Maths and Physics

7/17/2019 Maths and Physics

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1.1 Overview

1.1.1 Relation

A relation R from a non-empty set A to a non empty set B is a subset of the Cartesian product A × B. The set of all first elements of the ordered pairs in a relation R from a

set A to a set B is called the domain of the relation R. The set of all second elements in

a relation R from a set A to a set B is called the range of the relation R. The whole set

B is called the codomain of the relation R. Note that range is always a subset of

codomain.

1.1.2 Types of Relati ons

A relation R in a set A is subset of A × A. Thus empty set φ and A × A are two extreme

relations.

(i) A relation R in a set A is called empty relation, if no element of A is related to anyelement of A, i.e., R = φ ⊂

A × A.

(ii) A relation R in a set A is called universal relation, if each element of A is related

to every element of A, i.e., R = A × A.

(iii) A relation R in A is said to be reflexive if aR a for all a∈A, R is symmetric if

aR b ⇒ bR a, ∀ a, b ∈ A and it is said to be transitive if aR b and bR c ⇒ aR c

∀ a, b, c ∈ A. Any relation which is reflexive, symmetric and transitive is calledan equivalence relation.

Note: An important property of an equivalence relation is that it divides the set

into pairwise disjoint subsets called equivalent classes whose collection is calleda partition of the set. Note that the union of all equivalence classes gives

the whole set.

1.1.3 Types of Functions

(i) A function f : X → Y is defined to be one-one (or injective), if the images of

distinct elements of X under f are distinct, i.e.,

x1, x

2 ∈ X, f ( x

1) = f ( x

2) ⇒ x

1 = x

2.

(ii) A function f : X → Y is said to be onto (or surjective), if every element of Y is the

image of some element of X under f , i.e., for every y ∈ Y there exists an element

x ∈ X such that f ( x) = y.

Chapter 1RELATIONS AND FUNCTIONS



2 MATHEMATICS

(iii) A function f : X → Y is said to be one-one and onto (or bijective), if f is both one-

one and onto.

1.1.4 Compositi on of Functions

(i) Let f : A → B and g : B → C be two functions. Then, the composition of f and

g , denoted by g o f , is defined as the function g o f : A → C given by

g o f ( x) = g ( f ( x)), ∀ x ∈ A.

(ii) If f : A → B and g : B → C are one-one, then g o f : A → C is also one-one

(iii) If f : A → B and g : B → C are onto, then g o f : A → C is also onto.

However, converse of above stated results (ii) and (iii) need not be true. Moreover,we have the following results in this direction.

(iv) Let f : A → B and g : B → C be the given functions such that g o f is one-one.

Then f is one-one.

(v) Let f : A → B and g : B → C be the given functions such that g o f is onto. Then

g is onto.

1.1.5 I nverti ble Function

(i) A function f : X → Y is defined to be invertible, if there exists a function

g : Y → X such that g o f = Ix

and f o g = IY

. The function g is called the inverse

of f and is denoted by f – 1.

(ii) A function f : X → Y is invertible if and only if f is a bijective function.

(iii) I f f : X → Y, g : Y → Z and h : Z → S are functions, then

h o ( g o f ) = (h o g ) o f .

(iv) Let f : X → Y and g : Y → Z be two invertible functions. Then g o f is also

invertible with ( g o f ) –1 = f –1 o g –1.

1.1.6 Binary Operations

(i) A binary operation * on a set A is a function * : A × A → A. We denote * (a, b)

by a * b.

(ii) A binary operation * on the set X is called commutative, if a * b = b * a for every

a, b ∈ X.

(iii) A binary operation * : A × A → A is said to be associative if

(a * b) * c = a * (b * c), for every a, b, c ∈ A.

(iv) Given a binary operation * : A × A → A, an element e ∈ A, if it exists, is called

identity for the operation *, if a * e = a = e * a, ∀ a ∈ A.



RELATIONS AND FUNCTIONS 3

(v) Given a binary operation*

: A × A → A, with the identity element e in A, an

element a ∈ A, is said to be invertible with respect to the operation *, if there

exists an element b in A such that a * b = e = b * a and b is called the inverse of

a and is denoted by a –1.

1.2 Solved Examples

Short Answer (S.A.)

Example 1 Let A = {0, 1, 2, 3} and define a relation R on A as follows:

R = {(0, 0), (0, 1), (0, 3), (1, 0), (1, 1), (2, 2), (3, 0), (3, 3)}.

Is R reflexive? symmetric? transitive?

Solution R is reflexive and symmetric, but not transitive since for (1, 0) ∈ R and

(0, 3) ∈ R whereas (1, 3) ∉

R.

Example 2 For the set A = {1, 2, 3}, define a relation R in the set A as follows:

R = {(1, 1), (2, 2), (3, 3), (1, 3)}.

Write the ordered pairs to be added to R to make it the smallest equivalence relation.

Solution (3, 1) is the single ordered pair which needs to be added to R to make it the

smallest equivalence relation.

Example 3 Let R be the equivalence relation in the set Z of integers given by

R = {(a, b) : 2 divides a – b}. Write the equivalence class [0].

Solution [0] = {0, ± 2, ± 4, ± 6,...}

Example 4 Let the function f : R → R be defined by f ( x) = 4 x – 1, ∀ x ∈ R . Then,

show that f is one-one.

Solution For any two elements x1, x

2 ∈ R such that f ( x

1) = f ( x

2), we have

4 x1 – 1 = 4 x2 – 1

⇒ 4 x1 = 4 x

2, i.e., x

1 = x

2

Hence f is one-one.

Example 5 If f = {(5, 2), (6, 3)}, g = {(2, 5), (3, 6)}, write f o g .

Solution f o g = {(2, 2), (3, 3)}

Example 6 Let f : R → R be the function defined by f ( x) = 4 x – 3 ∀ x ∈ R . Then

write f –1.



4 MATHEMATICS

Solution Given that f ( x) = 4 x – 3 = y (say), then

4 x = y + 3

⇒ x =3

4

y +

Hence f –1 ( y) =3

4

y +⇒ f –1 ( x) =

3

4

y +

Example 7 Is the binary operation * defined on Z (set of integer) by

m * n = m – n + mn ∀ m, n ∈ Z commutative?

Solution No. Since for 1, 2 ∈ Z, 1 * 2 = 1 – 2 + 1.2 = 1 while 2 * 1 = 2 – 1 + 2.1 = 3

so that 1 * 2 ≠ 2 * 1.

Example 8 If f = {(5, 2), (6, 3)} and g = {(2, 5), (3, 6)}, write the range of f and g .

Solution The range of f = {2, 3} and the range of g = {5, 6}.

Example 9 If A = {1, 2, 3} and f , g are relations corresponding to the subset of A × A

indicated against them, which of f , g is a function? Why?

f = {(1, 3), (2, 3), (3, 2)} g = {(1, 2), (1, 3), (3, 1)}

Solution f is a function since each element of A in the first place in the ordered pairs

is related to only one element of A in the second place while g is not a function because

1 is related to more than one element of A, namely, 2 and 3.

Example 10 If A = {a, b, c, d } and f = {a, b), (b, d ), (c, a), (d , c)}, show that f is one-

one from A onto A. Find f –1.

Solution f is one-one since each element of A is assigned to distinct element of the set

A. Also, f is onto since f (A) = A. Moreover, f –1 = {(b, a), (d , b), (a, c), (c, d )}.

Example 11 In the set N of natural numbers, define the binary operation * by m * n =

g.c.d (m, n), m, n ∈ N. Is the operation * commutative and associative?

Solution The operation is clearly commutative since

m * n = g.c.d (m, n) = g.c.d (n, m) = n * m ∀ m, n ∈ N.

It is also associative because for l , m, n ∈ N, we have

l * (m * n) = g. c. d (l , g.c.d (m, n))

= g.c.d . ( g. c. d (l , m), n)

= (l * m)

* n.




Long Answer (L.A.)Example 12 In the set of natural numbers N, define a relation R as follows:

∀ n, m ∈ N, nR m if on division by 5 each of the integers n and m leaves the remainder

less than 5, i.e. one of the numbers 0, 1, 2, 3 and 4. Show that R is equivalence relation.

Also, obtain the pairwise disjoint subsets determined by R.

Solution R is reflexive since for each a ∈ N, aR a. R is symmetric since if aR b, then

bR a for a, b ∈ N. Also, R is transitive since for a, b, c ∈ N, if aR b and bR c, then aR c.

Hence R is an equivalence relation in N which will partition the set N into the pairwise

disjoint subsets. The equivalent classes are as mentioned below:

A0 = {5, 10, 15, 20 ...}A

1 = {1, 6, 11, 16, 21 ...}

A2 = {2, 7, 12, 17, 22, ...}

A3 = {3, 8, 13, 18, 23, ...}

A4 = {4, 9, 14, 19, 24, ...}

It is evident that the above five sets are pairwise disjoint and

A0 ∪ A

1 ∪ A

2 ∪ A

3 ∪ A

4 =

4

0Ai

i =∪ = N .

Example 13 Show that the function f : R → R defined by f ( x) = 2,

1

x x

x∀ ∈

+R , is

neither one-one nor onto.

Solution For x1, x

2 ∈ R , consider

f ( x1) = f ( x

2)

⇒ 1 2

2 21 21 1

x x

x x=

+ +

⇒ x1

22 x + x

1= x

2

21 x + x

2

⇒ x1 x

2( x

2 – x

1) = x

2 – x

1

⇒ x1= x

2or x

1 x

2 = 1

We note that there are point, x1 and x

2 with x

1≠ x

2 and f ( x

1) = f ( x

2), for instance, if

we take x1 = 2 and x

2=

1

2, then we have f ( x

1) =

2

5 and f ( x

2) =

2

5 but

12

2≠ . Hence

f is not one-one. Also, f is not onto for if so then for 1∈R ∃ x ∈ R such that f ( x) = 1



6 MATHEMATICS

which gives2

11

x x

=+

. But there is no such x in the domain R , since the equation

x2 – x + 1 = 0 does not give any real value of x.

Example 14 Let f , g : R → R be two functions defined as f ( x) = + x and

g ( x) = – x ∀ x ∈ R . Then, find f o g and g o f .

Solution Here f ( x) = + x which can be redefined as

f ( x) =2 if 0

0 if 0

x x

x

≥⎧⎨

<⎩

Similarly, the function g defined by g ( x) = x – x may be redefined as

g ( x) =0 if 0

–2 if 0

x

x x

≥⎧⎨

<⎩

Therefore, g o f gets defined as :For x ≥ 0, ( g o f ) ( x) = g ( f ( x) = g (2 x) = 0

and for x < 0, ( g o f ) ( x) = g ( f ( x) = g (0) = 0.

Consequently, we have ( g o f ) ( x) = 0, ∀ x ∈ R .

Similarly, f o g gets defined as:

For x ≥ 0, ( f o g ) ( x) = f ( g ( x) = f (0) = 0,

and for x < 0, ( f o g ) ( x) = f (g( x)) = f (–2 x) = – 4 x.

i.e.0, 0

( ) ( )4 , 0

x f o g x

x x

>⎧= ⎨

− <⎩

Example 15 Let R be the set of real numbers and f : R → R be the function defined

by f ( x) = 4 x + 5. Show that f is invertible and find f –1.

Solution Here the function f : R → R is defined as f ( x) = 4 x + 5 = y (say). Then

4 x = y – 5 or x =5

4

y −.




