find the conditional probability of obtaining the sum 8, given that the red die resulted in a number less than 4.

Question 11. A fair die is rolled. Consider events E = {1,3,5}, F = {2,3} and G = {2,3,4,5} Find
(i) P(E|F) and P(F|E)
(ii) P(E|G) and P(G|E)
(iii) P((E ∪ F)|G) and P ((E ∩ F)|G)

Question 12. Assume that each born child is equally likely to be a boy or a girl. If a family has two children, what is the conditional probability that both are girls given that (i) the youngest is a girl, (ii) at least one is a girl?

Question 13. An instructor has a question bank consisting of 300 easy True / False questions, 200 difficult True / False questions, 500 easy multiple choice questions and 400 difficult multiple choice questions. If a question is selected at random from the question bank, what is the probability that it will be an easy question given that it is a multiple choice question?

Question 14. Given that the two numbers appearing on throwing two dice are different. Find the probability of the event ‘the sum of numbers on the dice is 4’.

Question 15. Consider the experiment of throwing a die, if a multiple of 3 comes up, throw the die again and if any other number comes, toss a coin. Find the conditional probability of the event ‘the coin shows a tail’, given that ‘at least one die shows a 3’. In each of the Exercises 16 and 17 choose the correct answer:

Question 17. If A and B are events such that P(A|B) = P(B|A), then (
A) A ⊂ B but A ≠ B
(B) A = B
(C) A ∩ B = φ
(D) P(A) = P(B)

Question 2. Two cards are drawn at random and without replacement from a pack of 52 playing cards. Find the probability that both the cards are black.

Question 3. A box of oranges is inspected by examining three randomly selected oranges drawn without replacement. If all the three oranges are good, the box is approved for sale, otherwise, it is rejected. Find the probability that a box containing 15 oranges out of which 12 are good and 3 are bad ones will be approved for sale.

Question 4. A fair coin and an unbiased die are tossed. Let A be the event ‘head appears on the coin’ and B be the event ‘3 on the die’. Check whether A and B are independent events or not.

Question 5. A die marked 1, 2, 3 in red and 4, 5, 6 in green is tossed. Let A be the event, ‘the number is even,’ and B be the event, ‘the number is red’. Are A and B independent?

Question 12. A die is tossed thrice. Find the probability of getting an odd number at least once.

Question 13. Two balls are drawn at random with replacement from a box containing 10 black and 8 red balls. Find the probability that
(i) both balls are red.
(ii) first ball is black and second is red.
(iii) one of them is black and other is red.

Question 14. Probability of solving specific problem independently by A and B are 1 2 and 1 3 respectively. If both try to solve the problem independently, find the probability that (i) the problem is solved (ii) exactly one of them solves the problem.

Question 15. One card is drawn at random from a well shuffled deck of 52 cards. In which of the following cases are the events E and F independent ?
(i) E : ‘the card drawn is a spade’ F : ‘the card drawn is an ace’
(ii) E : ‘the card drawn is black’ F : ‘the card drawn is a king’
(iii) E : ‘the card drawn is a king or queen’ F : ‘the card drawn is a queen or jack’.

Question 16. In a hostel, 60% of the students read Hindi news paper, 40% read English news paper and 20% read both Hindi and English news papers. A student is selected at random.
(a) Find the probability that she reads neither Hindi nor English news papers.
(b) If she reads Hindi news paper, find the probability that she reads English news paper.
(c) If she reads English news paper, find the probability that she reads Hindi news paper. Choose the correct answer in Exercises 17 and 18.

Question 17. The probability of obtaining an even prime number on each die, when a pair of dice is rolled is
(A) 0
(B) 1 3
(C) 1 12
(D) 1 36

Question 18. Two events A and B will be independent, if
(A) A and B are mutually exclusive
(B) P(A′B′) = [1 – P(A)] [1 – P(B)]
(C) P(A) = P(B)
(D) P(A) + P(B) = 1

Question 16. In a hostel, 60% of the students read Hindi news paper, 40% read English news paper and 20% read both Hindi and English news papers. A student is selected at random.
(a) Find the probability that she reads neither Hindi nor English news papers.
(b) If she reads Hindi news paper, find the probability that she reads English news paper.
(c) If she reads English news paper, find the probability that she reads Hindi news paper. Choose the correct answer in Exercises 17 and 18.

Question 17. The probability of obtaining an even prime number on each die, when a pair of dice is rolled is
(A) 0
(B) 1 3
(C) 1 12
(D) 1 36

Question 18. Two events A and B will be independent, if
(A) A and B are mutually exclusive
(B) P(A′B′) = [1 – P(A)] [1 – P (B)]
(C) P(A) = P(B)
(D) P(A) + P(B) = 1

EXERCISE 13.3

Question 1. An urn contains 5 red and 5 black balls. A ball is drawn at random, its colour is noted and is returned to the urn. Moreover, 2 additional balls of the colour drawn are put in the urn and then a ball is drawn at random. What is the probability that the second ball is red?

Question 2. A bag contains 4 red and 4 black balls, another bag contains 2 red and 6 black balls. One of the two bags is selected at random and a ball is drawn from the bag which is found to be red. Find the probability that the ball is drawn from the first bag.

Question 3. Of the students in a college, it is known that 60% reside in hostel and 40% are day scholars (not residing in hostel). Previous year results report that 30% of all students who reside in hostel attain A grade and 20% of day scholars attain A grade in their annual examination. At the end of the year, one student is chosen at random from the college and he has an A grade, what is the probability that the student is a hostlier?

Question 4. In answering a question on a multiple choice test, a student either knows the answer or guesses. Let 3 4 be the probability that he knows the answer and 1 4 be the probability that he guesses. Assuming that a student who guesses at the answer will be correct with probability 1 4 . What is the probability that the student knows the answer given that he answered it correctly?

Question 5. A laboratory blood test is 99% effective in detecting a certain disease when it is in fact, present. However, the test also yields a false positive result for 0.5% of the healthy person tested (i.e. if a healthy person is tested, then, with probability 0.005, the test will imply he has the disease). If 0.1 percent of the population actually has the disease, what is the probability that a person has the disease given that his test result is positive ?

Question 6. There are three coins. One is a two headed coin (having head on both faces), another is a biased coin that comes up heads 75% of the time and third is an unbiased coin. One of the three coins is chosen at random and tossed, it shows heads, what is the probability that it was the two headed coin ?

Question 7. An insurance company insured 2000 scooter drivers, 4000 car drivers and 6000 truck drivers. The probability of an accidents are 0.01, 0.03 and 0.15 respectively. One of the insured persons meets with an accident. What is the probability that he is a scooter driver?

Question 8. A factory has two machines A and B. Past record shows that machine A produced 60% of the items of output and machine B produced 40% of the items. Further, 2% of the items produced by machine A and 1% produced by machine B were defective. All the items are put into one stockpile and then one item is chosen at random from this and is found to be defective. What is the probability that it was produced by machine B?

Question 9. Two groups are competing for the position on the Board of directors of a corporation. The probabilities that the first and the second groups will win are 0.6 and 0.4 respectively. Further, if the first group wins, the probability of introducing a new product is 0.7 and the corresponding probability is 0.3 if the second group wins. Find the probability that the new product introduced was by the second group.

Question 10. Suppose a girl throws a die. If she gets a 5 or 6, she tosses a coin three times and notes the number of heads. If she gets 1, 2, 3 or 4, she tosses a coin once and notes whether a head or tail is obtained. If she obtained exactly one head, what is the probability that she threw 1, 2, 3 or 4 with the die?

Question 11. A manufacturer has three machine operators A, B and C. The first operator A produces 1% defective items, where as the other two operators B and C produce 5% and 7% defective items respectively. A is on the job for 50% of the time, B is on the job for 30% of the time and C is on the job for 20% of the time. A defective item is produced, what is the probability that it was produced by A?

Question 12. A card from a pack of 52 cards is lost. From the remaining cards of the pack, two cards are drawn and are found to be both diamonds. Find the probability of the lost card being a diamond.

Question 13. Probability that A speaks truth is 4 5 . A coin is tossed. A reports that a head appears. The probability that actually there was head is
(A) 4 5
(B) 1 2
(C) 1 5
(D) 2 5

Question 14. If A and B are two events such that A ⊂ B and P(B) ≠ 0, then which of the following is correct?
(A) P(A| B) P(B) P(A) =
(B) P(A|B) < P(A)
(C) P(A|B) ≥ P(A)
(D) None of these

EXERCISE 13.4

Question 1. State which of the following are not the probability distributions of a random variable. Give reasons for your answer.

Question 2. An urn contains 5 red and 2 black balls. Two balls are randomly drawn. Let X represent the number of black balls. What are the possible values of X? Is X a random variable ?

Question 3. Let X represent the difference between the number of heads and the number of tails obtained when a coin is tossed 6 times. What are possible values of X?

Question 4. Find the probability distribution of (i) number of heads in two tosses of a coin. (ii) number of tails in the simultaneous tosses of three coins. (iii) number of heads in four tosses of a coin.

Question 5. Find the probability distribution of the number of successes in two tosses of a die, where a success is defined as (i) number greater than 4 (ii) six appears on at least one die.

Question 6. From a lot of 30 bulbs which include 6 defectives, a sample of 4 bulbs is drawn at random with replacement. Find the probability distribution of the number of defective bulbs.

Question 7. A coin is biased so that the head is 3 times as likely to occur as tail. If the coin is tossed twice, find the probability distribution of number of tails.

Question 8. A random variable X has the following probability distribution :

Question 9.Determine
(i) k
(ii) P(X < 3)
(iii) P(X > 6)
(iv) P(0 < X < 3)

EXERCISE 13.5

Question 1. A die is thrown 6 times. If ‘getting an odd number’ is a success, what is the probability of (i) 5 successes? (ii) at least 5 successes? (iii) at most 5 successes?

Question 2. A pair of dice is thrown 4 times. If getting a doublet is considered a success, find the probability of two successes.

Question 3. There are 5% defective items in a large bulk of items. What is the probability that a sample of 10 items will include not more than one defective item?

Question 4. Five cards are drawn successively with replacement from a well-shuffled deck of 52 cards. What is the probability that (i) all the five cards are spades? (ii) only 3 cards are spades? (iii) none is a spade?

Question 5. The probability that a bulb produced by a factory will fuse after 150 days of use is 0.05. Find the probability that out of 5 such bulbs (i) none (ii) not more than one (iii) more than one (iv) at least one will fuse after 150 days of use.

Question 6. A bag consists of 10 balls each marked with one of the digits 0 to 9. If four balls are drawn successively with replacement from the bag, what is the probability that none is marked with the digit 0?

Question 7. In an examination, 20 questions of true-false type are asked. Suppose a student tosses a fair coin to determine his answer to each question. If the coin falls heads, he answers 'true'; if it falls tails, he answers 'false'. Find the probability that he answers at least 12 questions correctly.

Question 8. Suppose X has a binomial distribution B 6, 1 2 . Show that X = 3 is the most likely outcome. (Hint : P(X = 3) is the maximum among all P(xi), xi = 0,1,2,3,4,5,6)

Question 9. On a multiple choice examination with three possible answers for each of the five questions, what is the probability that a candidate would get four or more correct answers just by guessing ?

Question 10. A person buys a lottery ticket in 50 lotteries, in each of which his chance of winning a prize is 1 100 . What is the probability that he will win a prize
(a) at least once
(b) exactly once
(c) at least twice?

Question 11. Find the probability of getting 5 exactly twice in 7 throws of a die.

Question 12. Find the probability of throwing at most 2 sixes in 6 throws of a single die.

Question 13. It is known that 10% of certain articles manufactured are defective. What is the probability that in a random sample of 12 such articles, 9 are defective? In each of the following, choose the correct answer:

Question 14. In a box containing 100 bulbs, 10 are defective. The probability that out of a sample of 5 bulbs, none is defective is

Miscellaneous Exercise

Question 1. A and B are two events such that P (A) ≠ 0. Find P(B|A), if (i) A is a subset of B (ii) A ∩ B = φ

Question 2. A couple has two children, (i) Find the probability that both children are males, if it is known that at least one of the children is male. (ii) Find the probability that both children are females, if it is known that the elder child is a female.

Question 3. Suppose that 5% of men and 0.25% of women have grey hair. A grey haired person is selected at random. What is the probability of this person being male? Assume that there are equal number of males and females.

Question 4. Suppose that 90% of people are right-handed. What is the probability that at most 6 of a random sample of 10 people are right-handed?

Question 5. An urn contains 25 balls of which 10 balls bear a mark 'X' and the remaining 15 bear a mark 'Y'. A ball is drawn at random from the urn, its mark is noted down and it is replaced. If 6 balls are drawn in this way, find the probability that
(i) all will bear 'X' mark.
(ii) not more than 2 will bear 'Y' mark.
(iii) at least one ball will bear 'Y' mark.
(iv) the number of balls with 'X' mark and 'Y' mark will be equal.

Question 6. In a hurdle race, a player has to cross 10 hurdles. The probability that he will clear each hurdle is 5 6 . What is the probability that he will knock down fewer than 2 hurdles?

Question 7. A die is thrown again and again until three sixes are obtained. Find the probability of obtaining the third six in the sixth throw of the die.

Question 8. If a leap year is selected at random, what is the chance that it will contain 53 tuesdays?

Question 9. An experiment succeeds twice as often as it fails. Find the probability that in the next six trials, there will be atleast 4 successes.

Question 10. How many times must a man toss a fair coin so that the probability of having at least one head is more than 90%?

Question 11. In a game, a man wins a rupee for a six and loses a rupee for any other number when a fair die is thrown. The man decided to throw a die thrice but to quit as and when he gets a six. Find the expected value of the amount he wins / loses.

Question 12. Suppose we have four boxes A,B,C and D containing coloured marbles as given below: One of the boxes has been selected at random and a single marble is drawn from it. If the marble is red, what is the probability that it was drawn from box A?, box B?, box C?

Question 13. Assume that the chances of a patient having a heart attack is 40%. It is also assumed that a meditation and yoga course reduce the risk of heart attack by 30% and prescription of certain drug reduces its chances by 25%. At a time a patient can choose any one of the two options with equal probabilities. It is given that after going through one of the two options the patient selected at random suffers a heart attack. Find the probability that the patient followed a course of meditation and yoga?

Question 14. If each element of a second order determinant is either zero or one, what is the probability that the value of the determinant is positive? (Assume that the individual entries of the determinant are chosen independently, each value being assumed with probability 1 2 ).

Question 15. An electronic assembly consists of two subsystems, say, A and B. From previous testing procedures, the following probabilities are assumed to be known: P(A fails) = 0.2 P(B fails alone) = 0.15 P(A and B fail) = 0.15 Evaluate the following probabilities (i) P(A fails|B has failed) (ii) P(A fails alone)

Question 16. Bag I contains 3 red and 4 black balls and Bag II contains 4 red and 5 black balls. One ball is transferred from Bag I to Bag II and then a ball is drawn from Bag II. The ball so drawn is found to be red in colour. Find the probability that the transferred ball is black. Choose the correct answer in each of the following:

Question 17. If A and B are two events such that P(A) ≠ 0 and P(B | A) = 1, then
(A) A ⊂ B
(B) B ⊂ A
(C) B = φ
(D) A = φ

Question 18. If P(A|B) > P(A), then which of the following is correct :
(A) P(B|A) < P
(B) (B) P(A ∩ B) < P(A) . P(B)
(C) P(B|A) > P(B)
(D) P(B|A) = P(B)

Question 19. If A and B are any two events such that P(A) + P(B) – P(A and B) = P(A), then
(A) P(B|A) = 1
(B) P(A|B) = 1
(C) P(B|A) = 0
(D) P(A|B) = 0

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Section – B, Writing (Short composition)

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(Download) CBSE Class-10 2016-17 Sample Paper (Spanish)

• Please check that this question paper contains 04 printed pages.

• Code number given on the question paper should be written on the title page of the answer-book by the candidate.

• Please check that this question paper contains 12 questions. Answer all questions in the target language.

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Time allowed: 03 hrs. Maximum Marks: 90

1. Lee el siguiente texto sobre el pintor colombiano Fernando Botero y señala si es verdadero (V) o falso (F). Read the following text about the Colombian painter, Fernando Botero, and reply TRUE/FALSE 05 Pocos artistas hispanoamericanos han logrado tanta repercusión a nivel internacional como el pintor y escultor colombiano Fernando Botero. Nacido en Medellín en 1932, Fernando Botero fue el segundo de los tres hijos de la pareja formada por David Botero Mejía y Flora Angulo de Botero

Aunque en su juventud estuvo durante un corto lapso de tiempo en la Academia de San Fernando en Madrid y en la de San Marcos en Florencia, su formación artística fue autodidacta. Sus primeras obras conocidas son las ilustraciones que publicó en el suplemento literario del diario El Colombiano, de su ciudad natal. A los 19 años viajó a Bogotá, donde hizo su primera exposición individual. Posteriormente viajó a Europa, donde residió por espacio de cuatro años, principalmente en Madrid, Barcelona, París y Florencia.

Aunque ingresó en las academias mencionadas, siguió formándose a base de leer, visitar museos y, sobre todo, pintar, como él mismo diría. Luego viajó a México, Nueva York y Washington en un período de febril creación y escasos recursos económicos, acompañado de su esposa Gloria Zea. Entre 1961 y 1973 fijó su residencia en Nueva York. Luego viviría en París, alternando su residencia en la capital francesa con largas estancias en Pietrasanta o su finca en el pueblo cundinamarqués de Tabio. En 1977 expuso sus bronces por primera vez en el Grand Palais de París. Tras cuatro decenios de labor ininterrumpida, su reconocimiento en el campo escultórico se hizo también universal. Convertido ya en uno de los artistas vivos más cotizados del mundo, Botero no ha dejado nunca, sin embargo, de alzar la voz contra la injusticia y de mantener su arte en línea con la realidad histórica y social.

a) Fernando Botero es un pintor hispanoamericano. V/F
b) Publicó sus primeras ilustraciones en un suplemento literario publicado en Medellín. V/F
c) Hizo su primera exposición individual en el año 1951. V/F
d) En el texto se afirma que Botero es un artista autodidacta. V/F
e) Según el texto, el artista actualmente vive en Nueva York. V/F

2. Lee el texto y contesta verdadero (V) o falso (F). Read the following text and reply TRUE/FALSE.  En estos tiempos, todos estamos de acuerdo en que hacer ejercicios es bueno para nuestra salud, pero a veces no aprovechamos todas sus ventajas al no hacerlo de manera correcta. Hay algunas normas para sacarle el máximo partido al ejercicio. En primer lugar, algunas personas piensan que deben seguir siempre una misma tabla de gimnasia, sin embargo, los músculos del cuerpo, después de unos meses de actividad, tienden a habituarse, haciendo que los ejercicios sean cada vez menos eficaces. Por eso conviene ir variando poco a poco la tabla de ejercicios.

Además de esto, es necesario cambiar esa imagen de una persona que suda y resopla mientras hace ejercicio. Es cierto que se necesita alcanzar un cierto ritmo que acelere la frecuencia de respiración, pero la intensidad debe ser la justa para que nos permita hablar a la vez que corremos, andamos, vamos en bicicleta, etc…Por otra parte, en cuanto a los ejercicios de suelo o con pesas, lo mejor es realizarlos despacio y con calma y prestar mucha atención al momento de volver a la posición inicial.

Es caso de que se realicen deportes asimétricos, como el tenis, es muy recomendable que, después del entrenamiento, hagamos otros ejercicios compensatorios, ya que en el tenis por ejemplo, se utilizan más los músculos de un lado que los del otro. Por último, lo que nunca debemos olvidad es que después de nuestra sesión de ejercicio, hay que hacer a menos 5 minutos de estiramientos. Estos sirven para evitar lesiones y molestias. (Adaptado de Revista Consumer)

a) Según el texto, no se debe cambiar la tabla de ejercicios cuando queremos desarrollar más los músculos. V/F
b) En el texto se recomienda que al correr, aceleramos la respiración. V/F
c) En el texto se afirma que los estiramientos actúan como ejercicios compensatorios en actividades asimétricas.
d) ¿Estás de acuerdo con el mensaje del texto? Sí/No, y ¿por qué? (Do you subscribe to the idea contained in the text? Yes/No, and Why?)

3. Lee el siguiente diálogo y contesta a las siguientes preguntas. Read the following dialogue and answer the following questions. 05

Pedro fue a la librería y compró La Eda de Oro, de José Martí. Cuando llegó al albergue, puso el libro sobre la mesa. Poco después entró Antonio y preguntó:

- ¿De quién es este libro?
- Es mío – le contestó Pedro-, y tuyo también – añadió.
- Gracias, Pedro. Entonces, como es nuestro, voy a leerlo otra vez. Me gusta muchísimo esta obra que Martí escribió para los niños de América.
- A mí también. Quiero leerla otra vez, porque siempre aprendo algo nuevo cuando releo cualquiera de sus obras.
-¿Tienes algún otro libro suyo? –le preguntó Pedro a su amigo Antonio.
- Sí, allí en aquel librero están sus Versos sencillos.
- ¡Qué bueno! ¿Me lo prestas para mi amigo Jesús? Él quiere aprender algunos de esos versos para cantarlos con la música de La Guantanamera.
- Sí, puedes cogerlo.

a) ¿Qué compró Pedro?
b) ¿Qué le preguntó Antonio cuando entró?
c) ¿Qué le contestó Antonio?
d) ¿Antonio ya leyó La Edad de Oro?
e) ¿Para quiénes Martí escribió esa obra?
 

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(Download) CBSE Class-10 2016-17 Sample Paper (Tangkhul)

• Please check that this question paper contains 8 pages.

• Code number given on the right hand side of the question paper should be written on the title page of the answer book by the candidate.

• Please check that this question paper contains 38 questions.

• Please write down the Serial Number of the question before attempting it.

• 15 minute time has been allotted to read this question paper. The question paper will be distributed at 10.15 a.m. From 10.15 a.m. to 10.30 a.m., the! students will read the question paper only and will not write any answer on the answer-book during this period.

Time allowed: 3 hours

Maximum Marks: 90

Khangahan hi akhum mati sada khaira.

A: READING
B: WRITING SKILL
C: GRAMMAR
D: LITERATURE

GENERAL INSTRUCTION:

1. Khangahan saikorg ngahankaserlu,
2. Nana ningkachang ~khum eina haophok paira.
3. Akhum chiwui khangahan saikorg chi meisa eina ngahankalu,

AKHUM-A

1. Mikahai tuikhurkha hi palaga khangahan bing chi A, B, C kala D wui eina kapangkhuilaga ngahankalu. Longyao eina Kamringphiwui leishi khangarok hili ningkachang kakharam kala kakahao mazangla kachangkhat leishi khangarokna da theikhui. Mayar ngala salakhava meisum shim ka.a longshimli yaothui khangarok hiya leisera. Longyao eina Karru.ngphi leishi khangarokhi Phungcham preinaosanna ngazekrop eina leikasana. Kamringphi Vareina arong, sam, phgphor, mailep kala pheithei pangtbei mayut kharirik eina samihai. Chiwui tungli ngashom ngaror kharnipana acham-aram, achei-arei kala shgngahan katheiwui vang kharamaosan mateila katongana leikashiva kala khangachanvg shanao akha sakasana. Longyao Phungcham Vashumwoshi meiphungwui kharara shimkhur akha wui yarkhoka akha kasawui tungli zakrnaila kala mirinla nganantap eina okthuisa haoda r£!s£! khaphali khikha hapkakhano maleikapaiya yaronaona. Ani khaniwui leishat pamngaret kajuili haikhami thai.

Khanghan:

A. Meisumshim kaji hiya kajina.

a. Meisaparn nganaokhavai shim.
b. La tamkhavai shim.
c. Longshim.
d. Ngalanao khalei shimli yaronao yaothuivada pamnganao khavai shim.

B. Khili Longshim sa khala?

a. Shim kahakli kahangna,
b. Longnao pikhavai shimli kahangna.
d. Awor tamkhavai shimli kahangna.
c. Bichar sakhavai shimli kahangna.

C. Kamringphi ngala khamatha phaphor khaung tungli ngashom ngaror khami chi khikhala?

a. .acham-aram
c. shangahan khamatha
b. achei-arei
d. katonga hi zangsera.

D. Longyao mayar khamatha tungli mirinla nganantap eina okthuisa haoda rasa katha kaphali khikha hapkakhano maleikapaiya yaronnaona kaji hili 'nganantap eina' kaji hiya _

a. mavatlakla kajina.
c. Ringshilak eina kajina.
b. rnawunnap eina kajina.
d. hekmaheilak eina kajina.

E. Longyao eina Kamringphiwui Ieishat pamngaret kajuili haikhami thai kajiwui kakhalatva ---------

a. kashunglaka kajina
c. ani chalaka kajina
b. katongana theisera kajina
d. ani juilaka kajina.

2. Kasa Akhavana apuk apakva semchang hailaga okathuili mangla kapaiya eina chipemhaida shiri-shira, sayur-vayur kala mungkhavai mikumoli chip am kahai thili, atam ngachak khavai athurnwui ningkachang sakhuiranu chihaoda mangla kapaiya saikorali ngahotlaga kaphaning ngahanuwa. Hunakhava masikthat kahai thahaowa khipanakha khak mashokthua. Khaleilaga kafa mikshana hithada hangshokrasai, kumkha ngashunsa kumkha ngayasa, Chi kasha eina yarui ngayeirei haowa. kathada kumkha mangasamlakla otsa kala kumkha pira khalada ujjina rnamayaphut thua. Kha ala hithasa kajila mahangsangthua. Kasa Akhavala kafa wui chi sakapai ot maningmana jihaowa. Saikora chi varilui haowa. chitharan thingtonli mashonshei eina pakasa chaklenna hithada hangshokrasai. Kasa Akhava kafawui chiya ithum masararmara, iwui kaphaningle hunakha chotshap otsa, hunakha kachot vanshap ngasamkhui kajina phameira. Kasa Akhavana awui thangkhameichi theilaga ali marankhame saranuda somihaowa. Chiwuivang eina aj~ rashungda mikumola chaklen khon khanganana kala thang zimikkha hi ngashun-ngayada khai kahaina. Kafala avaishatshat eina leikhurli ayamsang haida arui rashungda thingngayung phashailaga kahor kaho matheila okthuida khaleina. Khangahan: (1 x 5 = 5)

A. Kasa Akhavana okathui hi shiri-shira, sayur-vayur kala mangla kapaiya ayayav~ chipem hailaga khipali mungngasak khala?

a. Kazingkhali
b. Shangkhgli
c. Kazingraoli
d. Mikumoli

B. Kasa Akhavana okathuihi horkhavai zimik semhailaga khiwuivang tangkhamang ngaya leingasak khangai?

a. Ngamiting haida ot masapai khavai
b. Ngashun sana kahai chi suita khavai.
c. Ngasamkhui khavai.
d. Ot masapam shonpai khavai.