This leads to a function g : R → R defined as

g ( y) =5

4

y −.

Therefore, ( g o f ) ( x) = g ( f ( x) = g (4 x + 5)

=4 5 5

4

x + − = x

or g o f = IR

Similarly ( f o g ) ( y) = f ( g ( y))

=5

4

y f

−⎛ ⎞⎜ ⎟⎝ ⎠

=5

4 54

y −⎛ ⎞+⎜ ⎟

⎝ ⎠ = y

or f o g = IR .

Hence f is invertible and f –1 = g which is given by

f –1 ( x) =5

4

x −

Example 16 Let * be a binary operation defined on Q. Find which of the following

binary operations are associative

(i) a * b = a – b for a, b ∈ Q.

(ii) a * b =4

ab for a, b ∈ Q.

(iii) a * b = a – b + ab for a, b ∈ Q.

(iv) a * b = ab2 for a, b ∈ Q.

Solution

(i) * is not associative for if we take a = 1, b = 2 and c = 3, then

(a * b) * c = (1 * 2) * 3 = (1 – 2) * 3 = – 1 – 3 = – 4 and

a * (b * c) = 1 * (2 * 3) = 1 * (2 – 3) = 1 – ( – 1) = 2.



8 MATHEMATICS

Thus (a * b)

* c ≠ a

* (b

* c) and hence

* is not associative.

(ii) * is associative since Q is associative with respect to multiplication.

(iii) * is not associative for if we take a = 2, b = 3 and c = 4, then

(a * b) * c = (2 * 3) * 4 = (2 – 3 + 6) * 4 = 5 * 4 = 5 – 4 + 20 = 21, and

a * (b * c) = 2 * (3 * 4) = 2 * (3 – 4 + 12) = 2 * 11 = 2 – 11 + 22 = 13

Thus (a * b) * c ≠ a * (b * c) and hence * is not associative.

(iv) * is not associative for if we take a = 1, b = 2 and c = 3, then (a * b) * c =

(1 * 2) * 3 = 4 * 3 = 4 × 9 = 36 and a * (b * c) = 1 * (2 * 3) = 1 * 18 =

1 × 182 = 324.

Thus (a * b) * c ≠ a * (b * c) and hence * is not associative.

Objective Type Questions

Choose the correct answer from the given four options in each of the Examples 17 to 25.

Example 17 Let R be a relation on the set N of natural numbers defined by nR m if n

divides m. Then R is

(A) Reflexive and symmetric (B) Transitive and symmetric

(C) Equivalence (D) Reflexive, transitive but not

symmetricSolution The correct choice is (D).

Since n divides n, ∀ n ∈ N, R is reflexive. R is not symmetric since for 3, 6 ∈ N,

3 R 6 ≠ 6 R 3. R is transitive since for n, m, r whenever n/m and m/r ⇒ n/r , i.e., n

divides m and m divides r , then n will devide r .

Example 18 Let L denote the set of all straight lines in a plane. Let a relation R be

defined by l R m if and only if l is perpendicular to m ∀ l , m ∈ L. Then R is

(A) reflexive (B) symmetric

(C) transitive (D) none of these

Solution The correct choice is (B).

Example 19 Let N be the set of natural numbers and the function f : N → N be

defined by f (n) = 2n + 3 ∀ n ∈ N. Then f is

(A) surjective (B) injective

(C) bijective (D) none of these

Solution (B) is the correct option.

Example 20 Set A has 3 elements and the set B has 4 elements. Then the number of




injective mappings that can be defined from A to B is

(A) 144 (B) 12

(C) 24 (D) 64

Solution The correct choice is (C). The total number of injective mappings from the

set containing 3 elements into the set containing 4 elements is 4P3 = 4! = 24.

Example 21 Let f : R → R be defined by f ( x) = sin x and g : R → R be defined by

g ( x) = x2, then f o g is

(A) x2 sin x (B) (sin x)2

(C) sin x2 (D) 2

sin

Solution (C) is the correct choice.

Example 22 Let f : R → R be defined by f ( x) = 3 x – 4. Then f –1 ( x) is given by

(A)4

3

x +(B) – 4

3

x

(C) 3 x + 4 (D) None of these

Solution (A) is the correct choice.

Example 23 Let f : R → R be defined by f ( x) = x2 + 1. Then, pre-images of 17

and – 3, respectively, are

(A) φ, {4, – 4} (B) {3, – 3}, φ(C) {4, –4}, φ (D) {4, – 4, {2, – 2}

Solution (C) is the correct choice since for f –1 ( 17 ) = x ⇒ f ( x) = 17 or x2 + 1 = 17

⇒ x = ± 4 or f –1 ( 17 ) = {4, – 4} and for f –1 (–3) = x ⇒ f ( x) = – 3 ⇒ x2 + 1

= – 3 ⇒ x2 = – 4 and hence f –1 (– 3) = φ.

Example 24 For real numbers x and y, define xR y if and only if x – y + 2 is an

irrational number. Then the relation R is

(A) reflexive (B) symmetric

(C) transitive (D) none of these

Solution (A) is the correct choice.

Fill in the blanks in each of the Examples 25 to 30.

Example 25 Consider the set A = {1, 2, 3} and R be the smallest equivalence relation

on A, then R = ________



10 MATHEMATICS

Solution R = {(1, 1), (2, 2), (3, 3)}.

Example 26 The domain of the function f : R → R defined by f ( x) = 2 – 3 2 x x + is

________.

Solution Here x2 – 3 x + 2 ≥ 0

⇒ ( x – 1) ( x – 2) ≥ 0

⇒ x ≤ 1 or x ≥ 2

Hence the domain of f = (– ∞, 1] ∪ [2, ∞)

Example 27 Consider the set A containing n elements. Then, the total number of injective functions from A onto itself is ________.

Solution n!

Example 28 Let Z be the set of integers and R be the relation defined in Z such that

aR b if a – b is divisible by 3. Then R partitions the set Z into ________ pairwise

disjoint subsets.

Solution Three.

Example 29 Let R be the set of real numbers and * be the binary operation defined on

R as a *

b = a + b – ab

∀ a, b

∈ R . Then, the identity element with respect to the

binary operation * is _______.

Solution 0 is the identity element with respect to the binary operation *.

State True or False for the statements in each of the Examples 30 to 34.

Example 30 Consider the set A = {1, 2, 3} and the relation R = {(1, 2), (1, 3)}. R is a

transitive relation.

Solution True.

Example 31 Let A be a finite set. Then, each injective function from A into itself is not

surjective.Solution False.

Example 32 For sets A, B and C, let f : A → B, g : B → C be functions such that

g o f is injective. Then both f and g are injective functions.

Solution False.

Example 33 For sets A, B and C, let f : A → B, g : B → C be functions such that

g o f is surjective. Then g is surjective

Solution True.




Example 34 Let N be the set of natural numbers. Then, the binary operation* in N

defined as a * b = a + b, ∀ a, b ∈ N has identity element.

Solution False.

1.3 EXERCISE

Short Answer (S.A.)

1. Let A = {a, b, c} and the relation R be defined on A as follows:

R = {(a, a), (b, c), (a, b)}.

Then, write minimum number of ordered pairs to be added in R to make R

reflexive and transitive.

2. Let D be the domain of the real valued function f defined by f ( x) = 225 x− .

Then, write D.

3. Let f , g : R → R be defined by f ( x) = 2 x + 1 and g ( x) = x2 – 2, ∀ x ∈ R ,

respectively. Then, find g o f .

4. Let f : R →

R be the function defined by f ( x) = 2 x – 3 ∀ x ∈ R. write f –1.

5. If A = {a, b, c, d } and the function f = {(a, b), (b, d ), (c, a), (d , c)}, write f –1.

6. If f : R →

R is defined by f ( x) = x2 – 3 x + 2, write f ( f ( x)).

7. Is g = {(1, 1), (2, 3), (3, 5), (4, 7)} a function? If g is described by

g ( x) = α x + β, then what value should be assigned to α and β.

8. Are the following set of ordered pairs functions? If so, examine whether the

mapping is injective or surjective.

(i) {( x, y): x is a person, y is the mother of x}.

(ii){(a, b): a is a person, b is an ancestor of a}.

9. If the mappings f and g are given by

f = {(1, 2), (3, 5), (4, 1)} and g = {(2, 3), (5, 1), (1, 3)}, write f o g .

10. Let C be the set of complex numbers. Prove that the mapping f : C → R given by

f ( z ) = | z |, ∀ z ∈ C, is neither one-one nor onto.

11. Let the function f : R → R be defined by f ( x) = cos x, ∀ x ∈ R . Show that f is

neither one-one nor onto.

12. Let X = {1, 2, 3}and Y = {4, 5}. Find whether the following subsets of X ×Y are

functions from X to Y or not.

(i) f = {(1, 4), (1, 5), (2, 4), (3, 5)} (ii) g = {(1, 4), (2, 4), (3, 4)}

(iii) h = {(1,4), (2, 5), (3, 5)} (iv) k = {(1,4), (2, 5)}.

13. If functions f : A → B and g : B → A satisfy g o f = IA, then show that f is one-

one and g is onto.



12 MATHEMATICS

14. Let f : R → R be the function defined by f ( x) = 12–cos x

x R .Then, find

the range of f .

15. Let n be a fixed positive integer. Define a relation R in Z as follows: a, b Z ,

aR b if and only if a – b is divisible by n . Show that R is an equivalance relation.

Long Answer (L.A.)

16. If A = {1, 2, 3, 4 }, define relations on A which have properties of being:

(a) reflexive, transitive but not symmetric

(b) symmetric but neither reflexive nor transitive(c) reflexive, symmetric and transitive.

17. Let R be relation defined on the set of natural number N as follows:

R = {( x, y): x N, y N, 2 x + y = 41}. Find the domain and range of the

relation R. Also verify whether R is reflexive, symmetric and transitive.

18. Given A = {2, 3, 4}, B = {2, 5, 6, 7}. Construct an example of each of the

following:

(a) an injective mapping from A to B

(b) a mapping from A to B which is not injective(c) a mapping from B to A.

19. Give an example of a map

(i) which is one-one but not onto

(ii) which is not one-one but onto

(iii) which is neither one-one nor onto.

20. Let A = R – {3}, B = R – {1}. Let f : A → B be defined by f ( x) = – 2

–3

x

x A . Then show that f is bijective.21. Let A = [–1, 1]. Then, discuss whether the following functions defined on A are

one-one, onto or bijective:

(i) ( )2

x f x (ii) g ( x) = x

(iii) ( )h x x x (iv) k ( x) = x2.

22. Each of the following defines a relation on N:

(i) x is greater than y, x, y N

(ii) x + y = 10, x, y N




(iii) x y is square of an integer x, y

N

(iv) x + 4y = 10 x, y N.

Determine which of the above relations are reflexive, symmetric and transitive.

23. Let A = {1, 2, 3, ... 9} and R be the relation in A ×A defined by (a, b) R (c, d ) if

a + d = b + c for (a, b), (c, d ) in A ×A. Prove that R is an equivalence relation

and also obtain the equivalent class [(2, 5)].

24. Using the definition, prove that the function f : A → B is invertible if and only if

f is both one-one and onto.

25. Functions f , g : R → R are defined, respectively, by f ( x) = x2 + 3 x + 1,

g ( x) = 2 x – 3, find(i) f o g (ii) g o f (iii) f o f (iv) g o g

26. Let * be the binary operation defined on Q. Find which of the following binary

operations are commutative

(i) a * b = a – b a, b ∈ Q (ii) a * b = a2 + b2 a, b ∈ Q

(iii) a * b = a + ab a, b ∈ Q (iv) a * b = (a – b)2 a, b ∈ Q

27. Let * be binary operation defined on R by a * b = 1 + ab, a, b ∈ R . Then the

operation * is

(i) commutative but not associative(ii) associative but not commutative

(iii) neither commutative nor associative

(iv) both commutative and associative


Choose the correct answer out of the given four options in each of the Exercises from

28 to 47 (M.C.Q.).

28. Let T be the set of all triangles in the Euclidean plane, and let a relation R on T

be defined as aR b if a is congruent to b a, b ∈ T. Then R is

(A) reflexive but not transitive (B) transitive but not symmetric

(C) equivalence (D) none of these

29. Consider the non-empty set consisting of children in a family and a relation R

defined as aR b if a is brother of b. Then R is

(A) symmetric but not transitive (B) transitive but not symmetric

(C) neither symmetric nor transitive (D) both symmetric and transitive



14 MATHEMATICS

30. The maximum number of equivalence relations on the set A = {1, 2, 3} are

(A) 1 (B) 2

(C) 3 (D) 5

31. If a relation R on the set {1, 2, 3} be defined by R = {(1, 2)}, then R is

(A) reflexive (B) transitive

(C) symmetric (D) none of these

32. Let us define a relation R in R as aR b if a ≥ b. Then R is

(A) an equivalence relation (B) reflexive, transitive but not

symmetric

(C) symmetric, transitive but (D) neither transitive nor reflexive

not reflexive but symmetric.

33. Let A = {1, 2, 3} and consider the relation

R = {1, 1), (2, 2), (3, 3), (1, 2), (2, 3), (1,3)}.

Then R is

(A) reflexive but not symmetric (B) reflexive but not transitive

(C) symmetric and transitive (D) neither symmetric, nor transitive

34. The identity element for the binary operation * defined on Q ~ {0} as

a * b =2

ab a, b ∈ Q ~ {0} is

(A) 1 (B) 0

(C) 2 (D) none of these

35. If the set A contains 5 elements and the set B contains 6 elements, then thenumber of one-one and onto mappings from A to B is

(A) 720 (B) 120


36. Let A = {1, 2, 3, ...n} and B = {a, b}. Then the number of surjections from A into

B is

(A) nP2

(B) 2n – 2

(C) 2n – 1 (D) None of these




37. Let f : R → R be defined by f ( x) = 1 x

x ∈ R . Then f is

(A) one-one (B) onto

(C) bijective (D) f is not defined

38. Let f : R → R be defined by f ( x) = 3 x2 – 5 and g : R → R by g ( x) = 2 1

x

x +.

Then g o f is

(A)2

4 2

3 5

9 30 26

x

x x

−− +

(B)2

4 2

3 5

9 6 26

x

x x

−− +

(C)

2

4 2

3

2 4

x

x x+ −(D)

2

4 2

3

9 30 2

x

x x+ −

39. Which of the following functions from Z into Z are bijections?

(A) f ( x) = x3 (B) f ( x) = x + 2

(C) f ( x) = 2 x + 1 (D) f ( x) = x2 + 1

40. Let f : R → R be the functions defined by f ( x) = x3 + 5. Then f –1 ( x) is

(A)1

3( 5) x + (B)1

3( 5) x −

(C)1

3(5 )− (D) 5 – x

41. Let f : A →

B and g : B → C be the bijective functions. Then ( g o f ) –1 is

(A) f –1 o g –1 (B) f o g

(C) g –1 o f –1 (D) g o f

42. Let f :3

5

⎧ ⎫− ⎨ ⎬

⎩ ⎭R →

R be defined by f ( x) =3 2

5 3

x

x

+

− . Then

(A) f –1 ( x) = f ( x) (B) f –1 ( x) = – f ( x)

(C) ( f o f ) x = – x (D) f –1 ( x) =1

19 f ( x)

43. Let f : [0, 1] → [0, 1] be defined by f ( x) =, if isrational

1 , if isirrational

x x

x x

⎧⎨

−⎩



16 MATHEMATICS

Then ( f o f ) x is

(A) constant (B) 1 + x

(C) x (D) none of these

44. Let f : [2, ∞) → R be the function defined by f ( x) = x2 – 4 x + 5, then the range

of f is

(A) R (B) [1, ∞)

(C) [4, ∞) (B) [5, ∞)

45. Let f : N → R be the function defined by f ( x) =2 1

2

x − and g : Q → R be

another function defined by g ( x) = x + 2. Then ( g o f )3

2 is

(A) 1 (B) 1

(C)7

2(B) none of these

46. Let f : R → R be defined by

2

2 : 3

( ) :1 33 : 1

x x

f x x x x x

>⎧⎪

= < ≤⎨⎪ ≤⎩

Then f (– 1) + f (2) + f (4) is

(A) 9 (B) 14


47. Let f : R → R be given by f ( x) = tan x. Then f –1 (1) is

(A)4

π

(B) {n π +4

π

: n ∈ Z}

(C) does not exist (D) none of these

Fill in the blanks in each of the Exercises 48 to 52.

48. Let the relation R be defined in N by aR b if 2a + 3b = 30. Then R = ______ .

49. Let the relation R be defined on the set

A = {1, 2, 3, 4, 5} by R = {(a, b) : |a2 – b2| < 8. Then R is given by _______.

50. Let f = {(1, 2), (3, 5), (4, 1) and g = {(2, 3), (5, 1), (1, 3)}. Then g o f = ______

and f o g = ______ .




51. Let f : R → R be defined by ( )2

.1

x f x x

=+

Then ( f o f o f ) ( x) = _______

52. If f ( x) = (4 – ( x –7)3}, then f –1( x) = _______ .

State True or False for the statements in each of the Exercises 53 to 63.

53. Let R = {(3, 1), (1, 3), (3, 3)} be a relation defined on the set A = {1, 2, 3}. Then R

is symmetric, transitive but not reflexive.

54. Let f : R → R be the function defined by f ( x) = sin (3 x+2) x ∈ R. Then f is

invertible.55. Every relation which is symmetric and transitive is also reflexive.

56. An integer m is said to be related to another integer n if m is a integral multiple of

n. This relation in Z is reflexive, symmetric and transitive.

57. Let A = {0, 1} and N be the set of natural numbers. Then the mapping

f : N → A defined by f (2n –1) = 0, f (2n) = 1, n ∈ N, is onto.

58.The relation R on the set A = {1, 2, 3} defined as R = {{1, 1), (1, 2), (2, 1), (3, 3)}

is reflexive, symmetric and transitive.

59. The composition of functions is commutative.60. The composition of functions is associative.

61. Every function is invertible.

62. A binary operation on a set has always the identity element.



11.1 Overview

11.1.1 Direction cosines of a line are the cosines of the angles made by the line with

positive directions of the co-ordinate axes.

11.1.2 If l , m, n are the direction cosines of a line, then l 2 + m2 + n2 = 1

11.1.3 Direction cosines of a line joining two points P ( x1, y

1, z

1) and Q ( x

2, y

2, z

2) are

2 1 2 1 2 1, ,PQ PQ PQ

x y y z z − − −,

where 2 2 22 1 2 1 2 1PQ = ( – ) +( ) ( ) x y y z z − + −

11.1.4 Direction ratios of a line are the numbers which are proportional to the directioncosines of the line.

11.1.5 If l , m, n are the direction cosines and a, b, c are the direction ratios of a line,

then2 2 2 2 2 2 2 2 2

; ;a b c

l m na b c a b c a b c

11.1.6 Skew lines are lines in the space which are neither parallel nor interesecting.

They lie in the different planes.

11.1.7 Angle between skew lines is the angle between two intersecting lines drawnfrom any point (preferably through the origin) parallel to each of the skew lines.

11.1.8 If l 1, m

1, n

1 and l

2, m

2, n

2 are the direction cosines of two lines and θ is the

acute angle between the two lines, then

cosθ = 1 2 1 2 1 2l l m m n n+ +

11.1.9 If a1, b

1, c

1 and a

2, b

2, c

2 are the directions ratios of two lines and θ is the

acute angle between the two lines, then

Chapter 11THREE DIMENSIONAL GEOMETRY



THREE DIMENSIONAL GEOMETRY 221

1 2 1 2 1 2

2 2 2 2 2 21 2 3 1 2 3

cos.

a a b b c c

a a a b b b

+ +θ=+ + + +

11.1.10 Vector equation of a line that passes through the given point whose position

vector is a

and parallel to a given vector b

is r a b= +λ

.

11.1.11 Equation of a line through a point ( x1, y

1, z

1) and having directions cosines

l , m, n (or, direction ratios a, b and c) is

1 1 1 x y y z z

l m n

− − −= = or1 1 1 x x y y z z

a b c

− − −⎛ ⎞= =⎜ ⎟⎝ ⎠

.

11.1.12 The vector equation of a line that passes through two points whose positions

vectors are a

and b

is ( )r a b a= + λ −

.

11.1.13 Cartesian equation of a line that passes through two points ( x1, y

1, z

1) and

( x2, y

2, z

2) is

1 1 1

2 1 2 1 2 1

x x y y z z x x y y z z − − −= =− − − .

11.1.14 If θ is the acute angle between the lines1 1r a b= + λ

and

2 2r a b= + λ

, then

θ is given by1 2 1 2 –1

1 2 1 2

. .cos or cos

b b b b

b b b bθ= θ=

.

11.1.15 If 1 1 1

1 1 1

x x y y z z

l m n

− − −= = and 2 2 2

1 2 2

x y y z z

l m n

− − −= = are equations of two

lines, then the acute angle θ between the two lines is given by

cosθ = 1 2 1 2 1 2l l m m n n+ + .

11.1.16 The shortest distance between two skew lines is the length of the line segment

perpendicular to both the lines.

11.1.17 The shortest distance between the lines1 1r a b= + λ

and

2 2r a b= + λ

is



222 MATHEMATICS

1 2 2 1

1 2

. – b b a a

b b

.

11.1.18 Shortest distance between the lines:1 1 1

1 1 1

x y y z z

a b c

− − −= = and

2 2 2

2 2 2

x y y z z

a b c

− − −= = is

2 1 2 1 2 1

1 1 1

2 2 2

2 2 21 2 2 1 1 2 2 1 1 2 2 1( ) ( ) ( )

x x y y z z

a b c

a b c

b c b c c a c a a b a b

− − −

− + − + −

11.1.19 Distance between parallel lines1r a b

and

2r a b= + λ

is

2 1 – b a a

b

.

11.1.20 The vector equation of a plane which is at a distance p from the origin, where

n is the unit vector normal to the plane, is ˆ.r n p=

.

11.1.21 Equation of a plane which is at a distance p from the origin with direction

cosines of the normal to the plane as l , m, n is lx + my + nz = p.

11.1.22 The equation of a plane through a point whose position vector is a

and

perpendicular to the vector n

is ( – ). 0r a n =

or .r n d =

, where . .d a n=

11.1.23 Equation of a plane perpendicular to a given line with direction ratios a, b, c

and passing through a given point ( x1, y

1, z

1) is a ( x – x

1) + b ( y – y

1) + c ( z – z

1) = 0.