C. Kafali miksha kaho hi awui amik - - --------------------

a. Khamathingwui vangna.
c. Makhamathingwui vangna
b. Kateowui vangna.
d. Kahakwui vangna.

D. Kafana hithasa kaji hili mamayangai meikap kajipa chi ujjina. Ujji hi

a. Varana
b. Vareina
c. Sarcina
d. Sateona

E. Kasa Akhavana awui theithangkhamei chi theilaga ali maran khame saranuda somihaowa. Hili "thangkharnei" kaj i tui hi grammar eina _

a. Noun hoi
b. Pronoun hoi
c. Adjective hoi
d. Adverb hoi

3. Ihaowui phanitbingwui ngachaili akhamang kaji shanaona shar kasa phanit chiya Chumphana, Chumpha hiya zingkum chiwui heiwun chumli ungkazipser hailaga khuizaphokuga kaji eina zaramei khavai kasa Akhavali mingkazai phanitna. Shanaowui shar sathuda mayarnao shimli pamshara, zeikhai ngaha, khalen, kapa kathathala numneishara jihaida khararli ahaiser haowa. Varan kaji mayarnao saikora khavaknao ngakom khuingarok laga luili arnorsong laga lumshilao asathaiya. Shimshong preivangarana rakhong khonhailaga ungngarung kadhar "sora" asokkhuilaga chieina phahanguwa laga gahara ngarana khaireo av£ eina ava chi mali zangkotphumhai, zamkhor zangzang, mahawui khaikao, hanshi eina hanglaga chumli zangphaphayaya, Chumkhot luk eina ma khayamkhuida maphuikha chithang khuiphok haowa. Hithada shar saki kaji kupser kahai eina khongnai prei ngakomkhui ngaroklaga athumla lumlaophup phashak phaza laga lumlaothaiya. Khangahan: (1 x 5 = 5)

A. Zaramei khavai kasa Akhavali mingkazai phanitna. Hili "zaramei'tkaji hiya kajina. -----

a. Kachungkha zakhavai
b. Kakan mangava khavai
c. Kanshimei khavai
d. Zingkum peikhavai

B. Kalikhada mayamaowui khalen Chumphali shanaowui shar kasali zangnar haiakha khalen chi --------------

a. Mahaishuna
b. Mamahaithua
c. Makapngaiki jihaowa
d. Shimanhaowa

C. Varan kaji mayamao saikora luili apiser haiakha kachikatha mayamao shimli taira khala?

a. Naoshinao mang
c. awo kasarnaomang
b. Matailakmara
d. Mayamao naoshinao eina awo kasarnao mango

D. Ihaowui ngashanli shanaona shar kasa hi manglina.

a. Chumphg
c. Mangkhap
b. Mawonzai
c. Luishom

E. Chumkhot luk eina ma khayamkhuida maphuikha chithang khuiphok haowa. Hili "khayamkhuida" kaji hi kajina.

a. Sankhuiya
c. Homkhuiya
b. Sekkhuiya
d. Mathasek eina f£!sangkhuiya
 

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NCERT Chemistry Question Paper (Class - 12)

:: Chapter 1 - The Solid State ::

INTEXT QUESTIONS

Question 1.1: Why are solids rigid?

Question 1.2: Why do solids have a definite volume?

Question 1.3: Classify the following as amorphous or crystalline solids: Polyurethane, naphthalene, benzoic acid, teflon, potassium nitrate, cellophane, olyvinylchloride, fibre glass, copper.

Question 1.4: Why is glass considered a super cooled liquid?

Question 1.5: Refractive index of a solid is observed to have the same value along all directions. Comment on the nature of this solid. Would it show cleavage property?

Question 1.6: Classify the following solids in different categories based on the nature of intermolecular forces operating in them:
Potassium sulphate, tin, benzene, urea, ammonia, water, zinc sulphide, graphite, rubidium, argon, silicon carbide.

Question 1.7: Solid A is a very hard electrical insulator in solid as well as in molten state and melts at extremely high temperature. What type of solid is it?

Question 1.8: Ionic solids conduct electricity in molten state but not in solid state. Explain.

Question 1.9: What type of solids are electrical conductors, malleable and ductile?

Question 1.10: Give the significance of a ‘lattice point’.

Question 1.11: Name the parameters that characterize a unit cell.

Question 1.12: Distinguish between (i) Hexagonal and monoclinic unit cells (ii) Face−centred and end−centred unit cells.

Question 1.13: Explain how much portion of an atom located at (i) corner and (ii) body−centre of a cubic unit cell is part of its neighboring unit cell.

Question 1.14: What is the two dimensional coordination number of a molecule in square close packed layer?

Question 1.15: A compound forms hexagonal close−packed structure. What is the total number of voids in 0.5 mol of it? How many of these are tetrahedral voids?

Question 1.16: A compound is formed by two elements M and N. The element N forms ccp and atoms of M occupy 1/3rd of tetrahedral voids. What is the formula of the compound?

Question 1.17: Which of the following lattices has the highest packing efficiency (i) simple cubic (ii) body−centred cubic and (iii) hexagonal close−packed lattice?

Question 1.18: An element with molar mass 2.7 × 10−2 kg mol−1 forms a cubic unit cell with edge length 405 pm. If its density is 2.7 × 103 kg m−3, what is the nature of the cubic unit cell?

Question 1.19: What type of defect can arise when a solid is heated? Which physical property is effected by it and in what way?

Question 1.20: What type of stoichiometric defect is shown by: (i) ZnS (ii) AgBr

Question 1.21: Explain how vacancies are introduced in an ionic solid when a cation of higher valence is added as an impurity in it.

Question 1.22: Ionic solids, which have anionic vacancies due to metal excess defect, develop colour.Explain with the help of a suitable example.

Question 1.23: A group 14 element is to be converted into n−type semiconductor by doping it with a suitable impurity. To which group should this impurity belong?

Question 1.24:What type of substances would make better permanent magnets, ferromagnetic or ferrimagnetic. Justify your Solution:.

EXERCISE

Question 1:Define the term 'amorphous'. Give a few examples of amorphous solids.

Question 2. What makes a glass different from a solid such as quartz?

Question 3. Classify each of the following solids as ionic, metallic, molecular, network (covalent) or amorphous.

Question 4. (i) What is meant by the term 'coordination number' ? (ii) What is the coordination number of atoms (a) in a cubic close packed structure? (b) in a body–centered cubic structure?

Question 5. How can you determine the atomic mass of an unknown metal if you know its density?

Question 6. 'Stability of a crystal is reflected in the magnitude of its melting points'. Comment.
Collect melting points of solid water, ethyl alcohol, diethyl ether and methane from a data book. What can you say about the intermolecular forces between these molecules?

Question 7. How will you distinguish between the following pairs of terms (i) Hexagonal close packing and cubic close packing (ii) Crystal lattice and unit cell (iii) Tetrahedral void and octahedral void.

Question 8 How many lattice points are there in one unit cell of each of the following lattice? (i) Face–centred cubic (ii) Face–centred tetragonal (iii) Body–centred

Question 9. Explain (i) The basis of similarities and differences between metallic and ionic crystals. (ii) Ionic solids are hard and brittle.

Question 10 Calculate the efficiency of packing in case of a metal crystal for (i) simple cubic (ii) body–centred cubic (iii) face–centred cubic (with the assumptions that atoms are touching each other).

Question 11 Silver crystallises in fcc lattice. If edge length of the cell is 4.07 × 10-8 cm and density is 0.5 g cm3, calculate the atomic mass of silver.

Question 12 A cubic solid is made of two elements P and Q. Atoms of Q are at the corners of the cube and P at the body–centre. What is the formula of the compound? What are the coordination numbers of P and Q?

Question 13 Niobium crystallises in body–centred cubic structure. If density is 8.55 g cm–3, calculate atomic radius of niobium using its atomic mass 93 u.

Question 14 If the radius of the octahedral void is r and radius of the atoms in closepacking is R, derive relation between r and R.

Question 15 Copper crystallises into a fcc lattice with edge length 3.61 × 10–8 cm. Show that the calculated density is in agreement with its measured value of 8.92 g cm-3.

Question 16 Analysis shows that nickel oxide has the formula Ni0.98O1.00. What fractions of nickel exist as Ni2+ and Ni3+ ions?

Question 17 What is a semiconductor? Describe the two main types of semiconductors and contrast their conduction mechanism.

Question 18: Non–stoichiometric cuprous oxide, Cu2O can be prepared in laboratory. In this oxide, copper to oxygen ratio is slightly less than 2:1. Can you account for the fact that this substance is a p–type semiconductor?

Question 19: Ferric oxide crystallises in a hexagonal close–packed array of oxide ions with two out of every three octahedral holes occupied by ferric ions. Derive the formula of the ferric oxide.

Question 20: Classify each of the following as being either a p–type or an n–type semiconductor: (i) Ge doped with In (ii) B doped with Si.

Question 21: Gold (atomic radius = 0.144 nm) crystallises in a face–centred unit cell. What is the length of a side of the cell?

Question 22: In terms of band theory, what is the difference (i) Between a conductor and an insulator (ii) Between a conductor and a semiconductor

Question 23: Explain the following terms with suitable examples: (i) Schottky defect (ii) Frenkel defect (iii) Interstitials and (iv) F–centres

Question 24: Aluminium crystallises in a cubic close–packed structure. Its metallic radius is 125 pm. (i) What is the length of the side of the unit cell? (ii) How many unit cells are there in 1.00 cm3 of aluminum?

Question 25: If NaCl is doped with 10−3 mol % of SrCl2, what is the concentration of cation vacancies?

Question 26: Explain the following with suitable examples: (i) Ferromagnetism (ii)Paramagnetism (iii)Ferrimagnetism (iv)Antiferromagnetism (v)12–16 and 13–15 group compounds.

:: Chapter 2 - Solutions ::

EXERCISE

2.2 Give an example of a solid solution in which the solute is a gas.

2.3 Define the following terms:
(i) Mole fraction

2.3 Define the following terms:
(ii) Molality

2.3 Define the following terms:
(iii) Molarity

2.3 Define the following terms:
(iv) Mass percentage.

2.4 Concentrated nitric acid used in laboratory work is 68% nitric acid by mass in aqueous solution. What should be the molarity of such a sample of the acid if the density of the solution is 1.504 g mL–1?

2.5 A solution of glucose in water is labelled as 10% w/w, what would be the molality and mole fraction of each component in the solution? If the density of solution is 1.2 g mL–1, then what shall be the molarity of the solution?

2.6 How many mL of 0.1 M HCl are required to react completely with 1 g mixture of Na2CO3 and NaHCO3 containing equimolar amounts of both?

2.7 A solution is obtained by mixing 300 g of 25% solution and 400 g of 40% solution by mass. Calculate the mass percentage of the resulting solution.

2.8 An antifreeze solution is prepared from 222.6 g of ethylene glycol (C2H6O2) and 200 g of water. Calculate the molality of the solution. If the density of the solution is 1.072 g mL–1, then what shall be the molarity of the solution?

:: Chapter 3 - Electrochemistry ::

CONCEPT

 Electrochemistry and It's Uses

construction and functioning of Daniell cell

electrode potential, cell potential & Represent a galvanic cell

Structure and Working of Standard Hydrogen Electrode

 measure the standard potential of Cu²⁺

 Use of platinum or gold in standard hydrogen electrode

What is the Nernst Equation

 Relation between EӨcell and Kc

Nernst Equation for the given chemical reaction

Gibbs energy of reaction taking place in an electrochemical cell

Conductance of Electrolytic Solutions

What is cell constant

What is a superconductor

Factor effecting conductance & ionic conductance

What are Electronically conducting polymers and there advantages

Problems takes place in measuring of conductivity

What is a conductivity cell

Method to measure conductance using Wheatstone bridge

Concept of Molar Conductivity

Kohlrausch law of independent migration of ions

How to measure equilibrium constant and limiting molar conductivity of week electrolytite

Faraday’s Laws of Electrolysis

Explain type of cells

What is corrosion explain how corrosion works as a cell

What is hydrogen economy

EXERCISE

Question 1:Arrange the following metals in the order in which they displace each other from the solution of their salts.
Al, Cu, Fe, Mg and Zn

Question 2:Given the standard electrode potentials,
K⁺/K = −2.93V, Ag⁺/Ag = 0.80V,
Hg²⁺/Hg = 0.79V
Mg²⁺/Mg = −2.37 V, Cr³⁺/Cr = − 0.74V Arrange these metals in their increasing order of reducing power.

Question 3:Depict the galvanic cell in which the reaction Zn(s) + 2Ag⁺(aq) → Zn²⁺(aq) + 2Ag(s) takes place. Further show:

(i) Which of the electrode is negatively charged?
(ii) The carriers of the current in the cell.
(iii) Individual reaction at each electrode.

Question 4:Calculate the standard cell potentials of galvanic cells in which the following reactions take place:

(i) 2Cr(s) + 3Cd²⁺(aq) → 2Cr³⁺(aq) + 3Cd
(ii) Fe²⁺(aq) + Ag⁺(aq) → Fe³⁺(aq) + Ag(s)

Calculate the =∆rGθ and equilibrium constant of the reactions.

Question 5:Write the Nernst equation and emf of the following cells at 298 K:

(i) Mg(s) | Mg²⁺(0.001M) || Cu²⁺(0.0001 M) | Cu(s)
(ii) Fe(s) | Fe²⁺(0.001M) || H⁺(1M)|H₂(g)(1bar) | Pt(s)
(iii) Sn(s) | Sn²⁺(0.050 M) || H⁺(0.020 M) | H₂(g) (1 bar) | Pt(s)
(iv) Pt(s) | Br₂(l) | Br−(0.010 M) || H

Question 6:In the button cells widely used in watches and other devices the following reaction takesplace:

Zn(s) + Ag₂O(s) + H₂O(l) → Zn²+(aq) + 2Ag(s) + 2OH⁻(aq)  Determine and for the reaction.

Question 7:Define conductivity and molar conductivity for the solution of an electrolyte. Discuss their variation with concentration.

Question 8:The conductivity of 0.20 M solution of KCl at 298 K is 0.0248 Scm−1. Calculate its molar conductivity.

Question 9:The resistance of a conductivity cell containing 0.001M KCl solution at 298 K is 500M. What is the cell constant if conductivity of 0.001M KCl solution at 298 K is 0.146 × 10−3S cm^−1.

Question 10:The conductivity of sodium chloride at 298 K has been determined at different concentrations and the results are given below:

Question 11: Conductivity of 0.00241 M acetic acid is 7.896 × 10⁻⁵ S cm⁻¹. Calculate its molar conductivity and if for acetic acid is 390.5 S cm² mol⁻¹, what is its dissociation constant?

Question 12:How much charge is required for the following reductions:

(i) 1 mol of Al³⁺ to Al.
(ii) 1 mol of Cu²⁺ to Cu.
(iii) 1 mol of MnO^4– to Mn²⁺.

Question 13:How much electricity in terms of Faraday is required to produce

(i) 20.0 g of Ca from molten CaCl₂.
(ii) 40.0 g of Al from molten Al₂O₃.

Question 14: How much electricity is required in coulomb for the oxidation of (i) 1 mol of H2O to O2. (ii) 1 mol of FeO to Fe₂O₃.

Question 15: A solution of Ni(NO₃)2 is electrolysed between platinum electrodes using a current of 5 amperes for 20 minutes. What mass of Ni is deposited at the cathode?

Question 16:Three electrolytic cells A,B,C containing solutions of ZnSO₄, AgNO3 and CuSO₄, respectively are connected in series. A steady current of 1.5 amperes was passed through them until 1.4 g of silver deposited at the cathode of cell B. How long did the current flow? What mass of copper and zinc were deposited?

Question 17: Using the standard electrode potentials given in Table 3.1, predict if the reaction between the following is feasible:

(i) Fe₃₊(aq) and I⁻(aq)
(ii) A^g+ (aq) and Cu(s)
(iii) Fe³⁺ (aq) and Br−(aq)
(iv) Ag(s) and Fe³⁺(aq)
(v) Br² (aq) and Fe²⁺ (aq).

Question 18: Predict the products of electrolysis in each of the following:

(i) An aqueous solution of AgNO₃ with silver electrodes.
(ii) An aqueous solution of AgNO₃with platinum electrodes.
(iii) A dilute solution of H₂SO₄with platinum electrodes.
(iv) An aqueous solution of CuCl₂ with platinum electrodes.

IN TEXT SOLUTION

Question 3.1: How would you determine the standard electrode potential of the systemMg2+ | Mg?
Can you store copper sulphate solutions in a zinc pot?

Question 3.3:Consult the table of standard electrode potentials and suggest three substances that an oxidise ferrous ions under suitable conditions.

Question 3.4:Calculate the potential of hydrogen electrode in contact with a solution whose pH is 10.

Question 3.5: Calculate the emf of the cell in which the following reaction takes place:

Question 3.6: The cell in which the following reactions occurs:

Question 3.7: Why does the conductivity of a solution decrease with dilution?

Question 3.8:Suggest a way to determine the Λ°m value of water.

Question 3.9:The molar conductivity of 0.025 mol L−1 methanoic acid is 46.1 S cm2 mol−1. Calculate its degree of dissociation and dissociation constant. Given λ0(H+)= 349.6 S cm2 mol–1 and λ0(HCOO–) = 54.6 S cm2 mol–1

Question 3.10:If a current of 0.5 ampere flows through a metallic wire for 2 hours, then how many electrons would flow through the wire?

Question 3.11:Suggest a list of metals that are extracted electrolytically.

Question 3.12:Consider the reaction:
Cr2O72– + 14H+ + 6e– → 2Cr3+ + 8H2O
What is the quantity of electricity in coulombs needed to reduce 1 mol of Cr2O72– ?

Question 3.14:Suggest two materials other than hydrogen that can be used as fuels in fuel cells.

Question 3.15: Explain how rusting of iron is envisaged as setting up of an electrochemical cell.

:: Chapter 4 - Chemical Kinetics ::

Question 4.11 The following results have been obtained during the kinetic studies of the reaction: 2A + B → C + D Determine the rate law and the rate constant for the reaction.

Question 4.12 The reaction between A and B is first order with respect to A and zero order with respect to B. Fill in the blanks in the following table:

Question 4.13 Calculate the half-life of a first order reaction from their rate constants given below:

(i) 200 s–1
(ii) 2 min–1
(iii) 4 years–1

Question 4.14 The half-life for radioactive decay of 14C is 5730 years. An archaeological artifact containing wood had only 80% of the 14C found in a living tree. Estimate the age of the sample.

Question 4.15 The experimental data for decomposition of N2O5 [2N2O5 → 4NO2 + O2] in gas phase at 318K are given below:

(i) Plot [N2O5] against t.
(ii) Find the half-life period for the reaction.
(iii) Draw a graph between log[N2O5] and t.
(iv) What is the rate law ?
(v) Calculate the rate constant.
(vi) Calculate the half-life period from k and compare it with (ii).

Question 4.16 The rate constant for a first order reaction is 60 s–1. How much time will it take to reduce the initial concentration of the reactant to its 1/16th value?

Question 4.17 During nuclear explosion, one of the products is 90Sr with half-life of 28.1 years. If 1μg of 90Sr was absorbed in the bones of a newly born baby instead of calcium, how much of it will remain after 10 years and 60 years if it is not lost metabolically.

Question 4.18 For a first order reaction, show that time required for 99% completion is twice the time required for the completion of 90% of reaction.

Question 4.19 A first order reaction takes 40 min for 30% decomposition. Calculate t1/2.

Question 4.20 For the decomposition of azoisopropane to hexane and nitrogen at 543 K, the following data are obtained. Calculate the rate constant.

Question 4.21 The following data were obtained during the first order thermal decomposition of SO2Cl2 at a constant volume. SO2Cl2 (g) → SO2 (g) + Cl2 (g)

Question 4.22 The rate constant for the decomposition of N2O5 at various temperatures is given below: Draw a graph between ln k and 1/T and calculate the values of A and Ea. Predict the rate constant at 30° and 50°C.

Question 4.23 The rate constant for the decomposition of hydrocarbons is 2.418 × 10–5s–1 at 546 K. If the energy of activation is 179.9 kJ/mol, what will be the value of pre-exponential factor.

Question 4.24 Consider a certain reaction A → Products with k = 2.0 × 10–2s–1. Calculate the concentration of A remaining after 100 s if the initial concentration of A is 1.0 mol L–1.

Question 4.25 Sucrose decomposes in acid solution into glucose and fructose according to the first order rate law, with t1/2 = 3.00 hours. What fraction of sample of sucrose remains after 8 hours ?

Question 4.26 The decomposition of hydrocarbon follows the equation k = ( 4.5 × 1011s–1) e-28000K/T Calculate Ea.

Question 4.27 The rate constant for the first order decomposition of H2O2 is given by the following equation: log k = 14.34 – 1.25 × 104K/T Calculate Ea for this reaction and at what temperature will its half-period be 256 minutes?

Question 4.28 The decomposition of A into product has value of k as4.5 × 103 s–1 at 10°C and energy of activation 60 kJ mol–1. At what temperature would k be 1.5 × 104s–1?

Question 4.29 The time required for 10% completion of a first order reaction at 298K is equal to that required for its 25% completion at 308K. If the value of A is 4 × 1010s–1. Calculate k at 318K and Ea.

Question 4.30 The rate of a reaction quadruples when the temperature changes from 293 K to 313 K. Calculate the energy of activation of the reaction assuming that it does not change with temperature. 

:: Chapter 5 - Surface Chemistry ::

Question 5.1 Distinguish between the meaning of the terms adsorption and absorption. Give one example of each.

Question 5.2 What is the difference between physisorption and chemisorption?

Question 5.3 Give reason why a finely divided substance is more effective as an adsorbent.

Question 5.4 What are the factors which influence the adsorption of a gas on a solid?

Question 5.5 What is an adsorption isotherm? Describe Freundlich adsorption isotherm.

Question 5.6 What do you understand by activation of adsorbent? How is it achieved?

Question 5.7 What role does adsorption play in heterogeneous catalysis?

Question 5.8 Why is adsorption always exothermic ?

Question 5.9 How are the colloidal solutions classified on the basis of physical states of the dispersed phase and dispersion medium?

Question 5.10 Discuss the effect of pressure and temperature on the adsorption of gases on solids.

Question 5.11 What are lyophilic and lyophobic sols? Give one example of each type. Why are hydrophobic sols easily coagulated ?

Question 5.12 What is the difference between multimolecular and macromolecular colloids? Give one example of each. How are associated colloids different from these two types of colloids

Question 5.13 What are enzymes ? Write in brief the mechanism of enzyme catalysis.

Question 5.14 How are colloids classified on the basis of (i) physical states of components (ii) nature of dispersion medium and (iii) interaction between dispersed phase and dispersion medium

Question 5.15 Explain what is observed

(i) when a beam of light is passed through a colloidal sol.
(ii) an electrolyte, NaCl is added to hydrated ferric oxide sol.
(iii) electric current is passed through a colloidal sol?

Question 5.16 What are emulsions? What are their different types? Give example of each type.

Question 5.17 What is demulsification? Name two demulsifiers.

Question 5.18 Action of soap is due to emulsification and micelle formation. Comment.

Question 5.19 Give four examples of heterogeneous catalysis.

Question 5.20 What do you mean by activity and selectivity of catalysts?

Question 5.21 Describe some features of catalysis by zeolites.

Question 5.22 What is shape selective catalysis?

Question 5.23 Explain the following terms:

(i) Electrophoresis
(ii) Coagulation
(iii) Dialysis
(iv) Tyndall effect.

Question 5.24 Give four uses of emulsions.

Question 5.25 What are micelles? Give an example of a micellers system.

Question 5.26 Explain the terms with suitable examples:

(i) Alcosol
(ii) Aerosol
(iii) Hydrosol.

Question 5.27 Comment on the statement that “colloid is not a substance but a state of substance”

:: Chapter 6 - General Principles and Processes of Isolation of Elements ::

Question 6.1 Copper can be extracted by hydrometallurgy but not zinc. Explain.

Question 6.2 What is the role of depressant in froth floatation process?

Question 6.3 Why is the extraction of copper from pyrites more difficult than that from its oxide ore through reduction?

Question 6.4 Explain:
(i) Zone refining
(ii) Column chromatography.

Question 6.5 Out of C and CO, which is a better reducing agent at 673 K ?

Question 6.6 Name the common elements present in the anode mud in electrolytic refining of copper. Why are they so present ?

Question 6.7 Write down the reactions taking place in different zones in the blast furnace during the extraction of iron.

Question 6.8 Write chemical reactions taking place in the extraction of zinc from zinc blende.

Question 6.9 State the role of silica in the metallurgy of copper.

Question 6.10 What is meant by the term “chromatography”?

Question 6.11 What criterion is followed for the selection of the stationary phase in chromatography?

Question 6.12 Describe a method for refining nickel.

Question 6.13 How can you separate alumina from silica in a bauxite ore associated with silica? Give equations, if any.

Question 6.14 Giving examples, differentiate between ‘roasting’ and ‘calcination’.

Question 6.15 How is ‘cast iron’ different from ‘pig iron”?