11.1.24 Equation of a plane passing through three non-collinear points ( x1, y

1, z

1),

( x2, y2, z 2) and ( x3, y3, z 3) is




1 1 1

2 1 2 1 2 1

3 1 3 1 3 1

– – –

– – – 0

– – –

x x y y z z

x x y y z z

x x y y z z

=.

11.1.25Vector equation of a plane that contains three non-collinear points having position

vectors a

, b

, c

is ( – ). ( – ) ( – ) 0r a b a c a⎡ ⎤× =⎣ ⎦

11.1.26Equation of a plane that cuts the co-ordinates axes at (a, 0, 0), (0, b, 0) and

(0, 0, c ) is 1 x y z a b c

+ + = .

11.1.27Vector equation of any plane that passes through the intersection of planes

1 1.r n d =

and 2 2.r n d =

is 1 1 2 2( . ) ( . ) 0r n d r n d − + λ − =

, where λ is any non-zero

constant.

11.1.28Cartesian equation of any plane that passes through the intersection of two

given planes A1 x + B

1 y + C

1 z + D

1 = 0 and A

2 x + B

2 y + C

2 z + D

2 = 0 is

(A1 x + B

1 y + C

1 z + D

1) + λ ( A

2 x + B

2 y + C

2 z + D

2) = 0.

11.1.29Two lines1 1r a b

and

2 2r a b= + λ

are coplanar if2 1 1 2( – ) . ( ) 0a a b b× =

11.1.30Two lines 1 1 1

1 1 1

– – – x x y y z z

a b c and 2 2 2

2 2 2

– – – x x y y z z

a b c are coplanar if

2 1 2 1 2 1

1 1 1

2 2 2

– – –

0

x x y y z z

a b c

a b c

=,

11.1.31In vector form, if θ is the acute angle between the two planes, 1 1.r n d =

and

2 2.r n d =

, then 1 2 –1

1 2

.cos

.

n n

n nθ=

11.1.32The acute angle θ between the line r a b

and plane .r n d =

is given by



224 MATHEMATICS

.sin

.

b n

b nθ=

.

11.2 Solved Examples

Short Answer (S.A.)

Example 1 If the direction ratios of a line are 1, 1, 2, find the direction cosines

of the line.

Solution The direction cosines are given by

2 2 2 2 2 2 2 2 2, ,

a b cl m n

a b c a b c a b c= = =

+ + + + + +

Here a, b, c are 1, 1, 2, respectively.

Therefore,2 2 2 2 2 2 2 2 2

1 1 2, ,

1 1 2 1 1 2 1 1 2l m n= = =

+ + + + + +

i.e.,1 1 2

, ,6 6 6

l m n= = = i.e.1 1 2

, ,6 6 6

⎛ ⎞±⎜ ⎟

⎝ ⎠ are D.C’s of the line.

Example 2 Find the direction cosines of the line passing through the points

P (2, 3, 5) and Q (–1, 2, 4).

Solution The direction cosines of a line passing through the points P ( x1, y

1, z

1) and

Q ( x2, y

2, z

2) are

2 1 2 1 2 1, ,PQ PQ PQ

x y y z z − − −.

Here 2 2 22 1 2 1 2 1PQ ( ) ( ) ( ) x y y z z = − + − + −

= 2 2 2( 1 2) (2 3) (4 5)− − + − + − = 9 1 1+ + = 11

Hence D.C.’s are




± 3 1 1

, ,11 11 11

⎛ ⎞− − −⎜ ⎟⎜ ⎟⎝ ⎠

or3 1 1

, ,11 11 11

⎛ ⎞± ⎜ ⎟⎝ ⎠

.

Example 3 If a line makes an angle of 30°, 60°, 90° with the positive direction of

x, y, z -axes, respectively, then find its direction cosines.

Solution The direction cosines of a line which makes an angle of α, β, γ with the axes,

are cosα, cosβ, cosγ

Therefore, D.C.’s of the line are cos30°, cos60°, cos90° i.e., 3 1, , 02 2

⎛ ⎞± ⎜ ⎟⎜ ⎟

⎝ ⎠

Example 4 The x-coordinate of a point on the line joining the points Q (2, 2, 1) and

R (5, 1, –2) is 4. Find its z -coordinate.

Solution Let the point P divide QR in the ratio λ : 1, then the co-ordinate of P are

5 2 2 –2 1, ,

1 1 1

⎛ ⎞λ + λ+ λ +⎜ ⎟λ + λ + λ +⎝ ⎠

But x – coordinate of P is 4. Therefore,

5 24 2

1

λ += ⇒ λ =

λ +

Hence, the z -coordinate of P is2 1

–11

− λ +=

λ + .

Example 5 Find the distance of the point whose position vector is ˆˆ ˆ(2 – )i j k + from

the plane r . ( i – 2 ˆ j + 4 k ) = 9

Solution Here a

= ˆˆ ˆ2 – i j k + , ˆˆ ˆ – 2 4n i j k

and d = 9

So, the required distance is( ) ( )ˆ ˆˆ ˆ ˆ ˆ2 – . 2 4 9

1 4 16

i j k i j k + − + −

+ +



226 MATHEMATICS

=2 2 4 9 13

21 21− − − = .

Example 6 Find the distance of the point (– 2, 4, – 5) from the line3 4 8

3 5 6

x y z + − += =

Solution Here P (–2, 4, – 5) is the given point.

Any point Q on the line is given by (3λλλλλ –3, 5λλλλλ + 4,,,,, (6λλλλλ –8 ),

PQ

= (3λλλλλ –1) ˆˆ ˆ5 (6 3)i j k + λ + λ − .

Since PQ

⊥⊥⊥⊥⊥ ( )ˆˆ ˆ3 5 6i j k + + , we have

3 (3λλλλλ –1) + 5( 5λ)λ)λ)λ)λ) + 6 (6λλλλλ –3 ) = 0

9λλλλλ + 25λ λ λ λ λ + 36λλλλλ = 21, i.e. λ =λ =λ =λ =λ =3

10

Thus PQ = 1 15 12 ˆˆ ˆ10 10 10

i j k − + −

Hence1 37

PQ 1 225 14410 10

= + + =

.

Example 7 Find the coordinates of the point where the line through (3, – 4, – 5) and

(2, –3, 1) crosses the plane passing through three points (2, 2, 1), (3, 0, 1) and (4, –1, 0)

Solution Equation of plane through three points (2, 2, 1), (3, 0, 1) and (4, –1, 0) is

ˆ ˆˆ ˆ ˆ ˆ ˆ ˆ( – (2 2 ) . ( – 2 ) ( – – ) 0r i j k i j i j k ⎡ ⎤ ⎡ ⎤+ + × =⎣ ⎦ ⎣ ⎦

i.e. ˆˆ ˆ.(2 ) 7r i j k + + =

or 2 x + y + z – 7 = 0 ... (1)

Equation of line through (3, – 4, – 5) and (2, – 3, 1) is

3 4 5

1 1 6

y z − + += =

− ... (2)




Any point on line (2) is (– λ λ λ λ λ + 3, λ λ λ λ λ – 4, 6λ λ λ λ λ – 5). This point lies on plane (1). Therefore,2 (– λ λ λ λ λ + 3) + (λ λ λ λ λ – 4) + (6λ λ λ λ λ – 5) – 7 = 0, i.e., λ =λ =λ =λ =λ = z

Hence the required point is (1, – 2, 7).

Long Answer (L.A.)

Example 8 Find the distance of the point (–1, –5, – 10) from the point of intersection

of the line ˆ ˆˆ ˆ ˆ ˆ2 2 (3 4 2 )r i j k i j k = − + + λ + +

and the plane ˆˆ ˆ. ( ) 5r i j k − + =

.

Solution We have ˆ ˆˆ ˆ ˆ ˆ2 2 (3 4 2 )r i j k i j k = − + + λ + +

and ˆˆ ˆ. ( ) 5r i j k − + =

Solving these two equations, we get ˆ ˆ ˆˆ ˆ ˆ ˆ ˆ ˆ[(2 2 ) (3 4 2 )].( – ) 5i j k i j k i j k − + +λ + + + =

which gives λ =λ =λ =λ =λ = 0.

Therefore, the point of intersection of line and the plane is (2, 1, 2) and the other

given point is (– 1, – 5, – 10). Hence the distance between these two points is

2 2 22 ( 1) [ 1 5] [2 ( 10)] , i.e. 13

Example 9 A plane meets the co-ordinates axis in A, B, C such that the centroid

of the ∆ ABC is the point (α, β, γ ). Show that the equation of the plane is

α

x +

β +

γ

z = 3

Solution Let the equation of the plane be

x

a

+ y

b

+ z

c

= 1

Then the co-ordinate of A, B, C are (a, 0, 0), (0,b,0) and (0, 0, c) respectively. Centroid

of the ∆ ABC is

1 2 3 1 2 3 1 2 3, ,3 3 3

x x x y y y z z z i.e. , ,

3 3 3

a b c

But co-ordinates of the centroid of the ∆ ABC are (α, β, γ ) (given).



228 MATHEMATICS

Therefore, α =3a , β =

3b , γ =

3c , i.e. a = 3α, b = 3β, c = 3γ

Thus, the equation of plane is

α β γ

y z + + = 3

Example 10 Find the angle between the lines whose direction cosines are given by

the equations: 3l + m + 5n = 0 and 6mn – 2nl + 5lm = 0.

Solution Eliminating m from the given two equations, we get

⇒ 2n2 + 3 ln + l 2 = 0

⇒ (n + l ) (2n + l ) = 0

⇒ either n = – l or l = – 2n

Now if l = – n, then m = – 2n

and if l = – 2n, then m = n.

Thus the direction ratios of two lines are proportional to – n, –2n, n and –2n, n, n,i.e. 1, 2, –1 and –2, 1, 1.

So, vectors parallel to these lines are

a = i + 2 j – k and

b = –2i + j + k , respectively.

If θ is the angle between the lines, then

cos θ =

.a b

a b

=

( ) ( )2 2 2 2 2 2

2 – –2

1 2 (–1) (–2) 1 1

i j k i j k + ⋅ + +

+ + + +

=1

– 6

Hence θ = cos –1 1

– 6

.




Example 11 Find the co-ordinates of the foot of perpendicular drawn from the point A

(1, 8, 4) to the line joining the points B (0, –1, 3) and C (2, –3, –1).

Solution Let L be the foot of perpendicular drawn from the points A (1, 8, 4) to the line

passing through B and C as shown in the Fig. 11.2. The equation of line BC by using

formular =

a + λ (

b –

a ), the equation of the line BC is

r = ( ) ( ) – 3 2 – 2 – 4 j k i j k + + λ

⇒

xi yi zk =

2 – 2 1 3– 4

i i k Comparing both sides, we get

x = 2λ , y = – (2λ + 1), z = 3 – 4λ (1)

Thus, the co-ordinate of L are (2λ , – (2λ + 1), (3 – 4λ ),

so that the direction ratios of the line AL are (1 – 2λ ), 8 + (2λ + 1), 4 – (3 – 4λ ), i.e.

1 – 2λ , 2λ + 9, 1 + 4λ

Since AL is perpendicular to BC, we have,

(1 – 2λ ) (2 – 0) + (2λ + 9) (–3 + 1) + (4λ + 1) (–1 –3) = 0

⇒ λ = –5

6

The required point is obtained by substituting the value of λ , in (1), which is



230 MATHEMATICS

–5 2 19, ,3 3 3

⎛ ⎞⎜ ⎟⎝ ⎠

.

Example 12 Find the image of the point (1, 6, 3) in the line –1 – 2

1 2 3

x y z .

Solution Let P (1, 6, 3) be the given point and let L be the foot of perpendicular from

P to the given line.

The coordinates of a general point on the given line are

– 0 –1 – 2

1 2 3

x y z , i.e., x = λ , y = 2λ + 1, z = 3λ + 2.

If the coordinates of L are (λ , 2λ + 1, 3λ + 2), then the direction ratios of PL are

λ – 1, 2λ – 5, 3λ – 1.

But the direction ratios of given line which is perpendicular to PL are 1, 2, 3. Therefore,

(λ – 1) 1 + (2λ – 5) 2 + (3λ – 1) 3 = 0, which gives λ = 1. Hence coordinates of L are

(1, 3, 5).

Let Q ( x1 , y

1 , z

1) be the image of P (1, 6, 3) in the given line. Then L is the mid-point

of PQ. Therefore,1 1 11 6 3

1, 3 52 2 2

x y z

⇒ x1 = 1, y

1 = 0, z

1 = 7

Hence, the image of (1, 6, 3) in the given line is (1, 0, 7).




Example 13 Find the image of the point having position vector

3 4 i j k in the

plane

2 – 3 0r i j k .

Solution Let the given point be P

3 4 i j k and Q be the image of P in the plane

. 2 – 3 0r i j k as shown in the Fig. 11.4.

Then PQ is the normal to the plane. Since PQ passes through P and is normal to the

given plane, so the equation of PQ is given by

= 3 4 2 – r i j k i j k

Since Q lies on the line PQ, the position vector of Q can be expressed as

3 4 2 – i j k i j k , i.e., ( ) ( ) ( ) 1 2 3 – 4i j k + + + +λ λ λ

Since R is the mid point of PQ, the position vector of R is

( ) ( ) ( ) 1 2 3 – 4 3 4

2

i j k i j k ⎡ ⎤ ⎡ ⎤+ λ + λ + +λ + + +⎣ ⎦ ⎣ ⎦



232 MATHEMATICS

i.e., ( 1) 3– 4

2 2i j k λ λ⎛ ⎞ ⎛ ⎞λ + + + +⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠

Again, since R lies on the plane ( ) 2 – 3 0r i j k ⋅ + + =

, we have

( ) 1 3– 4 (2 – ) 3 02 2

j j ii k k ⎧ ⎫λ λ⎛ ⎞ ⎛ ⎞λ + + + + ⋅ + + =⎨ ⎬⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠⎩ ⎭

⇒ λ = –2

Hence, the position vector of Q is ( )3 4i j k + +

–2

2 – i j k , i.e. –3 5 2 ji k + + .


Choose the correct answer from the given four options in each of the Examples

14 to 19.

Example 14 The coordinates of the foot of the perpendicular drawn from the point

(2, 5, 7) on the x-axis are given by

(A) (2, 0, 0) (B) (0, 5, 0) (C) (0, 0, 7) (D) (0, 5, 7)

Solution (A) is the correct answer.

Example 15 P is a point on the line segment joining the points (3, 2, –1) and

(6, 2, –2). If x co-ordinate of P is 5, then its y co-ordinate is

(A) 2 (B) 1 (C) –1 (D) –2

Solution (A) is the correct answer. Let P divides the line segment in the ratio of λ : 1,

x - coordinate of the point P may be expressed as6 3

1 x

λ += λ + giving

6 351

λ +=λ + so that

λ = 2. Thus y-coordinate of P is2 2

21

λ +=

λ + .

Example 16 If α, β, γ are the angles that a line makes with the positive direction of x,

y, z axis, respectively, then the direction cosines of the line are.

(A) sin α, sin β, sin γ (B) cos α, cos β, cos γ

(C) tan α, tan β, tan γ (D) cos2

α, cos2

β, cos2

γ




Solution (B) is the correct answer.

Example 17 The distance of a point P (a, b, c) from x-axis is

(A) 2 2a c (B) 2 2a b

(C) 2 2b c (D) b2 + c2

Solution (C) is the correct answer. The required distance is the distance of P (a, b, c)

from Q (a, o, o), which is 2 2b c .

Example 18 The equations of x-axis in space are

(A) x = 0, y = 0 (B) x = 0, z = 0 (C) x = 0 (D) y = 0, z = 0

Solution (D) is the correct answer. On x-axis the y- co-ordinate and z - co-ordinates

are zero.

Example 19 A line makes equal angles with co-ordinate axis. Direction cosines of this

line are

(A) ± (1, 1, 1) (B)1 1 1, ,3 3 3

⎛ ⎞± ⎜ ⎟⎝ ⎠

(C)1 1 1

, ,3 3 3

⎛ ⎞± ⎜ ⎟⎝ ⎠

(D)1 1 1

, ,3 3 3

− −⎛ ⎞± ⎜ ⎟

⎝ ⎠

Solution (B) is the correct answer. Let the line makes angle α with each of the axis.

Then, its direction cosines are cos α, cos α, cos α.

Since cos2 α + cos2 α + cos2 α = 1. Therefore, cos α =1

3

Fill in the blanks in each of the Examples from 20 to 22.

Example 20 If a line makes angles3

,2 4

and4

with x, y, z axis, respectively, then

its direction cosines are _______



234 MATHEMATICS

Solution The direction cosines are cos2 , cos

3

4 , cos

4 , i.e.,

1 102 2

⎛ ⎞± ⎜ ⎟⎝ ⎠

, – .

Example 21 If a line makes angles α, β, γ with the positive directions of the coordinate

axes, then the value of sin2 α + sin2 β + sin2 γ is _______

Solution Note that

sin2 α + sin2 β + sin2 γ = (1 – cos2α) + (1 – cos2β) + (1 – cos2γ )

= 3 – (cos2α + cos2β + cos2γ ) = 2.

Example 22 If a line makes an angle of4

with each of y and z axis, then the angle

which it makes with x-axis is _________

Solution Let it makes angle α with x-axis. Then cos2α +2

4cos

+2

4cos

= 1

which after simplification gives α =2

.

State whether the following statements are True or False in Examples 23 and 24.

Example 23 The points (1, 2, 3), (–2, 3, 4) and (7, 0, 1) are collinear.

Solution Let A, B, C be the points (1, 2, 3), (–2, 3, 4) and (7, 0, 1), respectively.

Then, the direction ratios of each of the lines AB and BC are proportional to – 3, 1, 1.

Therefore, the statement is true.

Example 24 The vector equation of the line passing through the points (3,5,4) and

(5,8,11) is

3 5 4 2 3 7ˆ ˆˆ ˆ ˆ ˆ( )r i j k i j k r

Solution The position vector of the points (3,5,4) and (5,8,11) are

3 5 4 5 8 11ˆ ˆˆ ˆ ˆ ˆ,a i j k b i j k r

,

and therefore, the required equation of the line is given by

3 5 4 2 3 7ˆ ˆˆ ˆ ˆ ˆ( )r i j k i j k r

Hence, the statement is true.




11.3 EXERCISE

Short Answer (S.A.)

1. Find the position vector of a point A in space such that OA

is inclined at 60º to

OX and at 45° to OY and OA

= 10 units.

2. Find the vector equation of the line which is parallel to the vector ˆˆ ˆ3 2 6i j k

and which passes through the point (1,–2,3).

3. Show that the lines

1 2 3

2 3 4

x y z

and4 1

5 2

x y z

− −= = intersect.

Also, find their point of intersection.

4. Find the angle between the lines

ˆ ˆˆ ˆ ˆ ˆ3 2 6 (2 2 )r i j k i j k λ = − + + + +r

and ˆ ˆˆ ˆ ˆ(2 5 ) (6 3 2 )r j k i j k = − + + +μ r

5. Prove that the line through A (0, –1, –1) and B (4, 5, 1) intersects the line

through C (3, 9, 4) and D (– 4, 4, 4).

6. Prove that the lines x = py + q, z = ry + s and x = p′ y + q′, z = r ′ y + s′ are

perpendicular if pp′ + rr ′ + 1 = 0.

7. Find the equation of a plane which bisects perpendicularly the line joining the

points A (2, 3, 4) and B (4, 5, 8) at right angles.

8. Find the equation of a plane which is at a distance 3 3 units from origin and

the normal to which is equally inclined to coordinate axis.

9. If the line drawn from the point (–2, – 1, – 3) meets a plane at right angle at the

point (1, – 3, 3), find the equation of the plane.

10. Find the equation of the plane through the points (2, 1, 0), (3, –2, –2) and

(3, 1, 7).



236 MATHEMATICS

11. Find the equations of the two lines through the origin which intersect the line

3 3

2 1 1

x y z − −= = at angles of

3

πeach.

12. Find the angle between the lines whose direction cosines are given by the

equations l + m + n = 0, l 2 + m2 – n2 = 0.

13. If a variable line in two adjacent positions has direction cosines l , m, n and

l + δl , m + δm, n + δn, show that the small angle δθ between the two positions

is given by

δθ2 = δl 2 + δm2 + δn2

14. O is the origin and A is (a, b, c).Find the direction cosines of the line OA and

the equation of plane through A at right angle to OA.

15. Two systems of rectangular axis have the same origin. If a plane cuts them at

distances a, b, c and a′, b′, c′, respectively, from the origin, prove that

2 2 2 2 2 2

1 1 1 1 1 1

a b c a b c+ + = + +

′ ′ ′.

Long Answer (L.A.)

16. Find the foot of perpendicular from the point (2,3,–8) to the line

4 1

2 6 3

y z . Also, find the perpendicular distance from the given point

to the line.

17. Find the distance of a point (2,4,–1) from the line

5 3 6

1 4 –9

x y z

18. Find the length and the foot of perpendicular from the point3

1, ,22

⎛ ⎞⎜ ⎟⎝ ⎠

to the

plane 2 x – 2 y + 4 z + 5 = 0.

19. Find the equations of the line passing through the point (3,0,1) and parallel to

the planes x + 2 y = 0 and 3 y – z = 0.




20. Find the equation of the plane through the points (2,1,–1) and (–1,3,4), and

perpendicular to the plane x – 2 y + 4 z = 10.

21. Find the shortest distance between the lines given by ˆ ˆ(8 3 (9 16 )r i j= + λ − + λ

+

ˆ ˆ ˆˆ ˆ ˆ ˆ(10 7 ) and 15 29 5 (3 8 5 )k r i j k i j k + λ = + + +μ + −

.

22. Find the equation of the plane which is perpendicular to the plane

5 x + 3 y + 6 z + 8 = 0 and which contains the line of intersection of the planes

x + 2 y + 3 z – 4 = 0 and 2 x + y – z + 5 = 0.

23. The plane ax + by = 0 is rotated about its line of intersection with the plane z = 0 through an angle α. Prove that the equation of the plane in its new

position is ax + by 2 2( tan )a b± + α z = 0.

24. Find the equation of the plane through the intersection of the planes

r . ( i + 3 ˆ j ) – 6 = 0 and r

. (3 i – ˆ j – 4 k ) = 0, whose perpendicular

distance from origin is unity.

25. Show that the points ˆˆ ˆ( 3 )i j k − + and ˆˆ ˆ3( )i j k + + are equidistant from the plane

ˆˆ ˆ.(5 2 7 ) 9 0r i j k + − + = and lies on opposite side of it.