Question 6.16 Differentiate between “minerals” and “ores”.

Question 6.17 Why copper matte is put in silica lined converter?

Question 6.18 What is the role of cryolite in the metallurgy of aluminium ?

Question 6.19 How is leaching carried out in case of low grade copper ores?

Question 6.20 Why is zinc not extracted from zinc oxide through reduction using CO?

Question 6.21 The value of ΔfG0 for formation of Cr2 O3 is – 540 kJmol−1and that of Al2 O3 is – 827 kJmol−1. Is the reduction of Cr2 O3 possible with Al ?

Question 6.22 Out of C and CO, which is a better reducing agent for ZnO ?

Question 6.23 The choice of a reducing agent in a particular case depends on thermodynamic factor. How far do you agree with this statement? Support your opinion with two examples.

Question 6.24 Name the processes from which chlorine is obtained as a by-product. What will happen if an aqueous solution of NaCl is subjected to electrolysis?

Question 6.25 What is the role of graphite rod in the electrometallurgy of aluminium?

Question 6.27 Outline the principles of refining of metals by the following methods:

(i) Zone refining
(ii) Electrolytic refining
(iii) Vapour phase refining

Question 6.28 Predict conditions under which Al might be expected to reduce MgO. (Hint: See Intext question 6.4) 

:: Chapter 7 - The p-Block Elements ::

Question 7.1 Discuss the general characteristics of Group 15 elements with reference to their electronic configuration, oxidation state, atomic size, ionisation enthalpy and electronegativity.

Question 7.2 Why does the reactivity of nitrogen differ from phosphorus?

Question 7.3 Discuss the trends in chemical reactivity of group 15 elements.

Question 7.4 Why does NH3 form hydrogen bond but PH3 does not?

Question 7.5 How is nitrogen prepared in the laboratory? Write the chemical equations of the reactions involved.

Question 7.6 How is ammonia manufactured industrially?

Question 7.7 Illustrate how copper metal can give different products on reaction with HNO3.

Question 7.8 Give the resonating structures of NO2 and N2O5.

Question 7.9 The HNH angle value is higher than HPH, HAsH and HSbH angles. Why? [Hint: Can be explained on the basis of sp3 hybridisation in NH3 and only s–p bonding between hydrogen and other elements of the group].

Question 7.10 Why does R3P = O exist but R3N = O does not (R = alkyl group)?

Question 7.11 Explain why NH3 is basic while BiH3 is only feebly basic.

Question 7.12 Nitrogen exists as diatomic molecule and phosphorus as P4. Why?

Question 7.13 Write main differences between the properties of white phosphorus and red phosphorus.

Question 7.14 Why does nitrogen show catenation properties less than phosphorus?

Question 7.15 Give the disproportionation reaction of H3PO3.

Question 7.16 Can PCl5 act as an oxidising as well as a reducing agent? Justify.

Question 7.17 Justify the placement of O, S, Se, Te and Po in the same group of the periodic table in terms of electronic configuration, oxidation state and hydride formation.

Question 7.18 Why is dioxygen a gas but sulphur a solid?

Question 7.19 Knowing the electron gain enthalpy values for O → O– and O → O2– as –141 and 702 kJ mol–1 respectively, how can you account for the formation of a large number of oxides having O2– species and not O–? (Hint: Consider lattice energy factor in the formation of compounds).

Question 7.20 Which aerosols deplete ozone?

Question 7.21 Describe the manufacture of H2SO4 by contact process?

Question 7.22 How is SO2 an air pollutant?

Question 7.23 Why are halogens strong oxidising agents?

Question 7.24 Explain why fluorine forms only one oxoacid, HOF.

Question 7.25 Explain why inspite of nearly the same electronegativity, oxygen forms hydrogen bonding while chlorine does not.

Question 7.26 Write two uses of ClO2.

Question 7.27 Why are halogens coloured?

Question 7.28 Write the reactions of F2 and Cl2 with water.

Question 7.29 How can you prepare Cl2 from HCl and HCl from Cl2? Write reactions only.

Question 7.30 What inspired N. Bartlett for carrying out reaction between Xe and PtF6?

Question 7.31 What are the oxidation states of phosphorus in the following:

(i) H3PO3
(ii) PCl3
(iii) Ca3P2
(iv) Na3PO4
(v) POF3? Exercises Chemistry 208

Question 7.32 Write balanced equations for the following:

(i) NaCl is heated with sulphuric acid in the presence of MnO2.
(ii) Chlorine gas is passed into a solution of NaI in water.

Question 7.33 How are xenon fluorides XeF2, XeF4 and XeF6 obtained?

Question 7.34 With what neutral molecule is ClO– isoelectronic? Is that molecule a Lewis base?

Question 7.35 How are XeO3 and XeOF4 prepared?

Question 7.36 Arrange the following in the order of property indicated for each set:

(i) F2, Cl2, Br2, I2 - increasing bond dissociation enthalpy.
(ii) HF, HCl, HBr, HI - increasing acid strength.
(iii) NH3, PH3, AsH3, SbH3, BiH3 – increasing base strength.

Question 7.37 Which one of the following does not exist?

(i) XeOF4
(ii) NeF2
(iii) XeF2
(iv) XeF6

Question 7.38 Give the formula and describe the structure of a noble gas species which is isostructural with:

(i) ICl4 –
(ii) IBr2 –
(iii) BrO3 –

Question 7.39 Why do noble gases have comparatively large atomic sizes?

Question 7.40 List the uses of neon and argon gases.

:: Chapter 8 - The d- and f- Block Elements ::

Question 8.1 Write down the electronic configuration of:

(i) Cr3+
(iii) Cu+
(v) Co2 +
(vii) Mn2+
(ii) Pm3+
(iv) Ce4+
(vi) Lu2+
(viii) Th4+

Question 8.2 Why are Mn2+ compounds more stable than Fe2+ towards oxidation to their +3 state?

Question 8.3 Explain briefly how +2 state becomes more and more stable in the first half of the first row transition elements with increasing atomic number?

Question 8.4 To what extent do the electronic configurations decide the stability of oxidation states in the first series of the transition elements? Illustrate your answer with examples.

Question 8.5 What may be the stable oxidation state of the transition element with the following d electron configurations in the ground state of their atoms : 3d3, 3d5, 3d8 and 3d4?

Question 8.6 Name the oxometal anions of the first series of the transition metals in which the metal exhibits the oxidation state equal to its group number.

Question 8.7 What is lanthanoid contraction? What are the consequences of lanthanoid contraction?

Question 8.8 What are the characteristics of the transition elements and why are they called transition elements? Which of the d-block elements may not be regarded as the transition elements?

Question 8.9 In what way is the electronic configuration of the transition elements different from that of the non transition elements?

Question 8.10 What are the different oxidation states exhibited by the lanthanoids?

Question 8.11 Explain giving reasons:

(i) Transition metals and many of their compounds show paramagnetic behaviour.
(ii) The enthalpies of atomisation of the transition metals are high.
(iii) The transition metals generally form coloured compounds.
(iv) Transition metals and their many compounds act as good catalyst

Question 8.12 What are interstitial compounds? Why are such compounds well known for transition metals?

Question 8.13 How is the variability in oxidation states of transition metals different from that of the non transition metals? Illustrate with examples.

Question 8.14 Describe the preparation of potassium dichromate from iron chromite ore. What is the effect of increasing pH on a solution of potassium dichromate?

Question 8.15 Describe the oxidising action of potassium dichromate and write the ionic equations for its reaction with: (i) iodide (ii) iron(II) solution and (iii) H2S Exercises 235 The d- and f- Block Elements

Question 8.16 Describe the preparation of potassium permanganate. How does the acidified permanganate solution react with (i) iron(II) ions (ii) SO2 and (iii) oxalic acid? Write the ionic equations for the reactions.

Question 8.17 For M2+/M and M3+/M2+ systems the EV values for some metals are as follows: Cr2+/Cr -0.9V Cr3/Cr2+ -0.4 V Mn2+/Mn -1.2V Mn3+/Mn2+ +1.5 V Fe2+/Fe -0.4V Fe3+/Fe2+ +0.8 V Use this data to comment upon: (i) the stability of Fe3+ in acid solution as compared to that of Cr3+ or Mn3+ and (ii) the ease with which iron can be oxidised as compared to a similar process for either chromium or manganese metal.

Question 8.18 Predict which of the following will be coloured in aqueous solution? Ti3+, V3+, Cu+, Sc3+, Mn2+, Fe3+ and Co2+. Give reasons for each.

Question 8.19 Compare the stability of +2 oxidation state for the elements of the first transition series.

Question 8.20 Compare the chemistry of actinoids with that of the lanthanoids with special reference to: (i) electronic configuration (iii) oxidation state (ii) atomic and ionic sizes and (iv) chemical reactivity.

Question 8.21 How would you account for the following:

(i) Of the d4 species, Cr2+ is strongly reducing while manganese(III) is strongly oxidising.
(ii) Cobalt(II) is stable in aqueous solution but in the presence of complexing reagents it is easily oxidised.
(iii) The d1 configuration is very unstable in ions.

Question 8.22 What is meant by ‘disproportionation’? Give two examples of disproportionation reaction in aqueous solution.

Question 8.23 Which metal in the first series of transition metals exhibits +1 oxidation state most frequently and why?

Question 8.24 Calculate the number of unpaired electrons in the following gaseous ions: Mn3+, Cr3+, V3+ and Ti3+. Which one of these is the most stable in aqueous solution?

Question 8.25 Give examples and suggest reasons for the following features of the transition metal chemistry

(i) The lowest oxide of transition metal is basic, the highest is amphoteric/acidic.
(ii) A transition metal exhibits highest oxidation state in oxides and fluorides.
(iii) The highest oxidation state is exhibited in oxoanions of a metal.

Question 8.26 Indicate the steps in the preparation of:

(i) K2Cr2O7 from chromite ore.
(ii) KMnO4 from pyrolusite ore.

Question 8.27 What are alloys? Name an important alloy which contains some of the lanthanoid metals. Mention its uses.

Question 8.28 What are inner transition elements? Decide which of the following atomic numbers are the atomic numbers of the inner transition elements : 29, 59, 74, 95, 102, 104.

Question 8.29 The chemistry of the actinoid elements is not so smooth as that of the lanthanoids. Justify this statement by giving some examples from the oxidation state of these elements.

Question 8.30 Which is the last element in the series of the actinoids? Write the electronic configuration of this element. Comment on the possible oxidation state of this element.

:: Chapter 9 - Coordination Compounds ::

Question  9.1 Explain the bonding in coordination compounds in terms of Werner’s postulates.

Question  9.2 FeSO4 solution mixed with (NH4)2SO4 solution in 1:1 molar ratio gives the test of Fe2+ ion but CuSO4 solution mixed with aqueous ammonia in 1:4 molar ratio does not give the test of Cu2+ ion. Explain why?

Question  9.3 Explain with two examples each of the following: coordination entity, ligand, coordination number, coordination polyhedron, homoleptic and heteroleptic.

Question  9.4 What is meant by unidentate, didentate and ambidentate ligands? Give two examples for each.

Question  9.5 Specify the oxidation numbers of the metals in the following coordination entities:
(i) [Co(H2O)(CN)(en)2]2+
(ii) [CoBr2(en)2]+
(iii) [PtCl4]2–
(iv) K3[Fe(CN)6]
(v) [Cr(NH3)3Cl3 ]

Question  9.6 Using IUPAC norms write the formulas for the following:
(i) Tetrahydroxozincate(II)
(ii) Potassium tetrachloridopalladate(II)
(iii) Diamminedichloridoplatinum(II)
(iv) Potassium tetracyanonickelate(II)
(v) Pentaamminenitrito-O-cobalt(III)
(vi) Hexaamminecobalt(III) sulphate
(vii) Potassium tri(oxalato)chromate(III)
(viii) Hexaammineplatinum(IV)
(ix) Tetrabromidocuprate(II)
(x) Pentaamminenitrito-N-cobalt(III)

Question  9.7 Using IUPAC norms write the systematic names of the following:
(i) [Co(NH3)6]Cl3
(ii) [Pt(NH3)2Cl(NH2CH3)]Cl
(iii) [Ti(H2O)6]3+
(iv) [Co(NH3)4Cl(NO2)]Cl
(v) [Mn(H2O)6]2+
(vi) [NiCl4]2–
(vii) [Ni(NH3)6]Cl2
(viii) [Co(en)3]3+
(ix) [Ni(CO)4]

Question  9.8 List various types of isomerism possible for coordination compounds, giving an example of each.

Question  9.9 How many geometrical isomers are possible in the following coordination entities?
(i) [Cr(C2O4)3]3–
(ii) [Co(NH3)3Cl3]

Question  9.10 Draw the structures of optical isomers of:
(i) [Cr(C2O4)3]3– (ii) [PtCl2(en)2]2+
(iii) [Cr(NH3)2Cl2(en)]+ 259 Coordination Compounds

Question  9.11 Draw all the isomers (geometrical and optical) of:
(i) [CoCl2(en)2]+
(ii) [Co(NH3)Cl(en)2]2+
(iii) [Co(NH3)2Cl2(en)]+

Question  9.12 Write all the geometrical isomers of [Pt(NH3)(Br)(Cl)(py)] and how many of these will exhibit optical isomers?

Question  9.13 Aqueous copper sulphate solution (blue in colour) gives: (i) a green precipitate with aqueous potassium fluoride and (ii) a bright green solution with aqueous potassium chloride. Explain these experimental results.

Question  9.14 What is the coordination entity formed when excess of aqueous KCN is added to an aqueous solution of copper sulphate? Why is it that no precipitate of copper sulphide is obtained when H2S(g) is passed through this solution?

Question  9.15 Discuss the nature of bonding in the following coordination entities on the basis of valence bond theory:
(i) [Fe(CN)6]4–
(ii) [FeF6]3–
(iii) [Co(C2O4)3]3–
(iv) [CoF6]3–

Question  9.16 Draw figure to show the splitting of d orbitals in an octahedral crystal field.

Question  9.17 What is spectrochemical series? Explain the difference between a weak field ligand and a strong field ligand.

Question  9.18 What is crystal field splitting energy? How does the magnitude of Δo decide the actual configuration of d orbitals in a coordination entity?

Question  9.19 [Cr(NH3)6]3+ is paramagnetic while [Ni(CN)4]2– is diamagnetic. Explain why?

Question  9.20 A solution of [Ni(H2O)6]2+ is green but a solution of [Ni(CN)4]2– is colourless. Explain.

Question  9.21 [Fe(CN)6]4– and [Fe(H2O)6]2+ are of different colours in dilute solutions. Why?

Question  9.22 Discuss the nature of bonding in metal carbonyls.

Question  9.23 Give the oxidation state, d orbital occupation and coordination number of the central metal ion in the following complexes:
(i) K3[Co(C2O4)3]
(iii) (NH4)2[CoF4]
(ii) cis-[Cr(en)2Cl2]C l
(iv) [Mn(H2O)6]SO4

Question  9.24 Write down the IUPAC name for each of the following complexes and indicate the oxidation state, electronic configuration and coordination number. Also give stereochemistry and magnetic moment of the complex:
(i) K[Cr(H2O)2(C2O4)2].3H2O
(ii) [Co(NH3)5Cl-]Cl2
(iii) CrCl3(py)3
(iv) Cs[FeCl4]
(v) K4[Mn(CN)6]

Question  9.25 What is meant by stability of a coordination compound in solution? State the factors which govern stability of complexes.

Question  9.26 What is meant by the chelate effect? Give an example. 9

Question  9.27 Discuss briefly giving an example in each case the role of coordination compounds in:
(i) biological systems
(iii) analytical chemistry
(ii) medicinal chemistry
(iv) extraction/metallurgy of metals.

Question  9.28 How many ions are produced from the complex Co(NH3)6Cl2 in solution?
(i) 6
(ii) 4
(iii) 3
(iv) 2

Question  9.29 Amongst the following ions which one has the highest magnetic moment value?
(i) [Cr(H2O)6]3+
(ii) [Fe(H2O)6]2+
(iii) [Zn(H2O)6]2+

Question  9.30 The oxidation number of cobalt in K[Co(CO)4] is
(i) +1
(ii) +3
(iii) –1
(iv) –3

:: Chapter 10 - Haloalkanes and Haloarenes ::

Question 10.1 Name the following halides according to IUPAC system and classify them as alkyl, allyl, benzyl (primary, secondary, tertiary), vinyl or aryl halides:
(i) (CH3)2CHCH(Cl)CH3
(ii) CH3CH2CH(CH3)CH(C2H5)Cl
(iii) CH3CH2C(CH3)2CH2I
(iv) (CH3)3CCH2CH(Br)C6H5
(v) CH3CH(CH3)CH(Br)CH3
(vi) CH3C(C2H5)2CH2Br
(vii) CH3C(Cl)(C2H5)CH2CH3
(viii) CH3CH=C(Cl)CH2CH(CH3)2
(ix) CH3CH=CHC(Br)(CH3)2
(x) p-ClC6H4CH2CH(CH3)2
(xi) m-ClCH2C6H4CH2C(CH3)3
(xii) o-Br-C6H4CH(CH3)CH2CH3

Question 10.2 Give the IUPAC names of the following compounds:
(i) CH3CH(Cl)CH(Br)CH3
(ii) CHF2CBrClF
(iii) ClCH2C≡CCH2Br (iv) (CCl3)3CCl
(v) CH3C(p-ClC6H4)2CH(Br)CH3
(vi) (CH3)3CCH=ClC6H4I-p

Question 10.3 Write the structures of the following organic halogen compounds.
(i) 2-Chloro-3-methylpentane
(ii) p-Bromochlorobenzene
(iii) 1-Chloro-4-ethylcyclohexane
(iv) 2-(2-Chlorophenyl)-1-iodooctane
(v) Perfluorobenzene
(vi) 4-tert-Butyl-3-iodoheptane
(vii) 1-Bromo-4-sec-butyl-2-methylbenzene
(viii) 1,4-Dibromobut-2-ene 

Question 10.4 Which one of the following has the highest dipole moment?
(i) CH2Cl2
(ii) CHCl3
(iii) CCl4

Question 10.5 A hydrocarbon C5H10 does not react with chlorine in dark but gives a single monochloro compound C5H9Cl in bright sunlight. Identify the hydrocarbon.

Question 10.6 Write the isomers of the compound having formula C4H9Br.

Question 10.7 Write the equations for the preparation of 1-iodobutane from
(i) 1-butanol
(ii) 1-chlorobutane
(iii) but-1-ene.

Question 10.8 What are ambident nucleophiles? Explain with an example.

Question 10.9 Which compound in each of the following pairs will react faster in SN2 reaction with –OH?
(i) CH3Br or CH3I
(ii) (CH3)3CCl or CH3Cl

Question 10.10 Predict all the alkenes that would be formed by dehydrohalogenation of the following halides with sodium ethoxide in ethanol and identify the major alkene:
(i) 1-Bromo-1-methylcyclohexane
(ii) 2-Chloro-2-methylbutane
(iii) 2,2,3-Trimethyl-3-bromopentane.

Question 10.11 How will you bring about the following conversions?
(i) Ethanol to but-1-yne
(ii) Ethane to bromoethene
(iii) Propene to 1-nitropropane
(iv) Toluene to benzyl alcohol
(v) Propene to propyne
(vi) Ethanol to ethyl fluoride
(vii) Bromomethane to propanone
(viii) But-1-ene to but-2-ene
(ix) 1-Chlorobutane to n-octane
(x) Benzene to biphenyl.

Question 10.12 Explain why
(i) the dipole moment of chlorobenzene is lower than that of cyclohexyl chloride?
(ii) alkyl halides, though polar, are immiscible with water?
(iii) Grignard reagents should be prepared under anhydrous conditions? 1

Question 10.13 Give the uses of freon 12, DDT, carbon tetrachloride and iodoform. 1

Question 10.14 Write the structure of the major organic product in each of the following reactions:

(iii) 1-Bromobutane, 1-Bromo-2,2-dimethylpropane, 1-Bromo-2-methylbutane, 1-Bromo-3-methylbutane.

Question 10.17 Out of C6H5CH2Cl and C6H5CHClC6H5, which is more easily hydrolysed by aqueous KOH?

Question 10.18 p-Dichlorobenzene has higher m.p. and solubility than those of o- and m-isomers. Discuss.

Question 10.19 How the following conversions can be carried out?
(i) Propene to propan-1-ol
(ii) Ethanol to but-1-yne
(iii) 1-Bromopropane to 2-bromopropane
(iv) Toluene to benzyl alcohol
(v) Benzene to 4-bromonitrobenzene
(vi) Benzyl alcohol to 2-phenylethanoic acid
(vii) Ethanol to propanenitrile
(viii) Aniline to chlorobenzene
(ix) 2-Chlorobutane to 3, 4-dimethylhexane
(x) 2-Methyl-1-propene to 2-chloro-2-methylpropane
(xi) Ethyl chloride to propanoic acid
(xii) But-1-ene to n-butyliodide
(xiii) 2-Chloropropane to 1-propanol
(xiv) Isopropyl alcohol to iodoform
(xv) Chlorobenzene to p-nitrophenol
(xvi) 2-Bromopropane to 1-bromopropane
(xvii) Chloroethane to butane
(xviii) Benzene to diphenyl
(xix) tert-Butyl bromide to isobutyl bromide
(xx) Aniline to phenylisocyanide

Question 10.20 The treatment of alkyl chlorides with aqueous KOH leads to the formation of alcohols but in the presence of alcoholic KOH, alkenes are major products. Explain.

Question 10.21 Primary alkyl halide C4H9Br
(a) reacted with alcoholic KOH to give compound (b). Compound (b) is reacted with HBr to give (c) which is an isomer of (a). When (a) is reacted with sodium metal it gives compound (d), C8H18 which is different from the compound formed when n-butyl bromide is reacted with sodium. Give the structural formula of (a) and write the equations for all the reactions.

Question 10.22 What happens when
(i) n-butyl chloride is treated with alcoholic KOH
(ii) bromobenzene is treated with Mg in the presence of dry ether,
(iii) chlorobenzene is subjected to hydrolysis,
(iv) ethyl chloride is treated with aqueous KOH,
 (v) methyl bromide is treated with sodium in the presence of dry ether,
(vi) methyl chloride is treated with KCN? 

:: Chapter 11 - Alcohols, Phenols and Ethers ::

Question 11.2 Write structures of the compounds whose IUPAC names are as follows:
(i) 2-Methylbutan-2-ol
(ii) 1-Phenylpropan-2-ol
(iii) 3,5-Dimethylhexane –1, 3, 5-triol (iv) 2,3 – Diethylphenol
(v) 1 – Ethoxypropane
(vi) 2-Ethoxy-3-methylpentane
(vii) Cyclohexylmethanol
(viii) 3-Cyclohexylpentan-3-ol
(ix) Cyclopent-3-en-1-ol
(x) 3-Chloromethylpentan-1-ol. 1

Question 11..3 (i) Draw the structures of all isomeric alcohols of molecular formula C5H12O and give their IUPAC names. (ii) Classify the isomers of alcohols in question

Question 11.3 (i) as primary, secondary and tertiary alcohols.

Question 11.4 Explain why propanol has higher boiling point than that of the hydrocarbon, butane?

Question 11.5 Alcohols are comparatively more soluble in water than hydrocarbons of comparable molecular masses. Explain this fact.

Question 11.6 What is meant by hydroboration-oxidation reaction? Illustrate it with an example.

Question 11.7 Give the structures and IUPAC names of monohydric phenols of molecular formula, C7H8O.

Question 11.8 While separating a mixture of ortho and para nitrophenols by steam distillation, name the isomer which will be steam volatile. Give reason.

Question 11.9 Give the equations of reactions for the preparation of phenol from cumene.

Question 11.10 Write chemical reaction for the preparation of phenol from chlorobenzene.

Question 11.11 Write the mechanism of hydration of ethene to yield ethanol.

Question 11.12 You are given benzene, conc. H2SO4 and NaOH. Write the equations for the preparation of phenol using these reagents. 

Question 11.13 Show how will you synthesise:
(i) 1-phenylethanol from a suitable alkene .
(ii) cyclohexylmethanol using an alkyl halide by an SN2 reaction.
(iii) pentan-1-ol using a suitable alkyl halide?

Question 11.14 Give two reactions that show the acidic nature of phenol. Compare acidity of phenol with that of ethanol.

Question 11.15 Explain why is ortho nitrophenol more acidic than ortho methoxyphenol ?

Question 11.16 Explain how does the –OH group attached to a carbon of benzene ring activate it towards electrophilic substitution?

Question 11.17 Give equations of the following reactions:
(i) Oxidation of propan-1-ol with alkaline KMnO4 solution.
(ii) Bromine in CS2 with phenol.
(iii) Dilute HNO3 with phenol.
(iv) Treating phenol wih chloroform in presence of aqueous NaOH.

Question 11.18 Explain the following with an example.
(i) Kolbe’s reaction.
(ii) Reimer-Tiemann reaction.
(iii) Williamson ether synthesis.
(iv) Unsymmetrical ether.

Question 11.19 Write the mechanism of acid dehydration of ethanol to yield ethene.

Question 11.20 How are the following conversions carried out?
(i) Propene → Propan-2-ol.
(ii) Benzyl chloride → Benzyl alcohol.
(iii) Ethyl magnesium chloride → Propan-1-ol.
(iv) Methyl magnesium bromide → 2-Methylpropan-2-ol.

Question 11.21 Name the reagents used in the following reactions:
(i) Oxidation of a primary alcohol to carboxylic acid.
(ii) Oxidation of a primary alcohol to aldehyde.
(iii) Bromination of phenol to 2,4,6-tribromophenol.
(iv) Benzyl alcohol to benzoic acid. (v) Dehydration of propan-2-ol to propene.
(vi) Butan-2-one to butan-2-ol.

Question 11.22 Give reason for the higher boiling point of ethanol in comparison to methoxymethane. 