26. ˆ ˆˆ ˆ ˆ ˆAB 3 – and CD 3 2 4i j k i j k = + = − + +

are two vectors. The position vectors

of the points A and C are ˆ ˆˆ ˆ ˆ6 7 4 and – 9 2i j k j k + + + , respectively. Find the

position vector of a point P on the line AB and a point Q on the line CD such

that PQ

is perpendicular to AB

and CD

both.

27. Show that the straight lines whose direction cosines are given by

2l + 2m – n = 0 and mn + nl + lm = 0 are at right angles.

28. If l 1, m

1, n

1; l

2, m

2, n

2; l

3, m

3, n

3 are the direction cosines of three mutually

perpendicular lines, prove that the line whose direction cosines are proportional

to l 1 + l

2 + l

3, m

1 + m

2 + m

3, n

1 + n

2 + n

3 makes equal angles with them.


Choose the correct answer from the given four options in each of the Exercises from

29 to 36.

29. Distance of the point (α,β,γ ) from y-axis is



238 MATHEMATICS

(A) β (B) β (C) γ β + (D) 2 2γα +

30. If the directions cosines of a line are k,k,k, then

(A) k >0 (B) 0<k <1 (C) k =1 (D)1

3k or

1 –

3

31. The distance of the plane2 3 6 ˆˆ ˆ. 17 7 7

r i j k ⎛ ⎞+ − =⎜ ⎟⎝ ⎠

r from the origin is

(A) 1 (B) 7 (C)1

7(D) None of these

32. The sine of the angle between the straight line2 3 4

3 4 5

x y z and the

plane 2 x – 2 y + z = 5 is

(A)10

6 5

(B)4

5 2

(C)2 3

5

(D)2

10

33. The reflection of the point (α,β,γ ) in the xy– plane is

(A) (α,β,0) (B) (0,0,γ ) (C) (– α,– β,γ ) (D) (α,β,– γ )

34. The area of the quadrilateral ABCD, where A(0,4,1), B (2, 3, –1), C(4, 5, 0)

and D (2, 6, 2), is equal to

(A) 9 sq. units (B) 18 sq. units (C) 27 sq. units (D) 81 sq. units

35. The locus represented by xy + yz = 0 is

(A) A pair of perpendicular lines (B) A pair of parallel lines

(C) A pair of parallel planes (D) A pair of perpendicular planes

36. The plane 2 x – 3 y + 6 z – 11 = 0 makes an angle sin –1(α) with x-axis. The value

of α is equal to

(A)3

2(B)

2

3(C)

2

7(D)

3

7




Fill in the blanks in each of the Exercises 37 to 41.

37. A plane passes through the points (2,0,0) (0,3,0) and (0,0,4). The equation of

plane is __________.

38. The direction cosines of the vector ˆˆ ˆ(2 2 – )i j k are __________.

39. The vector equation of the line – 5 4 – 6

3 7 2

x y z is __________.

40. The vector equation of the line through the points (3,4,–7) and (1,–1,6) is

__________.

41. The cartesian equation of the plane ˆˆ ˆ.( – ) 2r i j k r

is __________.


42. The unit vector normal to the plane x + 2 y +3 z – 6 = 0 is

1 2 3 ˆˆ ˆ14 14 14

i j k .

43. The intercepts made by the plane 2 x – 3 y + 5 z +4 = 0 on the co-ordinate axis

are –2,4 4

,– 3 5

.

44. The angle between the line ˆ ˆˆ ˆ ˆ ˆ(5 – – 4 ) (2 – )r i j k i j k

and the plane

ˆˆ ˆ.(3 – 4 – ) 5 0r i j k

is –1 5

sin2 91

.

45. The angle between the planes ˆˆ ˆ.(2 – 3 ) 1r i j k

and ˆ ˆ.( – ) 4r i j is

–1 –5cos

58

.

46. The line ˆ ˆˆ ˆ ˆ ˆ2 – 3 – ( – 2 )r i j k i j k

lies in the plane ˆˆ ˆ.(3 – ) 2 0r i j k

.

47. The vector equation of the line – 5 4 – 6

3 7 2

x y z is



240 MATHEMATICS

ˆ ˆˆ ˆ ˆ ˆ5 – 4 6 (3 7 2 )r i j k i j k

.

48. The equation of a line, which is parallel to ˆˆ ˆ2 3i j k and which passes through

the point (5,–2,4), is – 5 2 – 4

2 –1 3

x y z .

49. If the foot of perpendicular drawn from the origin to a plane is (5, – 3, – 2),

then the equation of plane is ˆˆ ˆ.(5 – 3 2 ) 38r i j k − =

.



13.1 Overview

13.1.1 Conditional Probabili ty

If E and F are two events associated with the same sample space of a random

experiment, then the conditional probability of the event E under the condition that the

event F has occurred, written as P (E | F), is given by

P(E F)P(E | F) , P(F) 0

P(F)

∩= ≠

13.1.2 Properties of Conditi onal Probabil ity

Let E and F be events associated with the sample space S of an experiment. Then:

(i) P (S | F) = P (F | F) = 1

(ii) P [(A ∪ B) | F] = P (A | F) + P (B | F) – P [(A ∩ B | F)],

where A and B are any two events associated with S.

(iii) P (E′ | F) = 1 – P (E | F)

13.1.3 Multi plication Theorem on Probabili ty

Let E and F be two events associated with a sample space of an experiment. Then

P (E ∩ F) = P (E) P (F | E), P (E) ≠ 0

= P (F) P (E | F), P (F) ≠ 0

If E, F and G are three events associated with a sample space, then

P (E ∩ F ∩ G) = P (E) P (F | E) P (G | E ∩ F)

Chapter 13

PROBABILITY



PROBABILITY 259

13.1.4 I ndependent Events

Let E and F be two events associated with a sample space S. If the probability of

occurrence of one of them is not affected by the occurrence of the other, then we say

that the two events are independent. Thus, two events E and F will be independent, if

(a) P (F | E) = P (F), provided P (E) ≠ 0

(b) P (E | F) = P (E), provided P (F) ≠ 0

Using the multiplication theorem on probability, we have

(c) P (E ∩ F) = P (E) P (F)

Three events A, B and C are said to be mutually independent if all the following

conditions hold:

P (A ∩ B) = P (A) P (B)

P (A ∩ C) = P (A) P (C)

P (B ∩ C) = P (B) P (C)

and P (A ∩ B ∩ C) = P (A) P (B) P (C)

13.1.5 Parti tion of a Sample Space

A set of events E1, E

2,...., E

n is said to represent a partition of a sample space S if

(a) Ei ∩ E

j = φ, i ≠ j; i, j = 1, 2, 3,......, n

(b) Ei ∪ E

2∪ ... ∪ E

n = S, and

(c) Each Ei ≠ φ, i. e, P (E

i) > 0 for all i = 1, 2, ..., n

13.1.6 Theorem of Total Probabil ity

Let {E1, E, ..., E

n} be a partition of the sample space S. Let A be any event associated

with S, then

P (A) =1

P(E )P(A | E )n

j j

j=∑



260 MATHEMATICS

13.1.7 Bayes’ Theorem

If E1, E

2,..., E

n are mutually exclusive and exhaustive events associated with a sample

space, and A is any event of non zero probability, then

1

P(E )P(A | E )P(E | A)

P(E )P(A | E )

i ii n

i i

i =

=

∑

13.1.8 Random Variable and its Probabil ity Distribution

A random variable is a real valued function whose domain is the sample space of a

random experiment.

The probability distribution of a random variable X is the system of numbers

X : x1

x2

... xn

P (X) : p

1 p

2... p

n

where pi > 0, i =1, 2,..., n,

1

= 1

n

i

i

p=∑ .

13.1.9 Mean and Variance of a Random Vari able

Let X be a random variable assuming values x1, x

2,...., x

n with probabilities

p1, p

2, ..., p

n , respectively such that p

i ≥ 0,

1

= 1n

i

i

p=

∑ . Mean of X, denoted by µ [or

expected value of X denoted by E (X)] is defined as

1

μ = E (X) =n

i i

i

x p=

∑

and variance, denoted by σ2, is defined as

2 2 2 2

i i

1 1

= ( – ) = –n n

i i

i i

x p x p



PROBABILITY 261

or equivalently

σ2 = E (X – µ )2

Standard deviation of the random variable X is defined as

2

i

1

= variance (X) = ( – )n

i

i

x p

13.1.10 Bernoulli Trials

Trials of a random experiment are called Bernoulli trials, if they satisfy the followingconditions:

(i) There should be a finite number of trials

(ii) The trials should be independent

(iii) Each trial has exactly two outcomes: success or failure

(iv) The probability of success (or failure) remains the same in each trial.

13.1.11 Binomial Distribution

A random variable X taking values 0, 1, 2, ..., n is said to have a binomial distribution

with parameters n and p, if its probability distibution is given by

P (X = r ) = ncr pr qn–r ,

where q = 1 – p and r = 0, 1, 2, ..., n.

13.2 Solved Examples

Short Answer (S. A.)

Example 1 A and B are two candidates seeking admission in a college. The probability

that A is selected is 0.7 and the probability that exactly one of them is selected is 0.6.Find the probability that B is selected.

Solution Let p be the probability that B gets selected.

P (Exactly one of A, B is selected) = 0.6 (given)

P (A is selected, B is not selected; B is selected, A is not selected) = 0.6

P (A∩B′) + P (A′∩B) = 0.6



262 MATHEMATICS

P (A) P (B′) + P (A′) P (B) = 0.6

(0.7) (1 – p) + (0.3) p = 0.6

p = 0.25

Thus the probability that B gets selected is 0.25.

Example 2 The probability of simultaneous occurrence of at least one of two events

A and B is p. If the probability that exactly one of A, B occurs is q, then prove that

P (A′) + P (B′) = 2 – 2 p + q.

Solution Since P (exactly one of A, B occurs) = q (given), we get

P (A∪B) – P ( A∩B) = q

⇒ p – P (A∩B) = q

⇒ P (A∩B) = p – q

⇒ 1 – P (A′∪B′) = p – q

⇒ P (A′∪B′) = 1 – p + q

⇒ P (A′) + P (B′) – P (A′∩B′) = 1 – p + q

⇒ P (A′) + P (B′) = (1 – p + q) + P (A′ ∩ B′)

= (1 – p + q) + (1 – P (A ∪ B))

= (1 – p + q) + (1 – p)

= 2 – 2 p + q.

Example 3 10% of the bulbs produced in a factory are of red colour and 2% are red

and defective. If one bulb is picked up at random, determine the probability of its being

defective if it is red.

Solution Let A and B be the events that the bulb is red and defective, respectively.

10 1P (A) = =

100 10,

2 1P (A B) = =

100 50



PROBABILITY 263

P (A B) 1 10 1P (B | A) = =P (A) 50 1 5

∩ × =

Thus the probability of the picked up bulb of its being defective, if it is red, is1

5.

Example 4 Two dice are thrown together. Let A be the event ‘getting 6 on the first

die’ and B be the event ‘getting 2 on the second die’. Are the events A and B

independent?

Solution: A = {(6, 1), (6, 2), (6, 3), (6, 4), (6, 5), (6, 6)}

B = {(1, 2), (2, 2), (3, 2), (4, 2), (5, 2), (6, 2)}

A ∩ B = {(6, 2)}

6 1P(A)

36 6 ,

1P(B)

6 ,

1P(A B)

36

Events A and B will be independent if

P (A ∩ B) = P (A) P (B)

i.e.,1 1 1 1

LHS= P A B , RHS = P A P B36 6 6 36

Hence, A and B are independent.

Example 5 A committee of 4 students is selected at random from a group consisting 8

boys and 4 girls. Given that there is at least one girl on the committee, calculate the

probability that there are exactly 2 girls on the committee.

Solution Let A denote the event that at least one girl will be chosen, and B the eventthat exactly 2 girls will be chosen. We require P (B | A).

SinceA denotes the event that at least one girl will be chosen, A denotes that no girl

is chosen, i.e., 4 boys are chosen. Then

8

4

12

4

C 70 14P ( A )

C 495 99′ = = =



264 MATHEMATICS

14 85P (A) 1– 99 99

Now P (A ∩ B) = P (2 boys and 2 girls) =

8 4

2 2

12

4

C . C

C

6 28 56

495 165

×= =

Thus P (B | A)

P(A B) 56 99 168

P(A) 165 85 425

∩= = × =

Example 6 Three machines E1, E

2, E

3 in a certain factory produce 50%, 25% and

25%, respectively, of the total daily output of electric tubes. It is known that 4% of the

tubes produced one each of machines E1 and E

2 are defective, and that 5% of those

produced on E3 are defective. If one tube is picked up at random from a day’s production,

calculate the probability that it is defective.

Solution: Let D be the event that the picked up tube is defective

Let A1, A

2and A

3 be the events that the tube is produced on machines E

1, E

2 and E

3,

respectively .

P (D) = P (A1) P (D | A

1) + P (A

2) P (D | A

2) + P (A

3) P (D | A

3) (1)

P (A1) =

50

100 =

1

2, P (A

2) =

1

4, P (A

3) =

1

4

Also P (D | A1) = P (D | A

2) =

4

100 =

1

25

P (D | A3) =

5

100 =

1

20 .

Putting these values in (1), we get

P (D) =1

2 ×

1

25 +

1

4 ×

1

25 +

1

4 ×

1

20

=1

50 +

1

100 +

1

80 =

17

400 = .0425



PROBABILITY 265

Example 7 Find the probability that in 10 throws of a fair die a score which is a

multiple of 3 will be obtained in at least 8 of the throws.

Solution Here success is a score which is a multiple of 3 i.e., 3 or 6.

Therefore, p (3 or 6) =2 1

6 3

The probability of r successes in 10 throws is given by

P (r ) = 10Cr

10– 1 2

3 3

r r

Now P (at least 8 successes) = P (8) + P (9) + P (10)

8 2 9 1 10

10 10 10

8 9 10

1 2 1 2 1C C C

3 3 3 3 3

⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞= + +⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠

= 10

1

3[45 × 4 + 10 × 2 + 1] = 10

201

3.

Example 8 A discrete random variable X has the following probability distribution:

X 1 2 3 4 5 6 7

P (X) C 2C 2C 3C C2 2C2 7C2 + C

Find the value of C. Also find the mean of the distribution.

Solution Since Σ pi = 1, we have

C + 2C + 2C + 3C + C2 + 2C2 + 7C2 + C = 1

i.e., 10C2 + 9C – 1 = 0

i.e. (10C – 1) (C + 1) = 0

⇒ C =1

10, C = –1

Therefore, the permissible value of C =1

10 (Why?)



266 MATHEMATICS

Mean =1

n

i i

i

x p

=7

1

i i

i

x p

2 2 21 2 2 3 1 1 1 1

1 2 3 4 5 6 2 7 710 10 10 10 10 10 10 10

= × + × + × + × + + × + +⎛ ⎞⎛ ⎞ ⎛ ⎞ ⎛ ⎞⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎝ ⎠

1 4 6 12 5 12 49 7

10 10 10 10 100 100 100 10

= + + + + + + +

= 3.66.

Long Answer (L.A.)

Example 9 Four balls are to be drawn without replacement from a box containing

8 red and 4 white balls. If X denotes the number of red ball drawn, find the probability

distribution of X.

Solution Since 4 balls have to be drawn, therefore, X can take the values 0, 1, 2, 3, 4.

P (X = 0) = P (no red ball) = P (4 white balls)

4

4

1 2

4

C 1

C 4 9 5

P (X = 1) = P (1 red ball and 3 white balls)

8 4

1 3

12

4

C C 32

C 495

P (X = 2) = P (2 red balls and 2 white balls)

8 4

2 2

12

4

C C 168

C 495

P (X = 3) = P (3 red balls and 1 white ball)

8 4

3 1

12

4

C C 224

C 495



PROBABILITY 267

P (X = 4) = P (4 red balls)8

4

1 2

4

C 7 0C 4 9 5

.

Thus the following is the required probability distribution of X

X 0 1 2 3 4

P (X)1

495

32

495

168

495

224

495

70

495

Example 10 Determine variance and standard deviation of the number of heads inthree tosses of a coin.

Solution Let X denote the number of heads tossed. So, X can take the values 0, 1, 2,

3. When a coin is tossed three times, we get

Sample space S = {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT}

P (X = 0) = P (no head) = P (TTT) =1

8

P (X = 1) = P (one head) = P (HTT, THT, TTH) =3

8

P (X = 2) = P (two heads) = P (HHT, HTH, THH) =3

8

P (X = 3) = P (three heads) = P (HHH) =1

8

Thus the probability distribution of X is:

X 0 1 2 3

P (X)1

8

3

8

3

8

1

8

Variance of X = σ2 = Σ x2i p

i – µ2, (1)

where µ is the mean of X given by

µ = Σ xi p

i =

1 3 3 10 1 2 3

8 8 8 8



268 MATHEMATICS

= 32

(2)

Now

Σ x2i p

i =

2 2 2 21 3 3 10 1 2 3 3

8 8 8 8 (3)

From (1), (2) and (3), we get

σ2 =

23 3

3 –

2 4

Standard deviation2 3 3

4 2 .

Example 11 Refer to Example 6. Calculate the probability that the defective tube was

produced on machine E1.

Solution Now, we have to find P (A1/ D).

P (A1 / D) =

1 1 1P (A D) P (A ) P (D / A )

P(D) P (D)

=

1 1

82 2517 17

400

×=

.

Example 12 A car manufacturing factory has two plants, X and Y. Plant X manufactures

70% of cars and plant Y manufactures 30%. 80% of the cars at plant X and 90% of the

cars at plant Y are rated of standard quality. A car is chosen at random and is found to

be of standard quality. What is the probability that it has come from plant X?

Solution Let E be the event that the car is of standard quality. Let B1 and B

2 be the

events that the car is manufactured in plants X and Y, respectively. Now

P (B1) =

70 7

100 10= , P (B

2) =

30 3

100 10=

P (E | B1) = Probability that a standard quality car is manufactured in plant



PROBABILITY 269

= 80 8100 10

=

P (E | B2) =

90 9

100 10=

P (B1 | E) = Probability that a standard quality car has come from plant X

1 1

1 1 2 2

P (B ) × P (E | B )

P (B ) . P (E | B ) + P (B ) . P (E | B )=

7 85610 10

7 8 3 9 83

10 10 10 10

×= =

× + ×

Hence the required probability is56

83.


Choose the correct answer from the given four options in each of the Examples 13 to 17.

Example 13 Let A and B be two events. If P (A) = 0.2, P (B) = 0.4, P (A∪B) = 0.6,

then P (A | B) is equal to

(A) 0.8 (B) 0.5 (C) 0.3 (D) 0

Solution The correct answer is (D). From the given data P (A) + P (B) = P (A∪B).

This shows that P (A∩B) = 0. Thus P (A | B) =P (A B)

P (B)

= 0.

Example 14 Let A and B be two events such that P (A) = 0.6, P (B) = 0.2, andP (A | B) = 0.5.

Then P (A′ | B′) equals

(A)1

10(B)

3

10(C)

3

8(D)

6

7

Solution The correct answer is (C). P (A∩B) = P (A | B) P (B)

= 0.5 × 0.2 = 0.1



270 MATHEMATICS

P (A′ | B′) = ( )1– P A BP (A B ) P[(A B )]P (B ) P(B ) 1– P(B)

∪′ ′ ′∩ ∪= =′ ′

=1– P (A) – P (B) + P (A B)

1–0.2

∩ =

3

8.

Example 15 If A and B are independent events such that 0 < P (A) < 1 and

0 < P (B) < 1, then which of the following is not correct?

(A) A and B are mutually exclusive (B) A and B′ are independent

(C) A′ and B are independent (D) A′ and B′ are independent

Solution The correct answer is (A).

Example 16 Let X be a discrete random variable. The probability distribution of X is

given below:

X 30 10 – 10

P (X)1

5

3

10

1

2

Then E (X) is equal to

(A) 6 (B) 4 (C) 3 (D) – 5

Solution The correct answer is (B).

E (X) =1 3 1

30 10 –10 45 10 2

× + × × = .

Example 17 Let X be a discrete random variable assuming values x1, x

2, ..., x

n with

probabilities p1, p

2, ..., p

n, respectively. Then variance of X is given by

(A) E (X2) (B) E (X2) + E (X) (C) E (X2) – [E (X)]2

(D) 2 2E (X ) – [E (X)]

SolutionThe correct answer is (C).

Fill in the blanks in Examples 18 and 19

Example 18 If A and B are independent events such that P (A) = p, P (B) = 2 p and

P (Exactly one of A, B) =5

9, then p = __________



PROBABILITY 271

Solution p = 1 5,3 12

( )( ) ( ) 2 51– 2 1– 2 3 – 49

p p p p p p⎡ ⎤+ = =⎢ ⎥⎣ ⎦

Example 19 If A and B′ are independent events then P (A′∪B) = 1 – ________

Solution P (A′∪B) = 1 – P (A∩B′) = 1 – P (A) P (B′)(since A and B′ are independent).

State whether each of the statement in Examples 20 to 22 is True or False

Example 20 Let A and B be two independent events. Then P (A∩B) = P (A) + P (B)

Solution False, because P (A∩B) = P (A) . P(B) when events A and B are independent.

Example 21 Three events A, B and C are said to be independent if P (A∩B∩C) =

P (A) P (B) P (C).

Solution False. Reason is that A, B, C will be independent if they are pairwise

independent and P (A∩B∩C) = P (A) P (B) P (C).

Example 22 One of the condition of Bernoulli trials is that the trials are independent

of each other.

Solution:True.

13.3 EXERCISE

Short Answer (S.A.)

1. For a loaded die, the probabilities of outcomes are given as under:

P(1) = P(2) = 0.2, P(3) = P(5) = P(6) = 0.1 and P(4) = 0.3.

The die is thrown two times. Let A and B be the events, ‘same number each

time’, and ‘a total score is 10 or more’, respectively. Determine whether or not

A and B are independent.

2. Refer to Exercise 1 above. If the die were fair, determine whether or not the

events A and B are independent.

3. The probability that at least one of the two events A and B occurs is 0.6. If A and

B occur simultaneously with probability 0.3, evaluate P( A ) + P( B ).

4. A bag contains 5 red marbles and 3 black marbles. Three marbles are drawn one

by one without replacement. What is the probability that at least one of the three

marbles drawn be black, if the first marble is red?



272 MATHEMATICS

5. Two dice are thrown together and the total score is noted. The events E, F and

G are ‘a total of 4’, ‘a total of 9 or more’, and ‘a total divisible by 5’, respectively.

Calculate P(E), P(F) and P(G) and decide which pairs of events, if any, are

independent.

6. Explain why the experiment of tossing a coin three times is said to have binomial

distribution.