Question 11.24 Write the names of reagents and equations for the preparation of the following ethers by Williamson’s synthesis:
(i) 1-Propoxypropane
(ii) Ethoxybenzene
(iii) 2-Methoxy-2-methylpropane
(iv) 1-Methoxyethane

Question 11.25 Illustrate with examples the limitations of Williamson synthesis for the preparation of certain types of ethers.

Question 11.26 How is 1-propoxypropane synthesised from propan-1-ol? Write mechanism of this reaction.

Question 11.27 Preparation of ethers by acid dehydration of secondary or tertiary alcohols is not a suitable method. Give reason.

Question 11.28 Write the equation of the reaction of hydrogen iodide with:
(i) 1-propoxypropane (ii) methoxybenzene and (iii) benzyl ethyl ether.

Question 11.29 Explain the fact that in aryl alkyl ethers
(i) the alkoxy group activates the benzene ring towards electrophilic substitution and (ii) it directs the incoming substituents to ortho and para positions in benzene ring.

Question 11.30 Write the mechanism of the reaction of HI with methoxymethane.

Question 11.31 Write equations of the following reactions:
(i) Friedel-Crafts reaction – alkylation of anisole.
(ii) Nitration of anisole.
(iii) Bromination of anisole in ethanoic acid medium.
(iv) Friedel-Craft’s acetylation of anisole.

Question 11.32 Show how would you synthesise the following alcohols from appropriate

:: Chapter 12 - Aldehydes, Ketones and Carboxylic Acids ::

Question 12.1 What is meant by the following terms ? Give an example of the reaction in each case.

(i) Cyanohydrin
(ii) Acetal
(iii) Semicarbazone
(iv) Aldol
(v) Hemiacetal
(vi) Oxime
(vii) Ketal
(vii) Imine
(ix) 2,4-DNP-derivative
(x) Schiff’s base

Question 12.2 Name the following compounds according to IUPAC system of nomenclature:

(i) CH3CH(CH3)CH2CH2CHO
(ii) CH3CH2COCH(C2H5)CH2CH2Cl
(iii) CH3CH=CHCHO
(iv) CH3COCH2COCH3
(v) CH3CH(CH3)CH2C(CH3)2COCH3
(vi) (CH3)3CCH2COOH
(vii) OHCC6H4CHO-p

Question 12.3 Draw the structures of the following compounds.

(i) 3-Methylbutanal
(ii) p-Nitropropiophenone
(iii) p-Methylbenzaldehyde
(iv) 4-Methylpent-3-en-2-one
(v) 4-Chloropentan-2-one
(vi) 3-Bromo-4-phenylpentanoic acid
(vii) p,p’-Dihydroxybenzophenone
(viii) Hex-2-en-4-ynoic acid

Question 12.4 Write the IUPAC names of the following ketones and aldehydes. Wherever possible, give also common names.

(i) CH3CO(CH2)4CH3
(ii) CH3CH2CHBrCH2CH(CH3)CHO
(iii) CH3(CH2)5CHO
(iv) Ph-CH=CH-CHO

Question 12.5 Draw structures of the following derivatives.

(i) The 2,4-dinitrophenylhydrazone of benzaldehyde
(ii) Cyclopropanone oxime
(iii) Acetaldehydedimethylacetal
(iv) The semicarbazone of cyclobutanone
(v) The ethylene ketal of hexan-3-one
(vi) The methyl hemiacetal of formaldehyde

Question 12.6 Predict the products formed when cyclohexanecarbaldehyde reacts with following reagents.

(i) PhMgBr and then H3O+
(ii) Tollens’ reagent
(iii) Semicarbazide and weak acid
(iv) Excess ethanol and acid
(v) Zinc amalgam and dilute hydrochloric acid

Question 12.7 Which of the following compounds would undergo aldol condensation, which the Cannizzaro reaction and which neither? Write the structures of the expected products of aldol condensation and Cannizzaro reaction.

(i) Methanal
(ii) 2-Methylpentanal
(iii) Benzaldehyde
(iv) Benzophenone
(v) Cyclohexanone
(vi) 1-Phenylpropanone
(vii) Phenylacetaldehyde
(viii) Butan-1-ol
(ix) 2,2-Dimethylbutanal

Question 12.8 How will you convert ethanal into the following compounds?

(i) Butane-1,3-diol
(ii) But-2-enal
(iii) But-2-enoic acid

Question 12.9 Write structural formulas and names of four possible aldol condensation products from propanal and butanal. In each case, indicate which aldehyde acts as nucleophile and which as electrophile.

Question 12.10 An organic compound with the molecular formula C9H10O forms 2,4-DNP derivative, reduces Tollens’ reagent and undergoes Cannizzaro reaction. On vigorous oxidation, it gives 1,2-benzenedicarboxylic acid. Identify the compound.

Question 12.11 An organic compound

(A) (molecular formula C8H16O2) was hydrolysed with dilute sulphuric acid to give a carboxylic acid (B) and an alcohol (C). Oxidation of (C) with chromic acid produced (B). (C) on dehydration gives but-1-ene. Write equations for the reactions involved.

Question 12.12 Arrange the following compounds in increasing order of their property as indicated:

(i) Acetaldehyde, Acetone, Di-tert-butyl ketone, Methyl tert-butyl ketone (reactivity towards HCN)
(ii) CH3CH2CH(Br)COOH, CH3CH(Br)CH2COOH, (CH3)2CHCOOH, CH3CH2CH2COOH (acid strength)
(iii) Benzoic acid, 4-Nitrobenzoic acid, 3,4-Dinitrobenzoic acid, 4-Methoxybenzoic acid (acid strength)

Question 12.13 Give simple chemical tests to distinguish between the following pairs of compounds.

(i) Propanal and Propanone
(ii) Acetophenone and Benzophenone
(iii) Phenol and Benzoic acid
(iv) Benzoic acid and Ethyl benzoate
(v) Pentan-2-one and Pentan-3-one
(vi) Benzaldehyde and Acetophenone
(vii) Ethanal and Propanal

Question 12.14 How will you prepare the following compounds from benzene? You may use any inorganic reagent and any organic reagent having not more than one carbon atom

(i) Methyl benzoate
(ii) m-Nitrobenzoic acid
(iii) p-Nitrobenzoic acid
(iv) Phenylacetic acid
(v) p-Nitrobenzaldehyde.

Question 12.15 How will you bring about the following conversions in not more than two steps?

(i) Propanone to Propene
(ii) Benzoic acid to Benzaldehyde
(iii) Ethanol to 3-Hydroxybutanal
(iv) Benzene to m-Nitroacetophenone
(v) Benzaldehyde to Benzophenone
(vi) Bromobenzene to 1-Phenylethanol
(vii) Benzaldehyde to 3-Phenylpropan-1-ol
(viii) Benazaldehyde to α-Hydroxyphenylacetic acid
(ix) Benzoic acid to m- Nitrobenzyl alcohol

Question 12.16 Describe the following:

(i) Acetylation
(ii) Cannizzaro reaction
(iii) Cross aldol condensation
(iv) Decarboxylation

Question 12.18 Give plausible explanation for each of the following:

(i) Cyclohexanone forms cyanohydrin in good yield but 2,2,6-trimethylcyclohexanone does not.
(ii) There are two –NH2 groups in semicarbazide. However, only one is involved in the formation of semicarbazones.
(iii) During the preparation of esters from a carboxylic acid and an alcohol in the presence of an acid catalyst, the water or the ester should be removed as soon as it is formed.

Question 12.19 An organic compound contains 69.77% carbon, 11.63% hydrogen and rest oxygen. The molecular mass of the compound is 86. It does not reduce Tollens’ reagent but forms an addition compound with sodium hydrogensulphite and give positive iodoform test. On vigorous oxidation it gives ethanoic and propanoic acid. Write the possible structure of the compound.

Question 12.20 Although phenoxide ion has more number of resonating structures than carboxylate ion, carboxylic acid is a stronger acid than phenol. Why? 

:: Chapter 13 - Amines ::

Question 13.1 Write IUPAC names of the following compounds and classify them into primary, secondary and tertiary amines.

(i) (CH3)2CHNH2
(ii) CH3(CH2)2NH2
(iii) CH3NHCH(CH3)2
(iv) (CH3)3CNH2
(v) C6H5NHCH3
(vi) (CH3CH2)2NCH3
(vii) m–BrC6H4NH2

Question 13.2 Give one chemical test to distinguish between the following pairs of compounds.

(i) Methylamine and dimethylamine
(ii) Secondary and tertiary amines
(iii) Ethylamine and aniline
(iv) Aniline and benzylamine
(v) Aniline and N-methylaniline.

Question 13.3 Account for the following:

(i) pKb of aniline is more than that of methylamine.
(ii) Ethylamine is soluble in water whereas aniline is not .
(iii) Methylamine in water reacts with ferric chloride to precipitate hydrated ferric oxide.
(iv) Although amino group is o– and p– directing in aromatic electrophilic substitution reactions, aniline on nitration gives a substantial amount of m-nitroaniline.
(v) Aniline does not undergo Friedel-Crafts reaction.
(vi) Diazonium salts of aromatic amines are more stable th an those of aliphatic amines.
(vii) Gabriel phthalimide synthesis is preferred for synthesising primary amines.

Question 13.4 Arrange the following:

(i) In decreasing order of the pKb values: C2H5NH2, C6H5NHCH3, (C2H5)2NH and C6H5NH2
(ii) In increasing order of basic strength: C6H5NH2, C6H5N(CH3)2, (C2H5)2NH and CH3NH2
(iii) In increasing order of basic strength: (a) Aniline, p-nitroaniline and p-toluidine (b) C6H5NH2, C6H5NHCH3, C6H5CH2NH2.
(iv) In decreasing order of basic strength in gas phase: C2H5NH2, (C2H5)2NH, (C2H5)3N and NH3
(v) In increasing order of boiling point: C2H5OH, (CH3)2NH, C2H5NH2
(vi) In increasing order of solubility in water: C6H5NH2, (C2H5)2NH, C2H5NH2.

Question 13.5 How will you convert:

(i) Ethanoic acid into methanamine
(ii) Hexanenitrile into 1-aminopentane
(iii) Methanol to ethanoic acid
(iv) Ethanamine into methanamine
(v) Ethanoic acid into propanoic acid
(vi) Methanamine into ethanamine
(vii) Nitromethane into dimethylamine
(viii) Propanoic acid into ethanoic acid?

Question 13.6 Describe a method for the identification of primary, secondary and tertiary amines. Also write chemical equations of the reactions involved.

Question 13.7 Write short notes on the following:

(i) Carbylamine reaction
(ii) Diazotisation
(iii) Hofmann’s bromamide reaction
(iv) Coupling reaction
(v) Ammonolysis
(vi) Acetylation
(vii) Gabriel phthalimide synthesis.

Question 13.8 Accomplish the following conversions:

(i) Nitrobenzene to benzoic acid
(ii) Benzene to m-bromophenol
(iii) Benzoic acid to aniline
(iv) Aniline to 2,4,6-tribromofluorobenzene
(v) Benzyl chloride to 2-phenylethanamine
(vi) Chlorobenzene to p-chloroaniline
(vii) Aniline to p-bromoaniline
(viii) Benzamide to toluene
(ix) Aniline to benzyl alcoho

Question 13.10 An aromatic compound ‘A’ on treatment with aqueous ammonia and heating forms compound ‘B’ which on heating with Br2 and KOH forms a compound ‘C’ of molecular formula C6H7N. Write the structures and IUPAC names of compounds A, B and C.

Question 13.11 Complete the following reactions:

(i) C6H5NH2 + CHCl3 + alc.KOH →
(ii) C6H5N2Cl + H3PO2 + H2O →
(iii) ( ) C6H5NH2 + H2SO4 conc. →
(iv) C6H5N2Cl + C2H5OH →
(v) ( ) C6H5NH2 +Br2 aq →
(vi) ( ) 6 5 2 3 2 C H NH + CH CO O →
(vii) ( ) ( ) 4 2 i HBF 6 5 2 ii NaNO /Cu, C H N Cl Δ→

Question 13.12 Why cannot aromatic primary amines be prepared by Gabriel phthalimide synthesis?

Question 13.13 Write the reactions of (i) aromatic and (ii) aliphatic primary amines with nitrous acid.

Question 13.14 Give plausible explanation for each of the following:

(i) Why are amines less acidic than alcohols of comparable molecular masses?
(ii) Why do primary amines have higher boiling point than tertiary amines?
(iii) Why are aliphatic amines stronger bases than aromatic amines?

:: Chapter 14 - Biomolecules ::

Question 14.1 What are monosaccharides?

Question 14.2 What are reducing sugars?

Question 14.3 Write two main functions of carbohydrates in plants.

Question 14.4 Classify the following into monosaccharides and disaccharides. Ribose, 2-deoxyribose, maltose, galactose, fructose and lactose.

Question 14.5 What do you understand by the term glycosidic linkage?

Question 14.6 What is glycogen? How is it different from starch?

Question 14.7 What are the hydrolysis products of (i) sucrose and (ii) lactose?

Question 14.8 What is the basic structural difference between starch and cellulose?

Question 14.9 What happens when D-glucose is treated with the following reagents?

(i) HI
(ii) Bromine water
(iii) HNO3 

Question 14.10 Enumerate the reactions of D-glucose which cannot be explained by its open chain structure.

Question 14.11 What are essential and non-essential amino acids? Give two examples of each type.

Question 14.12 Define the following as related to proteins

(i) Peptide linkage
(ii) Primary structure
(iii) Denaturation.

Question 14.13 What are the common types of secondary structure of proteins?

Question 14.14 What type of bonding helps in stabilising the α-helix structure of proteins?

Question 14.15 Differentiate between globular and fibrous proteins.

Question 14.16 How do you explain the amphoteric behaviour of amino acids?

Question 14.17 What are enzymes?

Question 14.18 What is the effect of denaturation on the structure of proteins?

Question 14.19 How are vitamins classified? Name the vitamin responsible for the coagulation of blood.

Question 14.20 Why are vitamin A and vitamin C essential to us? Give their important sources.

Question 14.21 What are nucleic acids? Mention their two important functions.

Question 14.22 What is the difference between a nucleoside and a nucleotide?

Question 14.23 The two strands in DNA are not identical but are complementary. Explain.

Question 14.24 Write the important structural and functional differences between DNA and RNA.

Question 14.25 What are the different types of RNA found in the cell?

:: Chapter 15 - Polymers ::

Question 15.1 Explain the terms polymer and monomer.

Question 15.2 What are natural and synthetic polymers? Give two examples of each type.

Question 15.3 Distinguish between the terms homopolymer and copolymer and give an example of each.

Question 15.4 How do you explain the functionality of a monomer?

Question 15.5 Define the term polymerisation.

Question 15.6 Is ( NH-CHR-CO )n, a homopolymer or copolymer?

Question 15.7 In which classes, the polymers are classified on the basis of molecular forces? Question 15.8 How can you differentiate between addition and condensation polymerisation?

Question 15.9 Explain the term copolymerisation and give two examples.

Question 15.10 Write the free radical mechanism for the polymerisation of ethene.

Question 15.11 Define thermoplastics and thermosetting polymers with two examples of each.

Question 15.12 Write the monomers used for getting the following polymers.

(i) Polyvinyl chloride
(ii) Teflon
(iii) Bakelite

Question 15.13 Write the name and structure of one of the common initiators used in free radical addition polymerisation.

Question 15.14 How does the presence of double bonds in rubber molecules influence their structure and reactivity?

Question 15.15 Discuss the main purpose of vulcanisation of rubber.

Question 15.16 What are the monomeric repeating units of Nylon-6 and Nylon-6,6?

Question 15.17 Write the names and structures of the monomers of the following polymers:

(i) Buna-S
(ii) Buna-N
(iii) Dacron
(iv) Neoprene

Question 15.18 Identify the monomer in the following polymeric structures. 

Question 15.19 How is dacron obtained from ethylene glycol and terephthalic acid ?

Question 15.20 What is a biodegradable polymer ? Give an example of a biodegradable aliphatic polyester.

:: Chapter 16 - Chemistry in Everyday Life ::

Question 16.1 Why do we need to classify drugs in different ways ?

Question 16.2 Explain the term, target molecules or drug targets as used in medicinal chemistry.

Question 16.3 Name the macromolecules that are chosen as drug targets.

Question 16.4 Why should not medicines be taken without consulting doctors ?

Question 16.5 Define the term chemotherapy.

Question 16.6 Which forces are involved in holding the drugs to the active site of enzymes ?

Question 16.7 While antacids and antiallergic drugs interfere with the function of histamines, why do these not interfere with the function of each other ?

Question 16.8 Low level of noradrenaline is the cause of depression. What type of drugs are needed to cure this problem ? Name two drugs.

Question 16.9 What is meant by the term ‘broad spectrum antibiotics’ ? Explain.

Question 16.10 How do antiseptics differ from disinfectants ? Give one example of each.

Question 16.11 Why are cimetidine and ranitidine better antacids than sodium hydrogencarbonate or magnesium or aluminium hydroxide ?

Question 16.12 Name a substance which can be used as an antiseptic as well as disinfectant.\

Question 16.13 What are the main constituents of dettol ?

Question 16.14 What is tincture of iodine ? What is its use ?

Question 16.15 What are food preservatives ?

Question 16.16 Why is use of aspartame limited to cold foods and drinks ?

Question 16.17 What are artificial sweetening agents ? Give two examples.

Question 16.18 Name the sweetening agent used in the preparation of sweets for a diabetic patient.

Question 16.19 What problem arises in using alitame as artificial sweetener ?

Question 16.20 How are synthetic detergents better than soaps ?

Question 16.21 Explain the following terms with suitable examples
(i) cationic detergents
(ii) anionic detergents
(iii) non-ionic detergents.

Question 16.22 What are biodegradable and non-biodegradable detergents ? Give one example of each.

Question 16.23 Why do soaps not work in hard water ?

Question 16.24 Can you use soaps and synthetic detergents to check the hardness of water ?

Question 16.25 Explain the cleansing action of soaps. 

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CBSE warns schools to follow Only NCERT curriculum

The Central Board of Secondary Education (CBSE) has asked all schools affiliated to it to use only NCERT books from the academic session 2017-18.

The decision has been taken after recent reports of an eccentric experiment. A Class IV textbook asked students to suffocate a cat to death to understand the difference between living and non-living things. The book which is part of the curriculum of a Delhi school states "living things need air to breathe and non-living things can live without air for a few minutes." And then it asks the students to put two cats in two different boxes for the experiment.

The CBSE has now asked all its schools to follow only NCERT curriculum, both for textbooks and workbooks. The board has also asked NCERT to print adequate books before the commencement of the upcoming academic session.

Schools that are affiliated to CBSE and follow NCERT books, also use some books by private publisher, mostly for work-books and experiments. With this instruction, the practice is likely to go away from the new session, sources said.

"NCERT will be printing and supplying adequate quantity of NCERT textbooks for all classes from I to XII through its empaneled vendors and distributors," a circular sent by CBSE to all its affiliated schools read.

A link has also been created on the CBSE website where schools can fill up their demand of books.

"A review meeting was held recently with senior officials in the Ministry of HRD, and it was decided that all schools should follow NCERT curriculum as it will give uniformity to the syllabus taught in schools. Private publishers generally do not follow the guidelines while designing their syllabus, and hence instances like the recent cat-killing experiment surface," said a senior official in NCERT.

Read More....

Courtesy: DNA

CBSE Marking Scheme (Class X & XII)

Class-XII (12th) Marking Scheme 2016

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Model Answer by Candidate for Examination 2014

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• Mathematics
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• English Core
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CBSE (Class XII) Previous Year Papers Printed Books

Book Detail	ISBN	Page No	Buy Online
CBSE Chemistry Chapter wise Solved Paper 2016-2008 for Class XII by Arihant	9789350948873	438
CBSE English Core Chapter wise Solved Paper 2016-2008 for Class XII by Arihant	9789350948965	504
CBSE Physical Education Chapter wise Solved Paper 2016-2011 for Class XII by Arihant	9789350949030	232
CBSE Physics Chapter wise Solved Paper 2016-2009 for Class XII by Arihant	9789350948866	536
CBSE Computer Science Chapter wise Solved Paper 2016-2008 for Class XII by Arihant	9789350949047	448
CBSE Biology Chapter wise Solved Paper 2016-2009 for Class XII by Arihant	9789350948903	456
CBSE 10 Year Science Solved Paper for Class XII by Oswal Printers & Publishers	9789351780663	936
CBSE Economics Chapter wise Solved Paper 2016-2008 for Class XII by Arihant	9789350948927	424
CBSE Accountancy Chapter wise Solved Paper 2016-2008 for Class XII by Arihant	9789350948941	486

CBSE (Class X) Previous Year Papers Printed Books

Book Detail	ISBN	Page No	Buy Online
CBSE Mathematics Sample Question Paper for Class X by Oswaal Books	9789386177889	119
CBSE Science Sample Question Paper for Class X by Oswaal Books	9789386177865	111
CBSE English Communicative, Hindi, Science, Social Science & Maths Sample Question Paper for Class X by Oswaal Books	9788184818475	504
CBSE English Communicative Sample Question Paper for Class X by Arihant	9789351767312	104
CBSE Social Science Sample Question Paper for Class X by Arihant	9789351767305	78
CBSE Mathematics Sample Question Paper for Class X by Oswaal Books	9789386177278	220
CBSE Social Science Sample Question Paper for Class X by Oswaal Books	9789386177872	104
CBSE Science Sample Question Paper for Class X by Arihant	9789351767282	139
CBSE Mathematics Sample Question Paper for Class X by Arihant	9789351767299	152
CBSE Hindi Sample Question Paper for Class X by Arihant	9789351767350	120
CBSE Information Technology Sample Question Paper for Class X by Arihant	9789351767336	112

   

NCERT Physics Question Paper (Class - 12)

:: Chapter 1 - Electric Charges And Fields ::

List of Physics Formula

EXERCISE

Question 1: What is the force between two small charged spheres having charges of 2 × 10−7 C and 3 × 10−7 C placed 30 cm apart in air?

Question 2: The electrostatic force on a small sphere of charge 0.4 µC due to another small sphere of charge − 0.8 µC in air is 0.2 N. (a) What is the distance between the two spheres? (b) What is the force on the second sphere due to the first?

Question 3: Check that the ratio ke2/G memp is dimensionless. Look up a Table of physical constants and determine the value of this ratio. What does the ratio signify?

Question 4: (a) Explain the meaning of the statement ‘electric charge of a body is quantised’.
(b) Why can one ignore quantisation of electric charge when dealing with macroscopic i.e., large scale charges?

Question 5: When a glass rod is rubbed with a silk cloth, charges appear on both. A similar phenomenon is observed with many other pairs of bodies. Explain how this observation is consistent with the law of conservation of charge.

Question 6: Four point charges qA = 2 µC, qB = −5 µC, qC = 2 µC, and qD = −5 µC are located at the corners of a square ABCD of side 10 cm. What is the force on a charge of 1 µC placed at the centre of the square

Question 7: (a) An electrostatic field line is a continuous curve. That is, a field line cannot have sudden breaks. Why not? (b) Explain why two field lines never cross each other at any point?

Question 8: Two point charges qA = 3 µC and qB = −3 µC are located 20 cm apart in vacuum. (i) What is the electric field at the midpoint O of the line AB joining the two charges? (ii) If a negative test charge of magnitude 1.5 × 10−9 C is placed at this point, what is the force experienced by the test charge?

Question 9: A system has two charges qA = 2.5 × 10−7 C and qB = −2.5 × 10−7 C located at points A: (0, 0, − 15 cm) and B: (0, 0, + 15 cm), respectively. What are the total charge and electric dipole moment of the system?

:: Chapter 2 - Electrostatic Potential and Capacitance ::

EXERCISE

Question 2.1 Two charges 5 × 10–8 C and –3 × 10–8 C are located 16 cm apart. At what point (s) on the line joining the two charges is the electric potential zero? Take the potential at infinity to be zero.

Question 2.2 A regular hexagon of side 10 cm has a charge 5 μC at each of its vertices. Calculate the potential at the centre of the hexagon.

Question 2.3 Two charges 2 μC and –2 μC are placed at points A and B 6 cm apart.
(a) Identify an equipotential surface of the system.
(b) What is the direction of the electric field at every point on this surface?

Question 2.4 A spherical conductor of radius 12 cm has a charge of 1.6 × 10–7C distributed uniformly on its surface. What is the electric field
(a) inside the sphere
(b) just outside the sphere
(c) at a point 18 cm from the centre of the sphere?

Question 2.5 A parallel plate capacitor with air between the plates has a capacitance of 8 pF (1pF = 10–12 F). What will be the capacitance if the distance between the plates is reduced by half, and the space between them is filled with a substance of dielectric constant 6?

Question 2.6 Three capacitors each of capacitance 9 pF are connected in series.
(a) What is the total capacitance of the combination?
(b) What is the potential difference across each capacitor if the combination is connected to a 120 V supply?

Question 2.7 Three capacitors of capacitances 2 pF, 3 pF and 4 pF are connected in parallel.
(a) What is the total capacitance of the combination? (b) Determine the charge on each capacitor if the combination is connected to a 100 V supply.

Question 2.8 In a parallel plate capacitor with air between the plates, each plate has an area of 6 × 10–3 m2 and the distance between the plates is 3 mm. Calculate the capacitance of the capacitor. If this capacitor is connected to a 100 V supply, what is the charge on each plate of the capacitor?

Question 2.9 Explain what would happen if in the capacitor given in Exercise2.8, a 3 mm thick mica sheet (of dielectric constant = 6) were inserted between the plates,
(a) while the voltage supply remained connected.
(b) after the supply was disconnected.

Question 2.10 A 12pF capacitor is connected to a 50V battery. How much electrostatic energy is stored in the capacitor?

Question 2.11 A 600pF capacitor is charged by a 200V supply. It is then disconnected from the supply and is connected to another uncharged 600 pF capacitor. How much electrostatic energy is lost in the process?

ADDITIONAL EXERCISES QUESTIONS

Question 2.12 A charge of 8 mC is located at the origin. Calculate the work done in taking a small charge of –2 × 10–9 C from a point P (0, 0, 3 cm) to a point Q (0, 4 cm, 0), via a point R (0, 6 cm, 9 cm).

Question 2.13 A cube of side b has a charge q at each of its vertices. Determine the potential and electric field due to this charge array at the centre of the cube.

Question 2.14 Two tiny spheres carrying charges 1.5 μC and 2.5 μC are located 30 cm apart. Find the potential and electric field: (a) at the mid-point of the line joining the two charges, and (b) at a point 10 cm from this midpoint in a plane normal to the line and passing through the mid-point.

Question 2.15 A spherical conducting shell of inner radius r1 and outer radius r2 has a charge Q.
(a) A charge q is placed at the centre of the shell. What is the surface charge density on the inner and outer surfaces of the shell?
(b) Is the electric field inside a cavity (with no charge) zero, even if the shell is not spherical, but has any irregular shape? Explain.

Question 2.16 (a) Show that the normal component of electrostatic field has a discontinuity from one side of a charged surface to another given by 2 1 0 ( ) ˆ σ ε E − E n = where ˆn is a unit vector normal to the surface at a point and σ is the surface charge density at that point. (The direction of ˆn is from side 1 to side 2.) Hence show that just outside a conductor, the electric field is σ ˆn /ε0.
(b) Show that the tangential component of electrostatic field is continuous from one side of a charged surface to another. [Hint: For (a), use Gauss’s law. For, (b) use the fact that work done by electrostatic field on a closed loop is zero.]

Question 2.17 A long charged cylinder of linear charged density λ is surrounded by a hollow co-axial conducting cylinder. What is the electric field in the space between the two cylinders?

Question 2.18 In a hydrogen atom, the electron and proton are bound at a distance of about 0.53 Å:
(a) Estimate the potential energy of the system in eV, taking the zero of the potential energy at infinite separation of the electron from proton.
(b) What is the minimum work required to free the electron, given that its kinetic energy in the orbit is half the magnitude of potential energy obtained in (a)?
(c) What are the answers to (a) and (b) above if the zero of potential energy is taken at 1.06 Å separation?

Question 2.19 If one of the two electrons of a H2 molecule is removed, we get a hydrogen molecular ion H+ 2.In the ground state of an H+ 2, the two protons are separated by roughly 1.5 Å, and the electron is roughly 1 Å from each proton. Determine the potential energy of the system. Specify your choice of the zero of potential energy.

Question 2.20 Two charged conducting spheres of radii a and b are connected to each other by a wire. What is the ratio of electric fields at the surfaces of the two spheres? Use the result obtained to explain why charge density on the sharp and pointed ends of a conductor is higher than on its flatter portions.

Question 2.21 Two charges –q and +q are located at points (0, 0, –a) and (0, 0, a), respectively.
(a) What is the electrostatic potential at the points (0, 0, z) and (x, y, 0) ?
(b) Obtain the dependence of potential on the distance r of a point from the origin when r/a >> 1.
(c) How much work is done in moving a small test charge from the point (5,0,0) to (–7,0,0) along the x-axis? Does the answer change if the path of the test charge between the same points is not along the x-axis?

Question 2.22 Figure2.34 shows a charge array known as an electric quadrupole. For a point on the axis of the quadrupole, obtain the dependence of potential on r for r/a >> 1, and contrast your results with that due to an electric dipole, and an electric monopole (i.e., a single charge).

Question 2.23 An electrical technician requires a capacitance of 2 μF in a circuit across a potential difference of 1 kV. A large number of 1 μF capacitors are available to him each of which can withstand a potential difference of not more than 400 V. Suggest a possible arrangement that requires the minimum number of capacitors.

Question 2.24 What is the area of the plates of a 2 F parallel plate capacitor, given that the separation between the plates is 0.5 cm? [You will realise from your answer why ordinary capacitors are in the range of μF or less. However, electrolytic capacitors do have a much larger capacitance (0.1 F) because of very minute separation between the conductors.]

Question 2.25 Obtain the equivalent capacitance of the network in Fig. 2.35. For a 300 V supply, determine the charge and voltage across each capacitor.

Question 2.26 The plates of a parallel plate capacitor have an area of 90 cm2 each and are separated by 2.5 mm. The capacitor is charged by connecting it to a 400 V supply.
(a) How much electrostatic energy is stored by the capacitor?
(b) View this energy as stored in the electrostatic field between the plates, and obtain the energy per unit volume u. Hence arrive at a relation between u and the magnitude of electric field E between the plates.

Question 2.27 A 4 μF capacitor is charged by a 200 V supply. It is then disconnected from the supply, and is connected to another uncharged 2 μF capacitor. How much electrostatic energy of the first capacitor is lost in the form of heat and electromagnetic radiation?

Question 2.28 Show that the force on each plate of a parallel plate capacitor has a magnitude equal to (½) QE, where Q is the charge on the capacitor, and E is the magnitude of electric field between the plates. Explain the origin of the factor ½.

Question 2.29 A spherical capacitor consists of two concentric spherical conductors, held in position by suitable insulating supports (Fig2.36). Show that the capacitance of a spherical capacitor is given by 0 1 2 1 2 4 – r r C r r πε = where r1 and r2 are the radii of outer and inner spheres, respectively.

Question 2.30 A spherical capacitor has an inner sphere of radius 12 cm and an outer sphere of radius 13 cm. The outer sphere is earthed and the inner sphere is given a charge of 2.5 μC. The space between the concentric spheres is filled with a liquid of dielectric constant 32.
(a) Determine the capacitance of the capacitor.
(b) What is the potential of the inner sphere?
(c) Compare the capacitance of this capacitor with that of an isolated sphere of radius 12 cm. Explain why the latter is much smaller.s the magnitude of electrostatic force between them exactly given by Q1 Q2/4πε0r 2, where r is the distance between their centres?
(b) If Coulomb’s law involved 1/r3 dependence (instead of 1/r2), would Gauss’s law be still true ?
(c) A small test charge is released at rest at a point in an electrostatic field configuration. Will it travel along the field line passing through that point?
(d) What is the work done by the field of a nucleus in a complete circular orbit of the electron? What if the orbit is elliptical?
(e) We know that electric field is discontinuous across the surface of a charged conductor. Is electric potential also discontinuous there?
(f ) What meaning would you give to the capacitance of a single conductor?
(g) Guess a possible reason why water has a much greater dielectric constant (= 80) than say, mica (= 6).

Question 2.32 A cylindrical capacitor has two co-axial cylinders of length 15 cm and radii 1.5 cm and 1.4 cm. The outer cylinder is earthed and the inner cylinder is given a charge of 3.5 μC. Determine the capacitance of the system and the potential of the inner cylinder. Neglect end effects (i.e., bending of field lines at the ends).

Question 2.33 A parallel plate capacitor is to be designed with a voltage rating 1 kV, using a material of dielectric constant 3 and dielectric strength about 107 Vm–1. (Dielectric strength is the maximum electric field a material can tolerate without breakdown, i.e., without starting to conduct electricity through partial ionisation.) For safety, we should like the field never to exceed, say 10% of the dielectric strength. What minimum area of the plates is required to have a capacitance of 50 pF?

Question 2.34 Describe schematically the equipotential surfaces corresponding to
(a) a constant electric field in the z-direction,
(b) a field that uniformly increases in magnitude but remains in a constant (say, z) direction,
(c) a single positive charge at the origin, and
(d) a uniform grid consisting of long equally spaced parallel charged wires in a plane .

Question 2.35 In a Van de Graaff type generator a spherical metal shell is to be a 15 × 106 V electrode. The dielectric strength of the gas surrounding the electrode is 5 × 107 Vm–1. What is the minimum radius of the spherical shell required? (You will learn from this exercise why one cannot build an electrostatic generator using a very small shell which requires a small charge to acquire a high potential.)

Question 2.36 A small sphere of radius r1 and charge q1 is enclosed by a spherical shell of radius r2 and charge q2. Show that if q1 is positive, charge will necessarily flow from the sphere to the shell (when the two are connected by a wire) no matter what the charge q2 on the shell is.

Question 2.37 Answer the following:
(a) The top of the atmosphere is at about 400 kV with respect to the surface of the earth, corresponding to an electric field that decreases with altitude. Near the surface of the earth, the field is about 100 Vm–1. Why then do we not get an electric shock as we step out of our house into the open? (Assume the house to be a steel cage so there is no field inside!)
(b) A man fixes outside his house one evening a two metre high insulating slab carrying on its top a large aluminium sheet of area 1m 2.Will he get an electric shock if he touches the metal sheet next morning?
(c) The discharging current in the atmosphere due to the small conductivity of air is known to be 1800 A on an average over the globe. Why then does the atmosphere not discharge itself completely in due course and become electrically neutral? In other words, what keeps the atmosphere charged?
(d) What are the forms of energy into which the electrical energy of the atmosphere is dissipated during a lightning? (Hint: The earth has an electric field of about 100 Vm–1 at its surface in the downward direction, corresponding to a surface charge density = –10–9 C m– 2. Due to the slight conductivity of the atmosphere up to about 50 km (beyond which it is good conductor), about + 1800 C is pumped every second into the earth as a whole. The earth, however, does not get discharged since thunderstorms and lightning occurring continually all over the globe pump an equal amount of negative charge on the earth.)

:: Chapter 3 - Current Electricity ::

Question 3.1 The storage battery of a car has an emf of 12 V. If the internal resistance of the battery is 0.4 Ω, what is the maximum current that can be drawn from the battery?

Question 3.2 A battery of emf 10 V and internal resistance 3 Ω is connected to a resistor. If the current in the circuit is 0.5 A, what is the resistance of the resistor? What is the terminal voltage of the battery when the circuit is closed?

Question 3.3 (a) Three resistors 1 Ω, 2 Ω, and 3 Ω are combined in series. What is the total resistance of the combination? (b) If the combination is connected to a battery of emf 12 V and negligible internal resistance, obtain the potential drop across each resistor.

Question 3.4 (a) Three resistors 2 Ω, 4 Ω and 5 Ω are combined in parallel. What is the total resistance of the combination?
(b) If the combination is connected to a battery of emf 20 V and negligible internal resistance, determine the current through each resistor, and the total current drawn from the battery.

Question 3.5 At room temperature (27.0 °C) the resistance of a heating element is 100 Ω. What is the temperature of the element if the resistance is found to be 117 Ω, given that the temperature coefficient of the material of the resistor is 1.70 × 10–4 °C–1.

Question 3.6 A negligibly small current is passed through a wire of length 15 m and uniform cross-section 6.0 × 10–7 m2, and its resistance is measured to be 5.0 Ω. What is the resistivity of the material at the temperature of the experiment?

Question 3.7 A silver wire has a resistance of 2.1 Ω at 27.5 °C, and a resistance of 2.7 Ω at 100 °C. Determine the temperature coefficient of resistivity of silver.

Question 3.8 A heating element using nichrome connected to a 230 V supply draws an initial current of 3.2 A which settles after a few seconds to a steady value of 2.8 A. What is the steady temperature of the heating element if the room temperature is 27.0 °C? Temperature coefficient of resistance of nichrome averaged over the temperature range involved is 1.70 × 10–4 °C–1.

Question 3.9 Determine the current in each branch of the network shown in Fig. 3.30:

Question 3.10 (a) In a metre bridge [Fig. 3.7], the balance point is found to be at 39.5 cm from the end A, when the resistor Y is of 12.5 Ω. Determine the resistance of X. Why are the connections between resistors in a Wheatstone or meter bridge made of thick copper strips?
(b) Determine the balance point of the bridge above if X and Y are interchanged.
(c) What happens if the galvanometer and cell are interchanged at the balance point of the bridge? Would the galvanometer show any current?

Question 3.11 A storage battery of emf 8.0 V and internal resistance 0.5 Ω is being charged by a 120 V dc supply using a series resistor of 15.5 Ω. What is the terminal voltage of the battery during charging? What is the purpose of having a series resistor in the charging circuit?

Question 3.12 In a potentiometer arrangement, a cell of emf 1.25 V gives a balance point at 35.0 cm length of the wire. If the cell is replaced by another cell and the balance point shifts to 6 3.0 cm, what is the emf of the second cell?

Question 3. 13 The number density of free electrons in a copper conductor estimated in Example 3.1 is 8.5 × 1028 m–3. How long does an electron take to drift from one end of a wire 3.0 m long to its other end? The area of cross-section of the wire is 2.0 × 10–6 m2 and it is carrying a current of 3.0 A.

ADDITIONAL EXERCISES QUESTIONS

Question 3. 14 The earth’s surface has a negative surface charge density of 10–9 C m–2. The potential difference of 400 kV between the top of the atmosphere and the surface results (due to the low conductivity of the lower atmosphere) in a current of only 1800 A over the entire globe. If there were no mechanism of sustaining atmospheric electric field, how much time (roughly) would be required to neutralise the earth’s surface? (This never happens in practice because there is a mechanism to replenish electric charges, namely the continual thunderstorms and lightning in different parts of the globe). (Radius of earth = 6.37 × 106 m.)

Question 3.15 (a) Six lead-acid type of secondary cells each of emf 2.0 V and internal resistance 0.015 Ω are joined in series to provide a supply to a resistance of 8.5 Ω. What are the current drawn from the supply and its terminal voltage?
(b) A secondary cell after long use has an emf of 1.9 V and a large internal resistance of 380 Ω. What maximum current can be drawn from the cell? Could the cell drive the starting motor of a car?

Question 3.16 Two wires of equal length, one of aluminium and the other of copper have the same resistance. Which of the two wires is lighter? Hence explain why aluminium wires are preferred for overhead power cables. (ρAl = 2.63 × 10–8 Ω m, ρCu = 1.72 × 10–8 Ω m, Relative density of Al = 2.7, of Cu = 8.9.)

Question 3.17 What conclusion can you draw from the following observations on a resistor made of alloy manganin?

Question 3.18 Answer the following questions:
(a) A steady current flows in a metallic conductor of non-uniform cross-section. Which of these quantities is constant along the conductor: current, current density, electric field, drift speed?
(b) Is Ohm’s law universally applicable for all conducting elements? If not, give examples of elements which do not obey Ohm’s law.
(c) A low voltage supply from which one needs high currents must have very low internal resistance. Why?
(d) A high tension (HT) supply of, say, 6 kV must have a very large internal resistance. Why?

Question 3.19 Choose the correct alternative:
(a) Alloys of metals usually have (greater/less) resistivity than that of their constituent metals.
(b) Alloys usually have much (lower/higher) temperature coefficients of resistance than pure metals.
(c) The resistivity of the alloy manganin is nearly independent of/ increases rapidly with increase of temperature.
(d) The resistivity of a typical insulator (e.g., amber) is greater than that of a metal by a factor of the order of (1022/103).

Question 3.20 (a) Given n resistors each of resistance R, how will you combine them to get the
(i) maximum
(ii) minimum effective resistance? What is the ratio of the maximum to minimum resistance? (b) Given the resistances of 1 Ω, 2 Ω, 3 Ω, how will be combine them to get an equivalent resistance of (i) (11/3) Ω (ii) (11/5) Ω,
(iii) 6 Ω, (iv) (6/11) Ω? (c) Determine the equivalent resistance of networks shown in Fig. 3.31.

Question 3.22 Figure 3.33 shows a potentiometer with a cell of 2.0 V and internal resistance 0.40 Ω maintaining a potential drop across the resistor wire AB. A standard cell which maintains a constant emf of 1.02 V (for very moderate currents upto a few mA) gives a balance point at 67.3 cm length of the wire. To ensure very low currents drawn from the standard cell, a very high resistance of 600 kΩ is put in series with it, which is shorted close to the balance point. The standard cell is then replaced by a cell of unknown emf ε and the balance point found similarly, turns out to be at 82.3 cm length of the wire.
(c) Is the balance point affected by this high resistance?
(d) Is the balance point affected by the internal resistance of the driver cell?
(e) Would the method work in the above situation if the driver cell of the potentiometer had an emf of 1.0V instead of 2.0V? (f ) Would the circuit work well for determining an extremely small emf, say of the order of a few mV (such as the typical emf of a thermo-couple)? If not, how will you modify the circuit?

Question 3.23 Figure 3.34 shows a potentiometer circuit for comparison of two resistances. The balance point with a standard resistor R = 10.0 Ω is found to be 58.3 cm, while that with the unknown resistance X is 68.5 cm. Determine the value of X. What might you do if you failed to find a balance point with the given cell of emf ε ?

Question 3.24 Figure 3.35 shows a 2.0 V potentiometer used for the determination of internal resistance of a 1.5 V cell. The balance point of the cell in open circuit is 76.3 cm. When a resistor of 9.5 Ω is used in the external circuit of the cell, the balance point shifts to 64.8 cm length of the potentiometer wire. Determine the internal resistance of the cell.
 

:: Chapter 4 - Moving Charges and Magnetism ::

Question 4.1 A circular coil of wire consisting of 100 turns, each of radius 8.0 cm carries a current of 0.40 A. What is the magnitude of the magnetic field B at the centre of the coil?

Question 4.2 A long straight wire carries a current of 35 A. What is the magnitude of the field B at a point 20 cm from the wire?

Question 4.3 A long straight wire in the horizontal plane carries a current of 50 A in north to south direction. Give the magnitude and direction of B at a point 2.5 m east of the wire.

Question 4.4 A horizontal overhead power line carries a current of 90 A in east to west direction. What is the magnitude and direction of the magnetic field due to the current 1.5 m below the line?

Question 4.5 What is the magnitude of magnetic force per unit length ?

Question 4.6 A 3.0 cm wire carrying a current of 10 A is placed inside a solenoid perpendicular to its axis. The magnetic field inside the solenoid is given to be 0.27 T. What is the magnetic force on the wire?

Question 4.7 Two long and parallel straight wires A and B carrying currents of 8.0 A and 5.0 A in the same direction are separated by a distance of 4.0 cm. Estimate the force on a 10 cm section of wire A.

Question 4.8 A closely wound solenoid 80 cm long has 5 layers of windings of 400 turns each. The diameter of the solenoid is 1.8 cm. If the current carried is 8.0 A, estimate the magnitude of B inside the solenoid near its centre.

Question 4.9 A square coil of side 10 cm consists of 20 turns and carries a current of 12 A. The coil is suspended vertically and the normal to the plane of the coil makes an angle of 30º with the direction of a uniform horizontal magnetic field of magnitude 0.80 T. What is the magnitude of torque experienced by the coil?

Question 4.10 Two moving coil meters, M1 and M2 have the following particulars: R1 = 10 Ω, N1 = 30, A1 = 3.6 × 10–3 m2, B1 = 0.25 T R2 = 14 Ω, N2 = 42, A2 = 1.8 × 10–3 m2, B2 = 0.50 T (The spring constants are identical for the two meters). Determine the ratio of (a) current sensitivity and (b) voltage sensitivity of M2 and M1.

Question 4.11 In a chamber, a uniform magnetic field of 6.5 G (1 G = 10–4 T) is maintained. An electron is shot into the field
with a speed of 4.8 × 106 m s–1 normal to the field. Explain why the path of the electron is a circle. Determine the radius of the circular orbit. (e = 1.6 × 10–19 C, me = 9.1×10–31 kg)

Question 4.12 In Exercise 4.11 obtain the frequency of revolution of the electron in its circular orbit. Does the answer depend on the speed of the electron? Explain.

Question 4.13 (a) A circular coil of 30 turns and radius 8.0 cm carrying a current of 6.0 A is suspended vertically in a uniform horizontal magnetic field of magnitude 1.0 T. The field lines make an angle of 60º with the normal of the coil. Calculate the magnitude of the counter torque that must be applied to prevent the coil from turning.
(b) Would your answer change, if the circular coil in
(a) were replaced by a planar coil of some irregular shape that encloses the same area? (All other particulars are also unaltered.)

ADDITIONAL EXERCISES QUESTION

Question 4.14 Two concentric circular coils X and Y of radii 16 cm and 10 cm, respectively, lie in the same vertical plane containing the north to south direction. Coil X has 20 turns and carries a current of 16 A; coil Y has 25 turns and carries a current of 18 A. The sense of the current in X is anticlockwise, and clockwise in Y, for an observer looking at the coils facing west. Give the magnitude and direction of the net magnetic field due to the coils at their centre .

Question 4.15 A magnetic field of 100 G (1 G = 10–4 T) is required which is uniform in a region of linear dimension about 10 cm and area of cross-section about 10–3 m2. The maximum current-carrying capacity of a given coil of wire is 15 A and the number of turns per unit length that can be wound round a core is at most 1000 turns m–1. Suggest some appropriate design particulars of a solenoid for the required purpose. Assume the core is not ferromagnetic.

Question 4.16 For a circular coil of radius R and N turns carrying current I, the magnitude of the magnetic field at a point on its axis at a distance x from its centre is given by, ( ) 2 0 2 2 3/2 2 IR N B x R μ = +
(a) Show that this reduces to the familiar result for field at the centre of the coil.
(b) Consider two parallel co-axial circular coils of equal radius R, and number of turns N, carrying equal currents in the same direction, and separated by a distance R. Show that the field on the axis around the mid-point between the coils is uniform over a distance that is small as compared to R, and is given by, 0.72 0 NI B R μ = , approximately. [Such an arrangement to produce a nearly uniform magnetic field over a small region is known as Helmholtz coils.]

Question 4.17 A toroid has a core (non-ferromagnetic) of inner radius 25 cm and outer radius 26 cm, around which 3500 turns of a wire are wound. If the current in the wire is 11 A, what is the magnetic field (a) outside the toroid, (b) inside the core of the toroid, and (c) in the empty space surrounded by the toroid.

Question 4.18 Answer the following questions:
(a) A magnetic field that varies in magnitude from point to point but has a constant direction (east to west) is set up in a chamber. A charged particle enters the chamber and travels undeflectedalong a straight path with constant speed. What can you say about the initial velocity of the particle?
(b) A charged particle enters an environment of a strong and non-uniform magnetic field varying from point to point both in magnitude and direction, and comes out of it following a complicated trajectory. Would its final speed equal the initial speed if it suffered no collisions with the environment?
(c) An electron travelling west to east enters a chamber having a uniform electrostatic field in north to south direction. Specify the direction in which a uniform magnetic field should be set up to prevent the electron from deflecting from its straight line path.

Question 4.19 An electron emitted by a heated cathode and accelerated through a potential difference of 2.0 kV, enters a region with uniform magnetic field of 0.15 T. Determine the trajectory of the electron if the field
(a) is transverse to its initial velocity,
(b) makes an angle of 30º with the initial velocity.

Question 4.20 A magnetic field set up using Helmholtz coils (described in Exercise 4.16) is uniform in a small region and has a magnitude of 0.75 T. In the same region, a uniform electrostatic field is maintained in a direction normal to the common axis of the coils. A narrow beam of (single species) charged particles all accelerated through 15 kV enters this region in a direction perpendicular to both the axis of the coils and the electrostatic field. If the beam remains undeflected when the electrostatic field is 9.0 × 10–5 V m–1, make a simple guess as to what the beam contains. Why is the answer not unique?

Question 4.21 A straight horizontal conducting rod of length 0.45 m and mass 60 g is suspended by two vertical wires at its ends. A current of 5.0 A is set up in the rod through the wires.
(a) What magnetic field should be set up normal to the conductor in order that the tension in the wires is zero?
(b) What will be the total tension in the wires if the direction of current is reversed keeping the magnetic field same as before? (Ignore the mass of the wires.) g = 9.8 m s–2.

Question 4.22 The wires which connect the battery of an automobile to its starting motor carry a current of 300 A (for a short time). What is the force per unit length between the wires if they are 70 cm long and 1.5 cm apart? Is the force attractive or repulsive?

Question 4.23 A uniform magnetic field of 1.5 T exists in a cylindrical region of radius10.0 cm, its direction parallel to the axis along east to west. A wire carrying current of 7.0 A in the north to south direction passes through this region. What is the magnitude and direction of the force on the wire if,
(a) the wire intersects the axis,
(b) the wire is turned from N-S to northeast-northwest direction,
(c) the wire in the N-S direction is lowered from the axis by a distance of 6.0 cm?

Question 4.24 A uniform magnetic field of 3000 G is established along the positive z-direction. A rectangular loop of sides 10 cm and 5 cm carries a current of 12 A. What is the torque on the loop in the different cases shown in Fig. 4.28? What is the force on each case? Which case corresponds to stable equilibrium?

Question 4.25 A circular coil of 20 turns and radius 10 cm is placed in a uniform magnetic field of 0.10 T normal to the plane of the coil. If the current in the coil is 5.0 A, what is the
(a) total torque on the coil,
(b) total force on the coil,
(c) average force on each electron in the coil due to the magnetic field? (The coil is made of copper wire of cross-sectional area 10–5 m2, and the free electron density in copper is given to be about 1029 m–3.)

Question 4.26 A solenoid 60 cm long and of radius 4.0 cm has 3 layers of windings of 300 turns each. A 2.0 cm long wire of mass 2.5 g lies inside the solenoid (near its centre) normal to its axis; both the wire and the axis of the solenoid are in the horizontal plane. The wire is connected through two leads parallel to the axis of the solenoid to an external battery which supplies a current of 6.0 A in the wire. What value of current (with appropriate sense of circulation) in the windings of the solenoid can support the weight of the wire? g = 9.8 m s–2.

Question 4.27 A galvanometer coil has a resistance of 12 Ω and the metre shows full scale deflection for a current of 3 mA. How will you convert the metre into a voltmeter of range 0 to 18 V?

Question 4.28 A galvanometer coil has a resistance of 15 Ω and the metre shows full scale deflection for a current of 4 mA. How will you convert the metre into an ammeter of range 0 to 6 A? 

:: Chapter 5 - Magnetism and Matter ::

Question 5.1 Answer the following questions regarding earth’s magnetism:
(a) A vector needs three quantities for its specification. Name the three independent quantities conventionally used to specify the earth’s magnetic field.
(b) The angle of dip at a location in southern India is about 18º. Would you expect a greater or smaller dip angle in Britain?
(c) If you made a map of magnetic field lines at Melbourne in Australia, would the lines seem to go into the ground or come out of the ground?
(d) In which direction would a compass free to move in the vertical plane point to, if located right on the geomagnetic north or south pole?
(e) The earth’s field, it is claimed, roughly approximates the field due to a dipole of magnetic moment 8 × 1022 J T–1 located at its centre. Check the order of magnitude of this number in some way.
(f ) Geologists claim that besides the main magnetic N-S poles, there are several local poles on the earth’s surface oriented in different directions. How is such a thing possible at all?

Question 5.2 Answer the following questions:
(a) The earth’s magnetic field varies from point to point in space. Does it also change with time? If so, on what time scale does it change appreciably?
(b) The earth’s core is known to contain iron. Yet geologists do not regard this as a source of the earth’s magnetism. Why?
(c) The charged currents in the outer conducting regions of the earth’s core are thought to be responsible for earth’s magnetism. What might be the ‘battery’ (i.e., the source of energy) to sustain these currents?
(d) The earth may have even reversed the direction of its field several times during its history of 4 to 5 billion years. How can geologists know about the earth’s field in such distant past?
(e) The earth’s field departs from its dipole shape substantially at large distances (greater than about 30,000 km). What agencies may be responsible for this distortion?
(f ) Interstellar space has an extremely weak magnetic field of the order of 10–12 T. Can such a weak field be of any significant consequence?

Question 5.2 is meant mainly to arouse your curiosity. Answers to some questions above are tentative or unknown. Brief answers wherever possible are given at the end. For details, you should consult a good text on geomagnetism.]

Question 5.3 A short bar magnet placed with its axis at 30º with a uniform external magnetic field of 0.25 T experiences a torque of magnitude equal to 4.5 × 10–2 J. What is the magnitude of magnetic moment of the magnet?

Question 5.4 A short bar magnet of magnetic moment m = 0.32 JT–1 is placed in a uniform magnetic field of 0.15 T. If the bar is free to rotate in the plane of the field, which orientation wou

Question 5.5 A closely wound solenoid of 800 turns and area of cross section 2.5 × 10–4 m2 carries a current of 3.0 A. Explain the sense in which the solenoid acts like a bar magnet. What is its associated magnetic moment?

Question 5.6 If the solenoid in xercise 5.5 is free to turn about the vertical direction and a uniform horizontal magnetic field of 0.215 T is applied, what is the magnitude of torque on the solenoid when its axis makes an angle of 30° with the direction of applied field?

Question 5.7 A bar magnet of magnetic moment 1.5 J T–1 lies aligned with the direction of a uniform magnetic field of 0.22 T. (a) What is the amount of work required by an external torque to turn the magnet so as to align its magnetic moment:
(i) normal to the field direction , (ii) opposite to the field direction?
(b) What is the torque on the magnet in cases (i) and (ii)?

Question 5.8 A closely wound solenoid of 2000 turns and area of cross-section 1.6 × 10–4 m2, carrying a current of 4.0 A, is suspended through its centre allowing it to turn in a horizontal plane.
(a) What is the magnetic moment associated with the solenoid?
(b) What is the force and torque on the solenoid if a uniform horizontal magnetic field of 7.5 × 10–2 T is set up at an angle of 30º with the axis of the solenoid?

Question 5.9 A circular coil of 16 turns and radius 10 cm carrying a current of 0.75 A rests with its plane normal to an external field of magnitude 5.0 × 10–2 T. The coil is free to turn about an axis in its plane perpendicular to the field direction. When the coil is turned slightly and released, it oscillates about its stable equilibrium with a frequency of 2.0 s–1. What is the moment of inertia of the coil about its axis of rotation?

Question 5.10 A magnetic needle free to rotate in a vertical plane parallel to the magnetic meridian has its north tip pointing down at 22º with the horizontal. The horizontal component of the earth’s magnetic field at the place is known to be 0.35 G. Determine the magnitude of the earth’s magnetic field at the place.

Question 5.11 At a certain location in Africa, a compass points 12º west of the geographic north. The north tip of the magnetic needle of a dip circle placed in the plane of magnetic meridian points 60º above the horizontal. The horizontal component of the earth’s field is measured to be 0.16 G. Specify the direction and magnitude of the earth’s field at the location.

Question 5.12 A short bar magnet has a magnetic moment of 0.48 J T–1. Give the direction and magnitude of the magnetic field produced by the magnet at a distance of 10 cm from the centre of the magnet on (a) the axis,(b) the equatorial lines (normal bisector) of the magnet.

Question 5.13 A short bar magnet placed in a horizontal plane has its axis aligned along the magnetic north-south direction. Null points are found on the axis of the magnet at 14 cm from the centre of the magnet. The earth’s magnetic field at the place is 0.36 G and the angle of dip is zero. What is the total magnetic field on the normal bisector of the magnet at the same distance as the null–point (i.e., 14 cm) from the centre of the magnet? (At null points, field due to a magnet is equal and opposite to the horizontal component of earth’s magnetic field.)

Question 5.14 If the bar magnet in exercise 5.13 is turned around by 180º, where will the new null points be located?

Question 5.15 A short bar magnet of magnetic moment 5.25 × 10–2 J T–1 is placed with its axis perpendicular to the earth’s field direction. At what distance from the centre of the magnet, the resultant field is inclined at 45º with earth’s field on (a) its normal bisector and (b) its axis. Magnitude of the earth’s field at the place is given to be 0.42 G. Ignore the length of the magnet in comparison to the distances involved.

ADDITIONAL EXERCISES QUESTIONS

Question 5.16 Answer the following questions:
(a) Why does a paramagnetic sample display greater magnetisation (for the
(c) If a toroid uses bismuth for its core, will the field in the core be (slightly) greater or (slightly) less than when the core is empty?
(d) Is the permeability of a ferromagnetic material independent of the magnetic field? If not, is it more for lower or higher fields?
(e) Magnetic field lines are always nearly normal to the surface of a ferromagnet at every point. (This fact is analogous to the static electric field lines being normal to the surface of a conductor at every point.) Why?
(f ) Would the maximum possible magnetisation of a paramagnetic sample be of the same order of magnitude as the magnetisation of a ferromagnet?

Question 5.17 Answer the following questions:
(a) Explain qualitatively on the basis of domain picture the irreversibility in the magnetisation curve of a ferromagnet.
(b) The hysteresis loop of a soft iron piece has a much smaller area than that of a carbon steel piece. If the material is to go through repeated cycles of magnetisation, which piece will dissipate greater heat energy?
(c) ‘A system displaying a hysteresis loop such as a ferromagnet, is a device for storing memory?’ Explain the meaning of this statement.
(d) What kind of ferromagnetic material is used for coating magnetic tapes in a cassette player, or for building ‘memory stores’ in a modern computer?
(e) A certain region of space is to be shielded from magnetic fields. Suggest a method .

Question 5.18 A long straight horizontal cable carries a current of 2.5 A in the direction 10º south of west to 10º north of east. The magnetic meridian of the place happens to be 10º west of the geographic meridian. The earth’s magnetic field at the location is 0.33 G, and the angle of dip is zero. Locate the line of neutral points (ignore the thickness of the cable). (At neutral points, magnetic field due to a current-carrying cable is equal and opposite to the horizontal component of earth’s magnetic field.)

Question 5.19 A telephone cable at a place has four long straight horizontal wires carrying a current of 1.0 A in the same direction east to west. The earth’s magnetic field at the place is 0.39 G, and the angle of dip is 35º. The magnetic declination is nearly zero. What are the resultant magnetic fields at points 4.0 cm below the cable?

Question 5.20 A compass needle free to turn in a horizontal plane is placed at the centre of circular coil of 30 turns and radius 12 cm. The coil is in a vertical plane making an angle of 45º with the magnetic meridian. When the current in the coil is 0.35 A, the needle points west to east.
(a) Determine the horizontal component of the earth’s magnetic field at the location.
(b) The current in the coil is reversed, and the coil is rotated about its vertical axis by an angle of 90º in the anticlockwise sense looking from above. Predict the direction of the needle. Take the magnetic declination at the places to be zero.

Question 5.21 A magnetic dipole is under the influence of two magnetic fields. The angle between the field directions is 60º, and one of the fields has a magnitude of 1.2 × 10–2 T. If the dipole comes to stable equilibrium at an angle of 15º with this field, what is the magnitude of the other field?

Question 5.22 A monoenergetic (18 keV) electron beam initially in the horizontal direction is subjected to a horizontal magnetic field of 0.04 G normal to the initial direction. Estimate the up or down deflection of the beam over a distance of 30 cm (me = 9.11 × 10–19 C). [Note: Data in this exercise are so chosen that the answer will give you an idea of the effect of earth’s magnetic field on the motion of the electron beam from the electron gun to the screen in a TV set.]

Question 5.23 A sample of paramagnetic salt contains 2.0 × 1024 atomic dipoles each of dipole moment 1.5 × 10–23 J T–1. The sample is placed under a homogeneous magnetic field of 0.64 T, and cooled to a temperature of 4.2 K. The degree of magnetic saturation achieved is equal to 15%. What is the total dipole moment of the sample for a magnetic field of 0.98 T and a temperature of 2.8 K? (Assume Curie’s law

Question 5.24 A Rowland ring of mean radius 15 cm has 3500 turns of wire wound on a ferromagnetic core of relative permeability 800. What is the magnetic field B in the core for a magnetising current of 1.2 A?

Question 5.25 The magnetic moment vectors μs and μl associated with the intrinsic spin angular momentum S and orbital angular momentum l, respectively, of an electron are predicted by quantum theory (and verified experimentally to a high accuracy) to be given by: μs = –(e/m) S, μl = –(e/2m)l Which of these relations is in accordance with the result expected classically? Outline the derivation of the classical result

:: Chapter 6 - Electromagnetic Induction ::

Question 6.1 Predict the direction of induced current in the situations described by the following Figs. 6.18(a) to (f ).

Question 6.2 Use Lenz’s law to determine the direction of induced current in the situations described by Fig. 6.19:
(a) A wire of irregular shape turning into a circular shape;
(b) A circular loop being deformed into a narrow straight wire.

Question 6.3 A long solenoid with 15 turns per cm has a small loop of area 2.0 cm2 placed inside the solenoid normal to its axis. If the current carried by the solenoid changes steadily from 2.0 A to 4.0 A in 0.1 s, what is the induced emf in the loop while the current is changing?

Question 6.4 A rectangular wire loop of sides 8 cm and 2 cm with a small cut is moving out of a region of uniform magnetic field of magnitude 0.3 T directed normal to the loop. What is the emf developed across the cut if the velocity of the loop is 1 cm s–1 in a direction normal to the (a) longer side, (b) shorter side of the loop? For how long does the induced voltage last in each case?

Question 6.5 A 1.0 m long metallic rod is rotated with an angular frequency of 400 rad s–1 about an axis normal to the rod passing through its one end. The other end of the rod is in contact with a circular metallic ring. A constant and uniform magnetic field of 0.5 T parallel to the axis exists everywhere. Calculate the emf developed between the centre and the ring.

Question 6.6 A circular coil of radius 8.0 cm and 20 turns is rotated about its vertical diameter with an angular speed of 50 rad s–1 in a uniform horizontal magnetic field of magnitude 3.0 × 10–2 T. Obtain the maximum and average emf induced in the coil. If the coil forms a closed loop of resistance 10 Ω, calculate the maximum value of current in the coil. Calculate the average power loss due to Joule heating. Where does this power come from?

Question 6.7 A horizontal straight wire 10 m long extending from east to west is falling with a speed of 5.0 m s–1, at right angles to the horizontal component of the earth’s magnetic field, 0.30 × 10–4 Wb m–2.
(a) What is the instantaneous value of the emf induced in the wire?
(b) What is the direction of the emf?
(c) Which end of the wire is at the higher electrical potential?

Question 6.8 Current in a circuit falls from 5.0 A to 0.0 A in 0.1 s. If an average emf of 200 V induced, give an estimate of the self-inductance of the circuit.

Question 6.9 A pair of adjacent coils has a mutual inductance of 1.5 H. If the current in one coil changes from 0 to 20 A in 0.5 s, what is the change of flux linkage with the other coil?

Question 6.10 A jet plane is travelling towards west at a speed of 1800 km/h. What is the voltage difference developed between the ends of the wing

ADDITIONAL EXERCISES QUESTIONS

Question 6.11 Suppose the loop in Exercise Question 6.4 is stationary but the current feeding the electromagnet that produces the magnetic field is gradually reduced so that the field decreases from its initial value of 0.3 T at the rate of 0.02 T s–1. If the cut is joined and the loop has a resistance of 1.6 Ω, how much power is dissipated by the loop as heat? What is the source of this power?

Question 6.12 A square loop of side 12 cm with its sides parallel to X and Y axes is moved with a velocity of 8 cm s–1 in the positive x-direction in an environment containing a magnetic field in the positive z-direction. The field is neither uniform in space nor constant in time. It has a gradient of 10 –3 T cm–1 along the negative x-direction (that is it increases by 10 – 3 T cm–1 as one moves in the negative x-direction), and it is decreasing in time at the rate of 10 –3 T s–1. Determine the direction and magnitude of the induced current in the loop if its resistance is 4.50 mΩ.

Question 6.13 It is desired to measure the magnitude of field between the poles of a powerful loud speaker magnet. A small flat search coil of area 2 cm2 with 25 closely wound turns, is positioned normal to the field direction, and then quickly snatched out of the field region. Equivalently, one can give it a quick 90° turn to bring its plane parallel to the field direction). The total charge flown in the coil (measured by a ballistic galvanometer connected to coil) is 7.5 mC. The combined resistance of the coil and the galvanometer is 0.50 Ω. Estimate the field strength of magnet.

Question 6.14 Figure 6.20 shows a metal rod PQ resting on the smooth rails AB and positioned between the poles of a permanent magnet. The rails, the rod, and the magnetic field are in three mutual perpendicular directions. A galvanometer G connects the rails through a switch K. Length of the rod = 15 cm, B = 0.50 T, resistance of the closed loop containing the rod = 9.0 mΩ. Assume the field to be uniform.
(a) Suppose K is open and the rod is moved with a speed of 12 cm s–1 in the direction shown. Give the polarity and magnitude of the induced emf. experience magnetic force due to the motion of the rod. Explain.
(d) What is the retarding force on the rod when K is closed?
(e) How much power is required (by an external agent) to keep the rod moving at the same speed (=12 cm s–1) when K is closed? How much power is required when K is open?
(f ) How much power is dissipated as heat in the closed circuit? What is the source of this power?
(g) What is the induced emf in the moving rod if the magnetic field is parallel to the rails instead of being perpendicular?

Question 6.15 An air-cored solenoid with length 30 cm, area of cross-section 25 cm2 and number of turns 500, carries a current of 2.5 A. The current is suddenly switched off in a brief time of 10–3 s. How much is the average back emf induced across the ends of the open switch in the circuit? Ignore the variation in magnetic field near the ends of the solenoid.

Question 6.16 (a) Obtain an expression for the mutual inductance between a long straight wire and a square loop of side a as shown in Fig. 6.21.
(b) Now assume that the straight wire carries a current of 50 A and the loop is moved to the right with a constant velocity, v = 10 m/s. Calculate the induced emf in the loop at the instant when x = 0.2 m. Take a = 0.1 m and assume that the loop has a large resistance.

Question 6.17 A line charge λ per unit length is lodged uniformly onto the rim of a wheel of mass M and radius R. The wheel has light non-conducting spokes and is free to rotate without friction about its axis (Fig. 6.22). A uniform magnetic field extends over a circular region within the rim. It is given by, B = – B0 k (r ≤ a; a < R) = 0 (otherwise) What is the angular velocity of the wheel after the field is suddenly switched off?

:: Chapter 7 - Alternating Current ::

Question 7.1 A 100 Ω resistor is connected to a 220 V, 50 Hz ac supply.
(a) What is the rms value of current in the circuit?
(b) What is the net power consumed over a full cycle?

Question 7.2(a) The peak voltage of an ac supply is 300 V. What is the rms voltage?
(b) The rms value of current in an ac circuit is 10 A. What is the peak current?

Question 7.3 A 44 mH inductor is connected to 220 V, 50 Hz ac supply. Determine the rms value of the current in the circuit.

Question 7.4 A 60 μF capacitor is connected to a 110 V, 60 Hz ac supply. Determine the rms value of the current in the circuit.

Question 7.5 In Exercises 7.3 and 7.4, what is the net power absorbed by each circuit over a complete cycle. Explain your answer.

Question 7.6 Obtain the resonant frequency ωr of a series LCR circuit with L = 2.0H, C = 32 μF and R = 10 Ω. What is the Q-value of this circuit?

Question 7.7 A charged 30 μF capacitor is connected to a 27 mH inductor. What is the angular frequency of free oscillations of the circuit?

Question 7.8 Suppose the initial charge on the capacitor in Exercise

Question 7.7 is 6 mC. What is the total energy stored in the circuit initially? What is the total energy at later time?

Question 7.9 A series LCR circuit with R = 20 Ω, L = 1.5 H and C = 35 μF is connected to a variable-frequency 200 V ac supply. When the frequency of the supply equals the natural frequency of the circuit, what is the average power transferred to the circuit in one complete cycle?

Question 7.10 A radio can tune over the frequency range of a portion of MW broadcast band: (800 kHz to 1200 kHz). If its LC circuit has an effective inductance of 200 μH, what must be the range of its variable capacitor? [Hint: For tuning, the natural frequency i.e., the frequency of free oscillations of the LC circuit should be equal to the frequency of the radiowave.]

Question 7.11 Figure 7.21 shows a series LCR circuit connected to a variable frequency 230 V source. L = 5.0 H, C = 80μF, R = 40 Ω.
(a) Determine the source frequency which drives the circuit in resonance.
(b) Obtain the impedance of the circuit and the amplitude of current at the resonating frequency.
(c) Determine the rms potential drops across the three elements of the circuit. Show that the potential drop across the LC combination is zero at the resonating frequency.

ADDITIONAL EXERCISES QUESTIONS

Question 7.12 An LC circuit contains a 20 mH inductor and a 50 μF capacitor with an initial charge of 10 mC. The resistance of the circuit is negligible. Let the instant the circuit is closed be t = 0 .
(a) What is the total energy stored initially? Is it conserved during LC oscillations?
(b) What is the natural frequency of the circuit?
(c) At what time is the energy stored
(i) completely electrical (i.e., stored in the capacitor)?
(ii) completely magnetic (i.e., stored in the inductor)?
(d) At what times is the total energy shared equally between the inductor and the capacitor?
(e) If a resistor is inserted in the circuit, how much energy is eventually dissipated as heat?

Question 7.13 A coil of inductance 0.50 H and resistance 100 Ω is connected to a 240 V, 50 Hz ac supply.
(a) What is the maximum current in the coil?
(b) What is the time lag between the voltage maximum and the current maximum?

Question 7.14 Obtain the answers (a) to (b) in Exercise 7.13 if the circuit is connected to a high frequency supply (240 V, 10 kHz). Hence, explain the statement that at very high frequency, an inductor in a circuit nearly amounts to an open circuit. How does an inductor behave in a dc circuit after the steady state?

Question 7.15 A 100 μF capacitor in series with a 40 Ω resistance is connected to a 110 V, 60 Hz supply.
(a) What is the maximum current in the circuit?
(b) What is the time lag between the current maximum and the voltage maximum?

Question 7.16 Obtain the answers to (a) and (b) in Exercise 7.15 if the circuit is connected to a 110 V, 12 kHz supply? Hence, explain the statement that a capacitor is a conductor at very high frequencies. Compare this behaviour with that of a capacitor in a dc circuit after the steady state.

Question 7.17 Keeping the source frequency equal to the resonating frequency of the series LCR circuit, if the three elements, L, C and R are arranged in parallel, show that the total current in the parallel LCR circuit is minimum at this frequency. Obtain the current rms value in each branch of the circuit for the elements and source specified in Exercise 7.11 for this frequency.

Question 7.18 A circuit containing a 80 mH inductor and a 60 μF capacitor in series is connected to a 230 V, 50 Hz supply. The resistance of the circuit is negligible.
(a) Obtain the current amplitude and rms values
(b) Obtain the rms values of potential drops across each element.
(c) What is the average power transferred to the inductor?
(d) What is the average power transferred to the capacitor?
(e) What is the total average power absorbed by the circuit? [‘Average’ implies ‘averaged over one cycle’.]

Question 7.19 Suppose the circuit in Exercise 7.18 has a resistance of 15 Ω. Obtain the average power transferred to each element of the circuit, and the total power absorbed. Physics 268

Question 7.20 A series LCR circuit with L = 0.12 H, C = 480 nF, R = 23 Ω is connected to a 230 V variable frequency supply.
(a) What is the source frequency for which current amplitude is maximum. Obtain this maximum value.
(b) What is the source frequency for which average power absorbed by the circuit is maximum. Obtain the value of this maximum power.
(c) For which frequencies of the source is the power transferred to the circuit half the power at resonant frequency? What is the current amplitude at these frequencies?
(d) What is the Q-factor of the given circuit?

Question 7.21 Obtain the resonant frequency and Q-factor of a series LCR circuit with L = 3.0 H, C = 27 μF, and R = 7.4 Ω. It is desired to improve the sharpness of the resonance of the circuit by reducing its ‘full width at half maximum’ by a factor of 2. Suggest a suitable way.

Question 7.22 Answer the following questions:
(a) In any ac circuit, is the applied instantaneous voltage equal to the algebraic sum of the instantaneous voltages across the series elements of the circuit? Is the same true for rms voltage?
(b) A capacitor is used in the primary circuit of an induction coil.
(c) An applied voltage signal consists of a superposition of a dc voltage and an ac voltage of high frequency. The circuit consists of an inductor and a capacitor in series. Show that the dc signal will appear across C and the  ac signal across L.
(d) A choke coil in series with a lamp is connected to a dc line. The lamp is seen to shine brightly. Insertion of an iron core in the choke causes no change in the lamp’s brightness. Predict the corresponding observations if the connection is to an ac line.
(e) Why is choke coil needed in the use of fluorescent tubes with ac mains? Why can we not use an ordinary resistor instead of the choke coil?

Question 7.23 A power transmission line feeds input power at 2300 V to a stepdown transformer with its primary windings having 4000 turns. What should be the number of turns in the secondary in order to get output power at 230 V?

Question 7.24 At a hydroelectric power plant, the water pressure head is at a height of 300 m and the water flow available is 100 m3s–1. If the turbine generator efficiency is 60%, estimate the electric power available from the plant (g = 9.8 ms–2 ).

Question 7.25 A small town with a demand of 800 kW of electric power at 220 V is situated 15 km away from an electric plant generating power at 440 V. The resistance of the two wire line carrying power is 0.5 Ω per km. The town gets power from the line through a 4000-220 V step-down transformer at a sub-station in the town.
(a) Estimate the line power loss in the form of heat.
(b) How much power must the plant supply, assuming there is negligible power loss due to leakage?
(c) Characterise the step up transformer at the plant.

Question 7.26 Do the same exercise as above with the replacement of the earlier transformer by a 40,000-220 V step-down transformer (Neglect, as before, leakage losses though this may not be a good assumption any longer because of the very high voltage transmission involved). Hence, explain why high voltage transmission is preferred?

:: Chapter 8 - Electromagnetic Waves ::

 Question 8.1 Figure 8.6 shows a capacitor made of two circular plates each of radius 12 cm, and separated by 5.0 cm. The capacitor is being charged by an external source (not shown in the figure). The charging current is constant and equal to 0.15A.

(a) Calculate the capacitance and the rate of charge of potential difference between the plates.
(b) Obtain the displacement current across the plates.
(c) Is Kirchhoff’s first rule (junction rule) valid at each plate of the capacitor? Explain

Question 8.2 A parallel plate capacitor (Fig. 8.7) made of circular plates each of radius R = 6.0 cm has a capacitance C = 100 pF. The capacitor is connected to a 230 V ac supply with a (angular) frequency of 300 rad s–1.
(a) What is the rms value of the conduction current?
(b) Is the conduction current equal to the displacement current?
(c) Determine the amplitude of B at a point 3.0 cm from the axis between the plates.

Question 8.3 What physical quantity is the same for X-rays of wavelength 10–10 m, red light of wavelength 6800 Å and radiowaves of wavelength 500m?

Question 8.4 A plane electromagnetic wave travels in vacuum along z-direction. What can you say about the directions of its electric and magnetic field vectors? If the frequency of the wave is 30 MHz, what is its wavelength?

Question 8.5 A radio can tune in to any station in the 7.5 MHz to 12 MHz band. What is the corresponding wavelength band?

Question 8.6 A charged particle oscillates about its mean equilibrium position with a frequency of 109 Hz. What is the frequency of the electromagnetic waves produced by the oscillator?

Question 8.7 The amplitude of the magnetic field part of a harmonic electromagnetic wave in vacuum is B0 = 510 nT. What is the amplitude of the electric field part of the wave?

Question 8.8 Suppose that the electric field amplitude of an electromagnetic wave is E0 = 120 N/C and that its frequency is ν = 50.0 MHz.
(a) Determine, B0,ω, k, and λ
(b) Find expressions for E and B.

Question 8.9 The terminology of different parts of the electromagnetic spectrum is given in the text. Use the formula E = hν (for energy of a quantum of radiation: photon) and obtain the photon energy in units of eV for different parts of the electromagnetic spectrum. In what way are the different scales of photon energies that you obtain related to the sources of electromagnetic radiation?

Question 8.10 In a plane electromagnetic wave, the electric field oscillates sinusoidally at a frequency of 2.0 × 1010 Hz and amplitude 48 V m–1.
(a) What is the wavelength of the wave?
(b) What is the amplitude of the oscillating magnetic field? (c) Show that the average energy density of the E field equals the average energy density of the B field. [c = 3 × 108 m s–1.]

ADDITIONAL EXERCISES

Question 8.11 Suppose that the electric field part of an electromagnetic wave in vacuum is E = {(3.1 N/C) cos [(1.8 rad/m) y + (5.4 × 106 rad/s)t]}ˆi .
(a) What is the direction of propagation?
(b) What is the wavelength λ ?
(c) What is the frequency ν ?
(d) What is the amplitude of the magnetic field part of the wave?
(e) Write an expression for the magnetic field part of the wave.

Question 8.12 About 5% of the power of a 100 W light bulb is converted to visible radiation. What is the average intensity of visible radiation
(a) at a distance of 1m from the bulb?
(b) at a distance of 10 m? Assume that the radiation is emitted isotropically and neglect reflection.

Question 8.13 Use the formula λm T = 0.29 cmK to obtain the characteristic temperature ranges for different parts of the electromagnetic spectrum. What do the numbers that you obtain tell you?

Question 8.14 Given below are some famous numbers associated with electromagnetic radiations in different contexts in physics. State the part of the electromagnetic spectrum to which each belongs.
(a) 21 cm (wavelength emitted by atomic hydrogen in interstellar space).
(b) 1057 MHz (frequency of radiation arising from two close energy levels in hydrogen; known as Lamb shift).
(c) 2.7 K [temperature associated with the isotropic radiation filling all space-thought to be a relic of the ‘big-bang’ origin of the universe].
(d) 5890 Å - 5896 Å [double lines of sodium]
(e) 14.4 keV [energy of a particular transition in 57Fe nucleus associated with a famous high resolution spectroscopic method (Mössbauer spectroscopy)].

Question 8.15 Answer the following questions:
(a) Long distance radio broadcasts use short-wave bands. Why?
(b) It is necessary to use satellites for long distance TV transmission. Why?
(c) Optical and radiotelescopes are built on the ground but X-ray astronomy is possible only from satellites orbiting the earth. Why?
(d) The small ozone layer on top of the stratosphere is crucial for human survival. Why?
(e) If the earth did not have an atmosphere, would its average surface temperature be higher or lower than what it is now?
(f ) Some scientists have predicted that a global nuclear war on the earth would be followed by a severe ‘nuclear winter’ with a devastating effect on life on earth. What might be the basis of this prediction?

:: Chapter 9 - Ray Optics and Optical Instruments ::

Question 9.1 A small candle, 2.5 cm in size is placed at 27 cm in front of a concave mirror of radius of curvature 36 cm. At what distance from the mirror should a screen be placed in order to obtain a sharp image? Describe the nature and size of the image. If the candle is moved closer to the mirror, how would the screen have to be moved?

Question 9.2 A 4.5 cm needle is placed 12 cm away from a convex mirror of focal length 15 cm. Give the location of the image and the magnification. Describe what happens as the needle is moved farther from the mirror.

Question 9.3 A tank is filled with water to a height of 12.5 cm. The apparent depth of a needle lying at the bottom of the tank is measured by a microscope to be 9.4 cm. What is the refractive index of water? If water is replaced by a liquid of refractive index 1.63 up to the same height, by what distance would the microscope have to be moved to focus on the needle again?

Question 9.4 Figures 9.34(a) and (b) show refraction of a ray in air incident at 60° with the normal to a glass-air and water-air interface, respectively. Predict the angle of refraction in glass when the angle of incidence in water is 45º with the normal to a water-glass interface [Fig. 9.34(c)].

Question 9.5 A small bulb is placed at the bottom of a tank containing water to a depth of 80cm. What is the area of the surface of water through which light from the bulb can emerge out? Refractive index of water is 1.33. (Consider the bulb to be a point source.)

Question 9.6 A prism is made of glass of unknown refractive index. A parallel beam of light is incident on a face of the prism. The angle of minimum deviation is measured to be 40°. What is the refractive index of the material of the prism? The refracting angle of the prism is 60°. If the prism is placed in water (refractive index 1.33), predict the new angle of minimum deviation of a parallel beam of light.

Question 9.7 Double-convex lenses are to be manufactured from a glass of refractive index 1.55, with both faces of the same radius of curvature. What is the radius of curvature required if the focal length is to be 20cm?

Question 9.8 A beam of light converges at a point P. Now a lens is placed in the path of the convergent beam 12cm from P. At what point does the beam converge if the lens is (a) a convex lens of focal length 20cm, and (b) a concave lens of focal length 16cm?

Question 9.9 An object of size 3.0cm is placed 14cm in front of a concave lens of focal length 21cm. Describe the image produced by the lens. What happens if the object is moved further away from the lens?

Question 9.10 What is the focal length of a convex lens of focal length 30cm in contact with a concave lens of focal length 20cm?Is the system a converging or a diverging lens? Ignore thickness of the lenses .

Question 9.11 A compound microscope consists of an objective lens of focal length 2.0cm and an eyepiece of focal length 6.25cm separated by a distance of 15cm. How far from the objective should an object be placed in order to obtain the final image at (a) the least distance of distinct vision (25cm), and (b) at infinity? What is the magnifying power of the microscope in each case?

Question 9.12 A person with a normal near point (25cm) using a compound microscope with objective of focal length 8.0 mm and an eyepiece of focal length 2.5cm can bring an object placed at 9.0mm from the objective in sharp focus. What is the separation between the two lenses? Calculate the magnifying power of the microscope,

Question 9.13 A small telescope has an objective lens of focal length 144cm and an eyepiece of focal length 6.0cm. What is the magnifying power of the telescope? What is the separation between the objective and the eyepiece?

Question 9.14 (a) A giant refracting telescope at an observatory has an objective lens of focal length 15m. If an eyepiece of focal length 1.0cm is used, what is the angular magnification of the telescope?
(b) If this telescope is used to view the moon, what is the diameter of the image of the moon formed by the objective lens? The diameter of the moon is 3.48 × 106m, and the radius of lunar orbit is 3.8 × 108m.

Question 9.15 Use the mirror equation to deduce that:
(a) an object placed between f and 2f of a concave mirror produces a real image beyond 2f.
(b) a convex mirror always produces a virtual image independent of the location of the object.
(c) the virtual image produced by a convex mirror is always diminished in size and is located between the focus and the pole.
(d) an object placed between the pole and focus of a concave mirror produces a virtual and enlarged image. [Note: This exercise helps you deduce algebraically properties of images that one obtains from explicit ray diagrams.]

Question 9.16 A small pin fixed on a table top is viewed from above from a distance of 50cm. By what distance would the pin appear to be raised if it is viewed from the same point through a 15cm thick glass slab held parallel to the table? Refractive index of glass = 1.5. Does the answer depend on the location of the slab?

Question 9.17 (a) Figure 9.35 shows a cross-section of a ‘light pipe’ made of a glass fibre of refractive index 1.68. The outer covering of the pipe is made of a material of refractive index 1.44. What is the range of the angles of the incident rays with the axis of the pipe for which total reflections inside the pipe take place, as shown in the figure.
(b) What is the answer if there is no outer covering of the pipe?

Question 9.18 Answer the following questions:
(a) You have learnt that plane and convex mirrors produce virtual images of objects. Can they produce real images under some circumstances? Explain.
(b) A virtual image, we always say, cannot be caught on a screen. Yet when we ‘see’ a virtual image, we are obviously bringing it on to the ‘screen’ (i.e., the retina) of our eye. Is there a contradiction?
(c) A diver under water, looks obliquely at a fisherman standing on the bank of a lake. Would the fisherman look taller or shorter to the diver than what he actually is?
(d) Does the apparent depth of a tank of water change if viewed obliquely? If so, does the apparent depth increase or decrease?
(e) The refractive index of diamond is much greater than that of ordinary glass. Is this fact of some use to a diamond cutter?

Question 9.19 The image of a small electric bulb fixed on the wall of a room is to be obtained on the opposite wall 3m away by means of a large convex lens. What is the maximum possible focal length of the lens required for the purpose?

Question 9.20 A screen is placed 90cm from an object. The image of the object on the screen is formed by a convex lens at two different locations separated by 20cm. Determine the focal length of the lens.

Question 9.21 (a) Determine the ‘effective focal length’ of the combination of the two lenses in Exercise 9.10, if they are placed 8.0cm apart with their principal axes coincident. Does the answer depend on which side of the combination a beam of parallel light is incident? Is the notion of effective focal length of this system useful at all?
(b) An object 1.5 cm in size is placed on the side of the convex lens in the arrangement (a) above. The distance between the object and the convex lens is 40cm. Determine the magnification produced by the two-lens system, and the size of the image.

Question 9.22 At what angle should a ray of light be incident on the face of a prism of refracting angle 60° so that it just suffers total internal reflection at the other face? The refractive index of the material of the prism is 1.524.

Question 9.23 You are given prisms made of crown glass and flint glass with a wide variety of angles. Suggest a combination of prisms which will (a) deviate a pencil of white light without much dispersion, (b) disperse (and displace) a pencil of white light without much deviation.

Question 9.24 For a normal eye, the far point is at infinity and the near point of distinct vision is about 25cm in front of the eye. The cornea of the eye provides a converging power of about 40 dioptres, and the least converging power of the eye-lens behind the cornea is about 20 dioptres. From this rough data estimate the range of accommodation (i.e., the range of converging power of the eye-lens) of a normal eye.

Question 9.25 Does short-sightedness (myopia) or long-sightedness (hypermetropia) imply necessarily that the eye has partially lost its ability of accommodation? If not, what might cause these defects of vision?

Question 9.26 A myopic person has been using spectacles of power –1.0 dioptre for distant vision. During old age he also needs to use separate reading glass of power + 2.0 dioptres. Explain what may have happened.

Question 9.27 A person looking at a person wearing a shirt with a pattern comprising vertical and horizontal lines is able to see the vertical lines more distinctly than the horizontal ones. What is this defect due to? How is such a defect of vision corrected?

Question 9.28 A man with normal near point (25 cm) reads a book with small print using a magnifying glass: a thin convex lens of focal length 5 cm.
(a) What is the closest and the farthest distance at which he should keep the lens from the page so that he can read the book when viewing through the magnifying glass?
(b) What is the maximum and the minimum angular magnification (magnifying power) possible using the above simple microscope?

Question 9.29 A card sheet divided into squares each of size 1 mm2 is being viewed at a distance of 9 cm through a magnifying glass (a converging lens of focal length 9 cm) held close to the eye.
(a) What is the magnification produced by the lens? How much is the area of each square in the virtual image?
(b) What is the angular magnification (magnifying power) of the lens?
(c) Is the magnification in
(a) equal to the magnifying power in (b)? Explain.

Question 9.30 (a) At what distance should the lens be held from the figure in Exercise 9.29 in order to view the squares distinctly with the maximum possible magnifying power?
(b) What is the magnification in this case?
(c) Is the magnification equal to the magnifying power in this case? Explain. 9.31 What should be the distance between the object in Exercise

Question 9.30 and the magnifying glass if the virtual image of each square in the figure is to have an area of 6.25 mm2. Would you be able to see the squares distinctly with your eyes very close to the magnifier? [Note: Exercises 9.29 to

Question 9.31will help you clearly understand the tion (or magnifying power) of an instrument.]

Question 9.32 Answer the following questions:
(a) The angle subtended at the eye by an object is equal to the angle subtended at the eye by the virtual image produced by a magnifying glass. In what sense then does a magnifying glass provide angular magnification?
(b) In viewing through a magnifying glass, one usually positions one’s eyes very close to the lens. Does angular magnification change if the eye is moved back?
(c) Magnifying power of a simple microscope is inversely proportional to the focal length of the lens. What then stops us from using a convex lens of smaller and smaller focal length and achieving greater and greater magnifying power? (d) Why must both the objective and the eyepiece of a compound microscope have short focal lengths?
(e) When viewing through a compound microscope, our eyes should be positioned not on the eyepiece but a short distance away from it for best viewing. Why? How much should be that short distance between the eye and eyepiece?

Question 9.33 An angular magnification (magnifying power) of 30X is desired using an objective of focal length 1.25cm and an eyepiece of focal length 5cm. How will you set up the compound microscope?

Question 9.34 A small telescope has an objective lens of focal length 140cm and an eyepiece of focal length 5.0cm.
What is the magnifying power of the telescope for viewing distant objects when
(a) the telescope is in normal adjustment (i.e., when the final image is at infinity)?
(b) the final image is formed at the least distance of distinct vision (25cm)?

Question 9.35 (a) For the telescope described in Exercise 3.4 (a), what is the separation between the objective lens and the eyepiece? (b) If this telescope is used to view a 100 m tall tower 3 km away, what is the height of the image of the tower formed by the objective lens? (c) What is the height of the final image of the tower if it is formed at 25cm?

Question 9.36 A Cassegrain telescope uses two mirrors as shown in Fig. 9.33. Such a telescope is built with the mirrors 20mm apart. If the radius of curvature of the large mirror is 220mm and the small mirror is 140mm, where will the final image of an object at infinity be?

Question 9.37 Light incident normally on a plane mirror attached to a galvanometer coil retraces backwards as shown in Fig.

Question 9.36. A current in the coil produces a deflection of 3.5o of the mirror. What is the displacement of the reflected spot of light on a screen placed 1.5 m away?

Question 9.39 shows an equiconvex lens (of refractive index 1.50) in contact with a liquid layer on top of a plane mirror. A small needle with its tip on the principal axis is moved along the axis until its inverted image is found at the position of the needle. The distance of the needle from the lens is measured to be 45.0cm. The liquid is removed and the experiment is repeated. The new distance is measured to be 30.0cm. What is the refractive index of the liquid?    

:: Chapter 10 - Wave Optics ::

Question 10.1 Monochromatic light of wavelength 589 nm is incident from air on a water surface. What are the wavelength, frequency and speed of (a) reflected, and (b) refracted light? Refractive index of water is 1.33.

Question 10.2 What is the shape of the wavefront in each of the following cases:
(a) Light diverging from a point source.
(b) Light emerging out of a convex lens when a point source is placed at its focus.
(c) The portion of the wavefront of light from a distant star intercepted by the Earth.

Question 10.3 (a) The refractive index of glass is 1.5. What is the speed of light in glass? (Speed of light in vacuum is 3.0 × 108 m s–1) (b) Is the speed of light in glass independent of the colour of light? If not, which of the two colours red and violet travels slower in a glass prism?

Question 10.4 In a Young’s double-slit experiment, the slits are separated by 0.28 mm and the screen is placed 1.4 m away. The distance between the central bright fringe and the fourth bright fringe is measured to be 1.2 cm. Determine the wavelength of light used in the experiment.

Question 10.5 In Young’s double-slit experiment using monochromatic light of wavelength λ, the intensity of light at a point on the screen where path difference is λ, is K units. What is the intensity of light at a point where path difference is λ/3?

Question 10.6 A beam of light consisting of two wavelengths, 650 nm and 520 nm, is used to obtain interference fringes in a Young’s double-slit experiment.
(a) Find the distance of the third bright fringe on the screen from the central maximum for wavelength 650 nm.
(b) What is the least distance from the central maximum where the bright fringes due to both the wavelengths coincide?

Question 10.7 In a double-slit experiment the angular width of a fringe is found to be 0.2° on a screen placed 1 m away. The wavelength of light used is 600 nm. What will be the angular width of the fringe if the entire experimental apparatus is immersed in water? Take refractive index of water to be 4/3.

Question 10.8 What is the Brewster angle for air to glass transition? (Refractive index of glass = 1.5.)

Question 10.9 Light of wavelength 5000 Å falls on a plane reflecting surface. What are the wavelength and frequency of the reflected light? For what angle of incidence is the reflected ray normal to the incident ray?

Question 10.10 Estimate the distance for which ray optics is good approximation for an aperture of 4 mm and wavelength 400 nm.

ADDITIONAL EXERCISES QUESTIONS

Question 10.11 The 6563 Å Hα line emitted by hydrogen in a star is found to be redshifted by 15 Å. Estimate the speed with which the star is receding from the Earth.

Question 10.12 Explain how Corpuscular theory predicts the speed of light in a medium, say, water, to be greater than the speed of light in vacuum. Is the prediction confirmed by experimental determination of the speed of light in water? If not, which alternative picture of light is consistent with experiment?

Question 10.13 You have learnt in the text how Huygens’ principle leads to the laws of reflection and refraction. Use the same principle to deduce directly that a point object placed in front of a plane mirror produces a virtual image whose distance from the mirror is equal to the object distance from the mirror.

Question 10.14 Let us list some of the factors, which could possibly influence the speed of wave propagation:
(i) nature of the source.
(ii) direction of propagation.
(iii) motion of the source and/or observer.
(iv) wavelength.
(v) intensity of the wave. On which of these factors, if any, does (a) the speed of light in vacuum, (b) the speed of light in a medium (say, glass or water), depend?

Question 10.15 For sound waves, the Doppler formula for frequency shift differs slightly between the two situations:
(i) source at rest; observer moving, and (ii) source moving; observer at rest. The exact Doppler formulas for the case of light waves in vacuum are, however, strictly identical for these situations. Explain why this should be so. Would you expect the formulas to be strictly identical for the two situations in case of light travelling in a medium?

Question 10.16 In double-slit experiment using light of wavelength 600 nm, the angular width of a fringe formed on a distant screen is 0.1º. What is the spacing between the two slits?

Question 10.17 Answer the following questions:
(a) In a single slit diffraction experiment, the width of the slit is made double the original width. How does this affect the size and intensity of the central diffraction band?
(b) In what way is diffraction from each slit related to the interference pattern in a double-slit experiment?
(c) When a tiny circular obstacle is placed in the path of light from a distant source, a bright spot is seen at the centre of the shadow of the obstacle. Explain why?
(d) Two students are separated by a 7 m partition wall in a room 10 m high. If both light and sound waves can bend around obstacles, how is it that the students are unable to see each other even though they can converse easily.
(e) Ray optics is based on the assumption that light travels in a straight line. Diffraction effects (observed when light propagates through small apertures/slits or around small obstacles) disprove this assumption. Yet the ray optics assumption is so commonly used in understanding location and several other properties of images in optical instruments. What is the justification?

Question 10.18 Two towers on top of two hills are 40 km apart. The line joining them passes 50 m above a hill halfway between the towers. What is the longest wavelength of radio waves, which can be sent between the towers without appreciable diffraction effects?

Question 10.19 A parallel beam of light of wavelength 500 nm falls on a narrow slit and the resulting diffraction pattern is observed on a screen 1 m away. It is observed that the first minimum is at a distance of 2.5 mm from the centre of the screen. Find the width of the slit.

Question 10.20 Answer the following questions:
(a) When a low flying aircraft passes overhead, we sometimes notice a slight shaking of the picture on our TV screen. Suggest a possible explanation.
(b) As you have learnt in the text, the principle of linear superposition of wave displacement is basic to understanding intensity distributions in diffraction and interference patterns. What is the justification of this principle?

Question 10.21 In deriving the single slit diffraction pattern, it was stated that the intensity is zero at angles of nλ/a. Justify this by suitably dividing the slit to bring out the cancellation.

:: Chapter 11 - Dual nature of Radiation and Matter ::

Question 11.1 Find the (a) maximum frequency, and (b) minimum wavelength of X-rays produced by 30 kV electrons.

Question 11.2 The work function of caesium metal is 2.14 eV. When light of frequency 6 ×1014Hz is incident on the metal surface, photoemission of electrons occurs. What is the (a) maximum kinetic energy of the emitted electrons, (b) Stopping potential, and (c) maximum speed of the emitted photoelectrons?

Question 11.3 The photoelectric cut-off voltage in a certain experiment is 1.5 V. What is the maximum kinetic energy of photoelectrons emitted?

Question 11.4 Monochromatic light of wavelength 632.8 nm is produced by a helium-neon laser. The power emitted is 9.42 mW. (a) Find the energy and momentum of each photon in the light beam, (b) How many photons per second, on the average, arrive at a target irradiated by this beam? (Assume the beam to have uniform cross-section which is less than the target area), and (c) How fast does a hydrogen atom have to travel in order to have the same momentum as that of the photon?

Question 11.5 The energy flux of sunlight reaching the surface of the earth is 1.388 × 103 W/m2. How many photons (nearly) per square metre are incident on the Earth per second? Assume that the photons in the sunlight have an average wavelength of 550 nm.

Question 11.6 In an experiment on photoelectric effect, the slope of the cut-off voltage versus frequency of incident light is found to be 4.12 × 10–15 V s. Calculate the value of Planck’s constant.

Question 11.7 A 100W sodium lamp radiates energy uniformly in all directions. The lamp is located at the centre of a large sphere that absorbs all the sodium light which is incident on it. The wavelength of the sodium light is 589 nm. (a) What is the energy per photon associated with the sodium light? (b) At what rate are the photons delivered to the sphere?

Question 11.8 The threshold frequency for a certain metal is 3.3 × 1014 Hz. If light of frequency 8.2 × 1014 Hz is incident on the metal, predict the cutoff voltage for the photoelectric emission.

Question 11.9 The work function for a certain metal is 4.2 eV. Will this metal give photoelectric emission for incident radiation of wavelength 330 nm?

Question 11.10 Light of frequency 7.21 × 1014 Hz is incident on a metal surface. Electrons with a maximum speed of 6.0 × 105 m/s are ejected from the surface. What is the threshold frequency for photoemission of electrons?

Question 11.11 Light of wavelength 488 nm is produced by an argon laser which is used in the photoelectric effect. When light from this spectral line is incident on the emitter, the stopping (cut-off) potential of photoelectrons is 0.38 V. Find the work function of the material from which the emitter is made.

Question 11.12 Calculate the (a) momentum, and (b) de Broglie wavelength of the electrons accelerated through a potential difference of 56V.

Question 11.13 What is the (a) momentum, (b) speed, and (c) de Broglie wavelength of an electron with kinetic energy of 120 eV.

Question 11.14 The wavelength of light from the spectral emission line of sodium is 589 nm. Find the kinetic energy at which (a) an electron, and (b) a neutron, would have the same de Broglie wavelength.

Question 11.15 What is the de Broglie wavelength of (a) a bullet of mass 0.040 kg travelling at the speed of 1.0 km/s, (b) a ball of mass 0.060 kg moving at a speed of 1.0 m/s, and (c) a dust particle of mass 1.0 × 10–9 kg drifting with a speed of 2.2 m/s?

Question 11.16 An electron and a photon each have a wavelength of 1.00 nm. Find (a) their momenta, (b) the energy of the photon, and (c) the kinetic energy of electron.

Question 11.17 (a) For what kinetic energy of a neutron will the associated de Broglie wavelength be 1.40 × 10–10m? (b) Also find the de Broglie wavelength of a neutron, in thermal equilibrium with matter, having an average kinetic energy of (3/2) k T at 300 K.

Question 11.18 Show that the wavelength of electromagnetic radiation is equal to the de Broglie wavelength of its quantum (photon).

Question 11.19 What is the de Broglie wavelength of a nitrogen molecule in air at 300 K? Assume that the molecule is moving with the root-meansquare speed of molecules at this temperature. (Atomic mass of nitrogen = 14.0076 u)

ADDITIONAL EXERCISES QUESTIONS

Question 11.20 (a) Estimate the speed with which electrons emitted from a heated emitter of an evacuated tube impinge on the collector maintained at a potential difference of 500 V with respect to the emitter. Ignore the small initial speeds of the electrons. The specific charge of the electron, i.e., its e/m is given to be 1.76 × 1011 C kg–1. (b) Use the same formula you employ in (a) to obtain electron speed for an collector potential of 10 MV. Do you see what is wrong ? In what way is the formula to be modified?

Question 11.21 (a) A monoenergetic electron beam with electron speed of 5.20 × 106 m s–1 is subject to a magnetic field of 1.30 × 10–4 T normal to the beam velocity. What is the radius of the circle traced by the beam, given e/m for electron equals 1.76 × 1011C kg–1. (b) Is the formula you employ in (a) valid for calculating radius of the path of a 20 MeV electron beam? If not, in what way is it modified ? [Note: Exercises 11.20(b) and 11.21 (b) take you to relativistic mechanics which is beyond the scope of this book. They have been inserted here simply to emphasise the point that the formulas you use in part (a) of the exercises are not valid at very high speeds or energies. See answers at the end to know what ‘very high speed or energy’ means.

Question 11.22 An electron gun with its collector at a potential of 100 V fires out electrons in a spherical bulb containing hydrogen gas at low pressure (∼10–2 mm of Hg). A magnetic field of 2.83 × 10–4 T curves the path of the electrons in a circular orbit of radius 12.0 cm. (The path can be viewed because the gas ions in the path focus the beam by attracting electrons, and emitting light by electron capture; this method is known as the ‘fine beam tube’ method.) Determine e/m from the data.

Question 11.23 (a) An X-ray tube produces a continuous spectrum of radiation with its short wavelength end at 0.45 Å. What is the maximum energy of a photon in the radiation? (b) From your answer to (a), guess what order of accelerating voltage (for electrons) is required in such a tube?

Question 11.24 In an accelerator experiment on high-energy collisions of electrons with positrons, a certain event is interpreted as annihilation of an electron-positron pair of total energy 10.2 BeV into two γ-rays of equal energy. What is the wavelength associated with each γ-ray? (1BeV = 109 eV)

Question 11.25 Estimating the following two numbers should be interesting. The first number will tell you why radio engineers do not need to worry much about photons! The second number tells you why our eye can never ‘count photons’, even in barely detectable light. (a) The number of photons emitted per second by a Medium wave transmitter of 10 kW power, emitting radiowaves of wavelength 500 m.
(b) The number of photons entering the pupil of our eye per second corresponding to the minimum intensity of white light that we humans can perceive (∼10–10 W m–2). Take the area of the pupil to be about 0.4 cm2, and the average frequency of white light to be about 6 × 1014 Hz.

Question 11.26 Ultraviolet light of wavelength 2271 Å from a 100 W mercury source irradiates a photo-cell made of molybdenum metal. If the stopping potential is –1.3 V, estimate the work function of the metal. How would the photo-cell respond to a high intensity (∼105 W m–2) red light of wavelength 6328 Å produced by a He-Ne laser?

Question 11.27 Monochromatic radiation of wavelength 640.2 nm (1nm = 10–9 m) from a neon lamp irradiates photosensitive material made of caesium on tungsten. The stopping voltage is measured to be 0.54 V. The source is replaced by an iron source and its 427.2 nm line irradiates the same photo-cell. Predict the new stopping voltage.

Question 11.28 A mercury lamp is a convenient source for studying frequency dependence of photoelectric emission, since it gives a number of spectral lines ranging from the UV to the red end of the visible spectrum. In our experiment with rubidium photo-cell, the following lines from a mercury source were used: λ1 = 3650 Å, λ2= 4047 Å, λ3= 4358 Å, λ4= 5461 Å, λ5= 6907 Å, The stopping voltages, respectively, were measured to be: V01 = 1.28 V, V02 = 0.95 V, V03 = 0.74 V, V04 = 0.16 V, V05 = 0 V Determine the value of Planck’s constant h, the threshold frequency and work function for the material. [Note: You will notice that to get h from the data, you will need to know e (which you can take to be 1.6 × 10–19 C). Experiments of this kind on Na, Li, K, etc. were performed by Millikan, who, using his own value of e (from the oil-drop experiment) confirmed Einstein’s photoelectric equation and at the same time gave an independent estimate of the value of h.

Question 11.29 The work function for the following metals is given: Na: 2.75 eV; K: 2.30 eV; Mo: 4.17 eV; Ni: 5.15 eV. Which of these metals will not give photoelectric emission for a radiation of wavelength 3300 Å from a He-Cd laser placed 1 m away from the photocell? What happens if the laser is brought nearer and placed 50 cm away?

Question 11.30 Light of intensity 10–5 W m–2 falls on a sodium photo-cell of surface area 2 cm2. Assuming that the top 5 layers of sodium absorb the incident energy, estimate time required for photoelectric emission in the wave-picture of radiation. The work function for the metal is given to be about 2 eV. What is the implication of your answer?

Question 11.31 Crystal diffraction experiments can be performed using X-rays, or electrons accelerated through appropriate voltage. Which probe has greater energy? (For quantitative comparison, take the wavelength of the probe equal to 1 Å, which is of the order of inter-atomic spacing in the lattice) (me=9.11 × 10–31 kg).

11.32 (a) Obtain the de Broglie wavelength of a neutron of kinetic energy 150 eV. As you have seen in Exercise

Question 11.31, an electron beam of this energy is suitable for crystal diffraction experiments. Would a neutron beam of the same energy be equally suitable ? Explain. (mn = 1.675 × 10–27 kg) (b) Obtain the de Broglie wavelength associated with thermal neutrons at room temperature (27 ºC). Hence explain why a fast neutron beam needs to be thermalised with the environment before it can be used for neutron diffraction experiments.

Question 11.33 An electron microscope uses electrons accelerated by a voltage of 50 kV. Determine the de Broglie wavelength associated with the electrons. If other factors (such as numerical aperture, etc.) are taken to be roughly the same, how does the resolving power of an electron microscope compare with that of an optical microscope which uses yellow light?

Question 11.34 The wavelength of a probe is roughly a measure of the size of a structure that it can probe in some detail. The quark structure of protons and neutrons appears at the minute length-scale of 10–15 m or less. This structure was first probed in early 1970’s using high energy electron beams produced by a linear accelerator at Stanford, USA. Guess what might have been the order of energy of these electron beams. (Rest mass energy of electron = 0.511 MeV.)

Question 11.35 Find the typical de Broglie wavelength associated with a He atom in helium gas at room temperature (27 ºC) and 1 atm pressure; and compare it with the mean separation between two atoms under these conditions.

Question 11.36 Compute the typical de Broglie wavelength of an electron in a metal at 27 ºC and compare it with the mean separation between two electrons in a metal which is given to be about 2 × 10–10 m. [Note:

Exercises 11.35 and 11.36 reveal that while the wave-packets associated with gaseous molecules under ordinary conditions are non-overlapping, the electron wave-packets in a metal strongly overlap with one another. This suggests that whereas molecules in an ordinary gas can be distinguished apart, electrons in a metal cannot be distintguished apart from one another. This indistinguishibility has many fundamental implications which you will explore in more advanced Physics courses.]

Question 11.37 Answer the following questions:

(a) Quarks inside protons and neutrons are thought to carry fractional charges [(+2/3)e ; (–1/3)e]. Why do they not show up in Millikan’s oil-drop experiment?
(b) What is so special about the combination e/m? Why do we not simply talk of e and m separately?
(c) Why should gases be insulators at ordinary pressures and start conducting at very low pressures?
(d) Every metal has a definite work function. Why do all photoelectrons not come out with the same energy if incident radiation is monochromatic? Why is there an energy distribution of photoelectrons?
(e) The energy and momentum of an electron are related to the frequency and wavelength of the associated matter wave by the relations: E = h ν, p = λ h But while the value of λ is physically significant, the value of ν (and therefore, the value of the phase speed ν λ) has no physical significance. Why?

:: Chapter 12 - Atoms ::

Question 12.1 Choose the correct alternative from the clues given at the end of the each statement:

(a) The size of the atom in Thomson’s model is .......... the atomic size in Rutherford’s model. (much greater than/no different from/much less than.)
(b) In the ground state of .......... electrons are in stable equilibrium, while in .......... electrons always experience a net force. (Thomson’s model/ Rutherford’s model.)
(c) A classical atom based on .......... is doomed to collapse. (Thomson’s model/ Rutherford’s model.)
(d) An atom has a nearly continuous mass distribution in a .......... but has a highly non-uniform mass distribution in .......... (Thomson’s model/ Rutherford’s model.)
(e) The positively charged part of the atom possesses most of the mass in .......... (Rutherford’s model/both the models.)

Question 12.2 Suppose you are given a chance to repeat the alpha-particle scattering experiment using a thin sheet of solid hydrogen in place of the gold foil. (Hydrogen is a solid at temperatures below 14 K.) What results do you expect?

Question 12.3 What is the shortest wavelength present in the Paschen series of spectral lines?

Question 12.4 A difference of 2.3 eV separates two energy levels in an atom. What is the frequency of radiation emitted when the atom make a transition from the upper level to the lower level?

Question 12.5 The ground state energy of hydrogen atom is –13.6 eV. What are the kinetic and potential energies of the electron in this state?

Question 12.6 A hydrogen atom initially in the ground level absorbs a photon, which excites it to the n = 4 level. Determine the wavelength and frequency of photon.

Question 12.7 (a) Using the Bohr’s model calculate the speed of the electron in a hydrogen atom in the n = 1, 2, and 3 levels. (b) Calculate the orbital period in each of these levels.

Question 12.8 The radius of the innermost electron orbit of a hydrogen atom is 5.3×10–11 m. What are the radii of the n = 2 and n =3 orbits?

Question 12.9 A12.5 eV electron beam is used to bombard gaseous hydrogen at room temperature. What series of wavelengths will be emitted?

Question 12.10 In accordance with the Bohr’s model, find the quantum number that characterises the earth’s revolution around the sun in an orbit of radius 1.5 × 1011 m with orbital speed 3 × 104 m/s. (Mass of earth = 6.0 × 1024 kg.)

ADDITIONAL EXERCISES QUESTIONS

Question 12.11 Answer the following questions, which help you understand the difference between Thomson’s model and Rutherford’s model better.

(a) Is the average angle of deflection of α-particles by a thin gold foil predicted by Thomson’s model much less, about the same, or much greater than that predicted by Rutherford’s model?
(b) Is the probability of backward scattering (i.e., scattering of α-particles at angles greater than 90°) predicted by Thomson’s model much less, about the same, or much greater than that predicted by Rutherford’s model?
(c) Keeping other factors fixed, it is found experimentally that for small thickness t, the number of α-particles scattered at moderate angles is proportional to t. What clue does this linear dependence on t provide?
(d) In which model is it completely wrong to ignore multiple scattering for the calculation of average angle of scattering of α-particles by a thin foil?

Question 12.12 The gravitational attraction between electron and proton in a hydrogen atom is weaker than the coulomb attraction by a factor of about 10–40. An alternative way of looking at this fact is to estimate the radius of the first Bohr orbit of a hydrogen atom if the electron and proton were bound by gravitational attraction. You will find the answer interesting.

Question 12.13 Obtain an expression for the frequency of radiation emitted when a hydrogen atom de-excites from level n to level (n–1). For large n, show that this frequency equals the classical frequency of revolution of the electron in the orbit.

Question 12.14 Classically, an electron can be in any orbit around the nucleus of an atom. Then what determines the typical atomic size? Why is an atom not, say, thousand times bigger than its typical size? The question had greatly puzzled Bohr before he arrived at his famous model of the atom that you have learnt in the text. To simulate what he might well have done before his discovery, let us play as follows with the basic constants of nature and see if we can get a quantity with the dimensions of length that is roughly equal to the known size of an atom (~ 10–10m).

(a) Construct a quantity with the dimensions of length from the fundamental constants e, me, and c. Determine its numerical value.
(b) You will find that the length obtained in (a) is many orders of magnitude smaller than the atomic dimensions. Further, it involves c. But energies of atoms are mostly in non-relativistic domain where c is not expected to play any role. This is what may have suggested Bohr to discard c and look for ‘something else’ to get the right atomic size. Now, the Planck’s constant h had already made its appearance elsewhere. Bohr’s great insight lay in recognising that h, me, and e will yield the right atomic size. Construct a quantity with the dimension of length from h, me, and e and confirm that its numerical value has indeed the correct order of magnitude.

Question 12.15 The total energy of an electron in the first excited state of the hydrogen atom is about –3.4 eV.

(a) What is the kinetic energy of the electron in this state?
(b) What is the potential energy of the electron in this state?
(c) Which of the answers above would change if the choice of the zero of potential energy is changed?

Question 12.16 If Bohr’s quantisation postulate (angular momentum = nh/2π) is a basic law of nature, it should be equally valid for the case of planetary motion also. Why then do we never speak of quantisation of orbits of planets around the sun?

Question 12.17 Obtain the first Bohr’s radius and the ground state energy of a muonic hydrogen atom [i.e., an atom in which a negatively charged muon (μ–) of mass about 207me orbits around a proton].

:: Chapter 13 - Nuclei ::

Question 13.1 (a) Two stable isotopes of lithium 6 3 Li and 7 3 Li have respective abundances of 7.5% and 92.5%. These isotopes have masses 6.01512 u and 7.01600 u, respectively. Find the atomic mass of lithium.
(b) Boron has two stable isotopes, 10 5B and 11 5B. Their respective masses are 10.01294 u and 11.00931 u, and the atomic mass of boron is 10.811 u. Find the abundances of 10 5B and 11 5 B .

Question 13.2 The three stable isotopes of neon: 20 21 22 10 10 10 Ne, Ne and Ne have respective abundances of 90.51%, 0.27% and 9.22%. The atomic masses of the three isotopes are 19.99 u, 20.99 u and 21.99 u, respectively. Obtain the average atomic mass of neon.

Question 13.3 Obtain the binding energy (in MeV) of a nitrogen nucleus (14 ) 7N , given m (14 ) 7N =14.00307 u

Question 13.4 Obtain the binding energy of the nuclei 56 26Fe and 209 83 Bi in units of MeV from the following data:
m ( 56 26Fe ) = 55.934939 u m ( 209 83 Bi ) = 208.980388 u

Question 13.5 A given coin has a mass of 3.0 g. Calculate the nuclear energy that would be required to separate all the neutrons and protons from each other. For simplicity assume that the coin is entirely made of 63 29Cu atoms (of mass 62.92960 u).

Question 13.6 Write nuclear reaction equations for (i) α-decay of 226 88 Ra (ii) α-decay of 242 94 Pu (iii) β–-decay of 32 15 P (iv) β–-decay of 210 83 Bi (v) β+-decay of 11 6 C (vi) β+-decay of 97 43 Tc (vii) Electron capture of 120 54 Xe

Question 13.7 A radioactive isotope has a half-life of T years. How long will it take the activity to reduce to a) 3.125%, b) 1% of its original value?

Question 13.8 The normal activity of living carbon-containing matter is found to be about 15 decays per minute for every gram of carbon. This activity arises from the small proportion of radioactive 14 6C present with the stable carbon isotope 12 6C . When the organism is dead, its interaction with the atmosphere (which maintains the above equilibrium activity) ceases and its activity begins to drop. From the known half-life (5730 years) of 14 6C , and the measured activity, the age of the specimen can be approximately estimated. This is the principle of 14 6C dating used in archaeology. Suppose a specimen from Mohenjodaro gives an activity of 9 decays per minute per gram of carbon. Estimate the approximate age of the Indus-Valley civilisation.

Question 13.9 Obtain the amount of 60 27Co necessary to provide a radioactive source of 8.0 mCi strength. The half-life of 60 27Co is 5.3 years.

Question 13.10 The half-life of 90 38Sr is 28 years. What is the disintegration rate of 15 mg of this isotope?

Question 13.11 Obtain approximately the ratio of the nuclear radii of the gold isotope 197 79 Au and the silver isotope 107 47 Ag .

Question 13.12 Find the Q-value and the kinetic energy of the emitted α-particle in the α-decay of (a) 226 88 Ra and (b) 220 86 Rn . Given m ( 226 88 Ra ) = 226.02540 u, m ( 222 86 Rn ) = 222.01750 u, m ( 222 86 Rn ) = 220.01137 u, m ( 216 84 Po ) = 216.00189 u.

Question 13.13 The radionuclide 11C decays according to 11 11 + 6C → 5 B+e +ν : T1/2=20.3 min The maximum energy of the emitted positron is 0.960 MeV. Given the mass values: m ( 11 6C) = 11.011434 u and m ( 11 6B ) = 11.009305 u, calculate Q and compare it with the maximum energy of the positron emitted.

Question 13.14 The nucleus 23 10 Ne decays by β– emission. Write down the β-decay equation and determine the maximum kinetic energy of the electrons emitted. Given that: m ( 23 10 Ne ) = 22.994466 u m ( 23 11 Na ) = 22.089770 u.

Question 13.15 The Q value of a nuclear reaction A + b → C + d is defined by Q = [ mA + mb – mC – md]c2 where the masses refer to the respective nuclei. Determine from the given data the Q-value of the following reactions and state whether the reactions are exothermic or endothermic. (i) 1 3 2 2 1 1 1 1 H+ H → H+ H (ii) 12 12 20 4 6 6 10 2 C+ C → Ne+ He Atomic masses are given to be m ( 2 1H) = 2.014102 u m ( 3 1H) = 3.016049 u m ( 12 6C ) = 12.000000 u m ( 20 10 Ne ) = 19.992439 u

Question 13.16 Suppose, we think of fission of a 56 26Fe nucleus into two equal fragments, 28 13 Al . Is the fission energetically possible? Argue by working out Q of the process. Given m ( 56 26Fe ) = 55.93494 u and m ( 28 13 Al ) = 27.98191 u.

Question 13.17 The fission properties of 239 94 Pu are very similar to those of 235 92 U. The average energy released per fission is 180 MeV. How much energy, in MeV, is released if all the atoms in 1 kg of pure 239 94 Pu undergo fission?

Question 13.18 A 1000 MW fission reactor consumes half of its fuel in 5.00 y. How much 235 92 U did it contain initially? Assume that the reactor operates 80% of the time, that all the energy generated arises from the fission of 235 92 U and that this nuclide is consumed only by the fission process.

Question 13.19 How long can an electric lamp of 100W be kept glowing by fusion of 2.0 kg of deuterium? Take the fusion reaction as 2 2 3 1H+ 1H→ 2He+n+3.27 MeV?

Question 13.20 Calculate the height of the potential barrier for a head on collision of two deuterons. (Hint: The height of the potential barrier is given by the Coulomb repulsion between the two deuterons when they just touch each other. Assume that they can be taken as hard spheres of radius 2.0 fm.)

Question 13.21 From the relation R = R0A1/3, where R0 is a constant and A is the mass number of a nucleus, show that the nuclear matter density is nearly constant (i.e. independent of A).

Question 13.22 For the β+ (positron) emission from a nucleus, there is another competing process known as electron capture (electron from an inner orbit, say, the K–shell, is captured by the nucleus and a neutrino is emitted). 1 A A Z Z e+ X Y ν − + → + Show that if β+ emission is energetically allowed, electron capture is necessarily allowed but not vice–versa.

ADDITIONAL EXERCISES QUESTIONS

Question 13.23 In a periodic table the average atomic mass of magnesium is given as 24.312 u. The average value is based on their relative natural abundance on earth. The three isotopes and their masses are 24 12Mg (23.98504u), 25 12Mg (24.98584u) and 26 12Mg (25.98259u). The natural abundance of 24 12Mg is 78.99% by mass. Calculate the abundances of other two isotopes.

Question 13.24 The neutron separation energy is defined as the energy required to remove a neutron from the nucleus. Obtain the neutron separation energies of the nuclei 41 20Ca and 27 13 Al from the following data: m( 40 20Ca ) = 39.962591 u m( 41 20Ca ) = 40.962278 u m( 26 13 Al ) = 25.986895 u m( 27 13 Al ) = 26.981541 u?

Question 13.25 A source contains two phosphorous radio nuclides 32 15P (T1/2 = 14.3d) and 33 15P (T1/2 = 25.3d). Initially, 10% of the decays come from 33 15P . How long one must wait until 90% do so?

Question 13.26 Under certain circumstances, a nucleus can decay by emitting a particle more massive than an α-particle. Consider the following decay processes: 223 209 14 88 82 6 Ra→ Pb + C 223 219 4 88Ra→ 86Rn + 2He Calculate the Q-values for these decays and determine that both are energetically allowed.

Question 13.27 Consider the fission of 238 92U by fast neutrons. In one fission event, no neutrons are emitted and the final end products, after the beta decay of the primary fragments, are 140 58Ce and 99 44Ru . Calculate Q for this fission process. The relevant atomic and particle masses are m( 238 92U ) =238.05079 u m( 140 58Ce ) =139.90543 u m( 99 44Ru ) = 98.90594 u

Question 13.28 Consider the D–T reaction (deuterium–tritium fusion) 2 3 4 1 1 2 H+ H→ He + n (a) Calculate the energy released in MeV in this reaction from the data: m( 2 1H )=2.014102 u m( 3 1H ) =3.016049 u (b) Consider the radius of both deuterium and tritium to be approximately 2.0 fm. What is the kinetic energy needed to overcome the coulomb repulsion between the two nuclei? To what temperature must the gas be heated to initiate the reaction? (Hint: Kinetic energy required for one fusion event =average thermal kinetic energy available with the interacting particles = 2(3kT/2); k = Boltzman’s constant, T = absolute temperature.)

Question 13.29 Obtain the maximum kinetic energy of β-particles, and the radiation frequencies of γ decays in the decay scheme shown in Fig. 13.6. You are given that m(198Au) = 197.968233 u m(198Hg) =197.966760 u

Question 13.30 Calculate and compare the energy released by a) fusion of 1.0 kg of hydrogen deep within Sun and b) the fission of 1.0 kg of 235U in a fission reactor.

Question 13.31 Suppose India had a target of producing by 2020 AD, 200,000 MW of electric power, ten percent of which was to be obtained from nuclear power plants. Suppose we are given that, on an average, the efficiency of utilization (i.e. conversion to electric energy) of thermal energy produced in a reactor was 25%. How much amount of fissionable uranium would our country need per year by 2020? Take the heat energy per fission of 235U to be about 200MeV.

:: Chapter 14 - Semiconductor Electronics: Materials, Devices and Simple Circuits ::

Question 14.1 In an n-type silicon, which of the following statement is true:
(a) Electrons are majority carriers and trivalent atoms are the dopants.
(b) Electrons are minority carriers and pentavalent atoms are the dopants.
(c) Holes are minority carriers and pentavalent atoms are the dopants.
(d) Holes are majority carriers and trivalent atoms are the dopants.

Question 14.2 Which of the statements given in Exercise 14.1 is true for p-type semiconductos.

Question 14.3 Carbon, silicon and germanium have four valence electrons each. These are characterised by valence and conduction bands separated by energy band gap respectively equal to (Eg)C, (Eg)Si and (Eg)Ge. Which of the following statements is true?

(a) (Eg)Si < (Eg)Ge < (Eg)C (b) (Eg)C < (Eg)Ge > (Eg)Si (c) (Eg)C > (Eg)Si > (Eg)Ge (d) (Eg)C = (Eg)Si = (Eg)Ge

Question 14.4 In an unbiased p-n junction, holes diffuse from the p-region to n-region because
(a) free electrons in the n-region attract them.
(b) they move across the junction by the potential difference .
(c) hole concentration in p-region is more as compared to n-region.
(d) All the above.

Question 14.5 When a forward bias is applied to a p-n junction, it (a) raises the potential barrier.
(b) reduces the majority carrier current to zero.
(c) lowers the potential barrier.
(d) None of the above.

Question 14.6 For transistor action, which of the following statements are correct:
(a) Base, emitter and collector regions should have similar size and doping concentrations.
(b) The base region must be very thin and lightly doped.
(c) The emitter junction is forward biased and collector junction is reverse biased.
(d) Both the emitter junction as well as the collector junction are forward biased.

Question 14.7 For a transistor amplifier, the voltage gain (a) remains constant for all frequencies.
(b) is high at high and low frequencies and constant in the middle frequency range.
(c) is low at high and low frequencies and constant at mid frequencies.
(d) None of the above.

Question 14.8 In half-wave rectification, what is the output frequency if the input frequency is 50 Hz. What is the output frequency of a full-wave rectifier for the same input frequency.

Question 14.9 For a CE-transistor amplifier, the audio signal voltage across the collected resistance of 2 kΩ is 2 V. Suppose the current amplification factor of the transistor is 100, find the input signal voltage and base current, if the base resistance is 1 kΩ.

Question 14.10 Two amplifiers are connected one after the other in series (cascaded). The first amplifier has a voltage gain of 10 and the second has a voltage gain of 20. If the input signal is 0.01 volt, calculate the output ac signal.

Question 14.11 A p-n photodiode is fabricated from a semiconductor with band gap of 2.8 eV. Can it detect a wavelength of 6000 nm?

ADDITIONAL EXERCISES QUESTIONS

Question 14.12 The number of silicon atoms per m3 is 5 × 1028. This is doped simultaneously with 5 × 1022 atoms per m3 of Arsenic and 5 × 1020 per m3 atoms of Indium. Calculate the number of electrons and holes. Given that ni = 1.5 × 1016 m–3. Is the material n-type or p-type?

Question 14.13 In an intrinsic semiconductor the energy gap Eg is 1.2eV. Its hole mobility is much smaller than electron mobility and independent of temperature. What is the ratio between conductivity at 600K and that at 300K? Assume that the temperature dependence of intrinsic carrier concentration ni is given by 0 exp – 2 g i B E n n k T = where n0 is a constant.

Question 14.14 In a p-n junction diode, the current I can be expressed as 0 exp – 1 2 B eV I I k T where I0 is called the reverse saturation current, V is the voltage across the diode and is positive for forward bias and negative for reverse bias, and I is the current through the diode, kB is the Boltzmann constant (8.6×10–5 eV/K) and T is the absolute temperature. If for a given diode I0 = 5 × 10–12 A and T = 300 K, then (a) What will be the forward current at a forward voltage of 0.6 V?

(b) What will be the increase in the current if the voltage across the diode is increased to 0.7 V?
(c) What is the dynamic resistance?
(d) What will be the current if reverse bias voltage changes from 1 V to 2 V?

Question 14.15 You are given the two circuits as shown in Fig.14.44. Show that circuit (a) acts as OR gate while the circuit (b) acts as AND gate.

Question 14.16 Write the truth table for a NAND gate connected as given in Fig. 14.45.Hence identify the exact logic operation carried out by this circuit.

Question 14.17 You are given two circuits as shown in Fig. 14.46, which consist of NAND gates. Identify the logic operation carried out by the two circuits.

Question 14.18 Write the truth table for circuit given in Fig. 14.47 below consisting of NOR gates and identify the logic operation (OR, AND, NOT) which this circuit is performing. (Hint: A = 0, B = 1 then A and B inputs of second NOR gate will be 0 and hence Y=1. Similarly work out the values of Y for other combinations of A and B. Compare with the truth table of OR, AND, NOT gates and find the correct one.)

Question 14.19 Write the truth table for the circuits given in Fig. 14.48 consisting of NOR gates only. Identify the logic operations (OR, AND, NOT) performed by the two circuits.

:: Chapter 15 - Communication Systems ::

Question 15.1 Which of the following frequencies will be suitable for beyond-thehorizon communication using sky waves? (a) 10 kHz (b) 10 MHz (c) 1 GHz (d) 1000 GHz

Question 15.2 Frequencies in the UHF range normally propagate by means of:

(a) Ground waves.
(b) Sky waves.
(c) Surface waves.
(d) Space waves.

Question 15.3 Digital signals (i) do not provide a continuous set of values, (ii) represent values as discrete steps, (iii) can utilize binary system, and (iv) can utilize decimal as well as binary systems. Which of the above statements are true?

(a) (i) and (ii) only
(b) (ii) and (iii) only
(c) (i), (ii) and (iii) but not (iv)
(d) All of (i), (ii), (iii) and (iv).

Question 15.4 Is it necessary for a transmitting antenna to be at the same height as that of the receiving antenna for line-of-sight communication? A TV transmitting antenna is 81m tall. How much service area can it cover if the receiving antenna is at the ground level?

Question 15.5 A carrier wave of peak voltage 12V is used to transmit a message signal. What should be the peak voltage of the modulating signal in order to have a modulation index of 75%?

Question 15.6 A modulating signal is a square wave, as shown in Fig. 15..14. The carrier wave is given by c (t ) = 2sin(8πt ) volts. (i) Sketch the amplitude modulated waveform (ii) What is the modulation index?

Question 15.7 For an amplitude modulated wave, the maximum amplitude is found to be 10V while the minimum amplitude is found to be 2V. Determine the modulation index, μ. What would be the value of μ if the minimum amplitude is zero volt?

Question 15.8 Due to economic reasons, only the upper sideband of an AM wave is transmitted, but at the receiving station, there is a facility for generating the carrier. Show that if a device is available which can multiply two signals, then it is possible to recover the modulating signal at the receiver station

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