7. A and B are two events such that P(A) =1

2, P(B) =

1

3 and P(A ∩ B)=

1

4.

Find :

(i) P(A|B) (ii) P(B|A) (iii) P(A'|B) (iv) P(A'|B')

8. Three events A, B and C have probabilities2

5,

1

3 and

1

2, respectively. Given

that P(A ∩ C) =1

5 and P(B ∩ C) =

1

4, find the values of P(C | B) and P(A' ∩ C').

9. Let E1 and E

2 be two independent events such that p(E

1) = p

1 and P(E

2) = p

2.

Describe in words of the events whose probabilities are:

(i) p1 p

2(ii) (1– p

1) p

2(iii) 1–(1– p

1)(1– p

2) (iv) p

1 + p

2 – 2 p

1 p

2

10. A discrete random variable X has the probability distribution given as below:

X 0.5 1 1.5 2

P(X) k k 2 2k 2 k

(i) Find the value of k

(ii) Determine the mean of the distribution.

11. Prove that

(i) P(A) = P(A ∩ B) + P(A ∩ B )

(ii) P(A ∪ B) = P(A ∩ B) + P(A ∩ B ) + P( A ∩ B)

12. If X is the number of tails in three tosses of a coin, determine the standard

deviation of X.

13. In a dice game, a player pays a stake of Re1 for each throw of a die. She

receives Rs 5 if the die shows a 3, Rs 2 if the die shows a 1 or 6, and nothing



PROBABILITY 273

otherwise. What is the player’s expected profit per throw over a long series of

throws?

14. Three dice are thrown at the sametime. Find the probability of getting three

two’s, if it is known that the sum of the numbers on the dice was six.

15. Suppose 10,000 tickets are sold in a lottery each for Re 1. First prize is of

Rs 3000 and the second prize is of Rs. 2000. There are three third prizes of Rs.

500 each. If you buy one ticket, what is your expectation.

16. A bag contains 4 white and 5 black balls. Another bag contains 9 white and 7 black balls. A ball is transferred from the first bag to the second and then a ball

is drawn at random from the second bag. Find the probability that the ball drawn

is white.

17. Bag I contains 3 black and 2 white balls, Bag II contains 2 black and 4 white

balls. A bag and a ball is selected at random. Determine the probability of selecting

a black ball.

18. A box has 5 blue and 4 red balls. One ball is drawn at random and not replaced.

Its colour is also not noted. Then another ball is drawn at random. What is the

probability of second ball being blue?

19. Four cards are successively drawn without replacement from a deck of 52 playing

cards. What is the probability that all the four cards are kings?

20. A die is thrown 5 times. Find the probability that an odd number will come up

exactly three times.

21. Ten coins are tossed. What is the probability of getting at least 8 heads?

22. The probability of a man hitting a target is 0.25. He shoots 7 times. What is the

probability of his hitting at least twice?

23. A lot of 100 watches is known to have 10 defective watches. If 8 watches are

selected (one by one with replacement) at random, what is the probability that

there will be at least one defective watch?



274 MATHEMATICS

24. Consider the probability distribution of a random variable X:

X 0 1 2 3 4

P(X) 0.1 0.25 0.3 0.2 0.15

Calculate (i)X

V2

⎛ ⎞⎜ ⎟⎝ ⎠

(ii) Variance of X.

25. The probability distribution of a random variable X is given below:

X 0 1 2 3

P(X) k 2

k

4

k

8

k

(i) Determine the value of k.

(ii) Determine P(X ≤ 2) and P(X > 2)

(iii) Find P(X ≤ 2) + P (X > 2).

26. For the following probability distribution determine standard deviation of therandom variable X.

X 2 3 4

P(X) 0.2 0.5 0.3

27. A biased die is such that P(4) =1

10 and other scores being equally likely. The die

is tossed twice. If X is the ‘number of fours seen’, find the variance of the

random variable X.

28. A die is thrown three times. Let X be ‘the number of twos seen’. Find the

expectation of X.

29. Two biased dice are thrown together. For the first die P(6) =1

2, the other scores

being equally likely while for the second die, P(1) =2

5 and the other scores are



PROBABILITY 275

equally likely. Find the probability distribution of ‘the number of ones seen’.

30. Two probability distributions of the discrete random variable X and Y are given

below.

X 0 1 2 3 Y 0 1 2 3

P(X)1

5

2

5

1

5

1

5P(Y)

1

5

3

10

2

5

1

10

Prove that E(Y2) = 2 E(X).

31. A factory produces bulbs. The probability that any one bulb is defective is1

50

and they are packed in boxes of 10. From a single box, find the probability

that

(i) none of the bulbs is defective

(ii) exactly two bulbs are defective

(iii) more than 8 bulbs work properly

32. Suppose you have two coins which appear identical in your pocket. You know

that one is fair and one is 2-headed. If you take one out, toss it and get a head,

what is the probability that it was a fair coin?

33. Suppose that 6% of the people with blood group O are left handed and 10% of

those with other blood groups are left handed 30% of the people have blood

group O. If a left handed person is selected at random, what is the probability

that he/she will have blood group O?

34. Two natural numbers r , s are drawn one at a time, without replacement from

the set S=

1, 2, 3 , ...., n . Find P [ ]|r p s p≤ ≤ , where p∈S .

35. Find the probability distribution of the maximum of the two scores obtained

when a die is thrown twice. Determine also the mean of the distribution.

36. The random variable X can take only the values 0, 1, 2. Given that P(X = 0) =

P (X = 1) = p and that E(X2) = E[X], find the value of p.



276 MATHEMATICS

37. Find the variance of the distribution:

x 0 1 2 3 4 5

P( x)1

6

5

18

2

9

1

6

1

9

1

18

38. A and B throw a pair of dice alternately. A wins the game if he gets a total of

6 and B wins if she gets a total of 7. It A starts the game, find the probability of

winning the game by A in third throw of the pair of dice.

39. Two dice are tossed. Find whether the following two events A and B are

independent:

A = ( , ) : + =11 x y x y B = ( , ) : 5 x y x

where ( x, y) denotes a typical sample point.

40. An urn contains m white and n black balls. A ball is drawn at random and is put

back into the urn along with k additional balls of the same colour as that of the

ball drawn. A ball is again drawn at random. Show that the probability of

drawing a white ball now does not depend on k .

Long Answer (L.A.)

41. Three bags contain a number of red and white balls as follows:

Bag 1 : 3 red balls, Bag 2 : 2 red balls and 1 white ball

Bag 3 : 3 white balls.

The probability that bag i will be chosen and a ball is selected from it is6

i,

i = 1, 2, 3. What is the probability that

(i) a red ball will be selected? (ii) a white ball is selected?

42. Refer to Question 41 above. If a white ball is selected, what is the probability

that it came from

(i) Bag 2 (ii) Bag 3

43. A shopkeeper sells three types of flower seeds A1, A

2 and A

3. They are sold as

a mixture where the proportions are 4:4:2 respectively. The germination rates

of the three types of seeds are 45%, 60% and 35%. Calculate the probability

(i) of a randomly chosen seed to germinate



PROBABILITY 277

(ii) that it will not germinate given that the seed is of type A3

,

(iii) that it is of the type A2 given that a randomly chosen seed does not germinate.

44. A letter is known to have come either from TATA NAGAR or from

CALCUTTA. On the envelope, just two consecutive letter TA are visible. What

is the probability that the letter came from TATA NAGAR.

45. There are two bags, one of which contains 3 black and 4 white balls while the

other contains 4 black and 3 white balls. A die is thrown. If it shows up 1 or 3,

a ball is taken from the Ist bag; but it shows up any other number, a ball is

chosen from the second bag. Find the probability of choosing a black ball.

46. There are three urns containing 2 white and 3 black balls, 3 white and 2 black

balls, and 4 white and 1 black balls, respectively. There is an equal probability

of each urn being chosen. A ball is drawn at random from the chosen urn and it

is found to be white. Find the probability that the ball drawn was from the

second urn.

47. By examining the chest X ray, the probability that TB is detected when a person

is actually suffering is 0.99. The probability of an healthy person diagnosed to

have TB is 0.001. In a certain city, 1 in 1000 people suffers from TB. A person

is selected at random and is diagnosed to have TB. What is the probability thathe actually has TB?

48. An item is manufactured by three machines A, B and C. Out of the total number

of items manufactured during a specified period, 50% are manufactured on A,

30% on B and 20% on C. 2% of the items produced on A and 2% of items

produced on B are defective, and 3% of these produced on C are defective. All

the items are stored at one godown. One item is drawn at random and is found

to be defective. What is the probability that it was manufactured on

machine A?

49. Let X be a discrete random variable whose probability distribution is defined as

follows:

( 1)for 1,2,3,4

(X ) 2 for 5,6,7

0 otherwise

k x x

P x kx x

+ =⎧⎪

= = =⎨⎪⎩



278 MATHEMATICS

where k is a constant. Calculate

(i) the value of k (ii) E (X) (iii) Standard deviation of X.

50. The probability distribution of a discrete random variable X is given as under:

X 1 2 4 2A 3A 5A

P(X)1

2

1

5

3

25

1

10

1

25

1

25

Calculate :

(i) The value of A if E(X) = 2.94

(ii) Variance of X.

51. The probability distribution of a random variable x is given as under:

P( X = x ) =

2 for 1,2,3

2 for 4,5,6

0 otherwise

kx x

kx x

⎧ =⎪

=⎨⎪⎩

where k is a constant. Calculate

(i) E(X) (ii) E (3X2) (iii) P(X ≥ 4)

52. A bag contains (2n + 1) coins. It is known that n of these coins have a head on

both sides where as the rest of the coins are fair. A coin is picked up at random

from the bag and is tossed. If the probability that the toss results in a head is

31

42, determine the value of n.

53. Two cards are drawn successively without replacement from a well shuffleddeck of cards. Find the mean and standard variation of the random variable X

where X is the number of aces.

54. A die is tossed twice. A ‘success’ is getting an even number on a toss. Find the

variance of the number of successes.

55. There are 5 cards numbered 1 to 5, one number on one card. Two cards are

drawn at random without replacement. Let X denote the sum of the numbers on

two cards drawn. Find the mean and variance of X.



PROBABILITY 279


Choose the correct answer from the given four options in each of the exercises from

56 to 82.

56. If P(A) =4

5, and P(A ∩ B) =

7

10, then P(B | A) is equal to

(A)1

10(B)

1

8(C)

7

8(D)

17

20

57. If P(A ∩ B) =7

10 and P(B) =

17

20, then P (A | B) equals

(A)14

17(B)

17

20(C)

7

8(D)

1

8

58. If P(A) =3

10, P (B) =

2

5 and P(A∪B) =

3

5, then P (B | A) + P (A | B) equals

(A)1

4(B)

1

3(C)

5

12(D)

7

2

59. If P(A) =2

5, P(B) =

3

10 and P (A ∩ B) =

1

5, then P(A | B ).P(B ' | A ') is equal

to

(A)5

6(B)

5

7(C)

25

42(D) 1

60. If A and B are two events such that P(A) =1

2, P(B) =

1

3, P(A/B)=

1

4, then

P(A B )′ ′∩ equals

(A)1

12(B)

3

4(C)

1

4(D)

3

16



280 MATHEMATICS

61. If P(A) = 0.4, P(B) = 0.8 and P(B | A) = 0.6, then P(A ∪ B) is equal to

(A) 0.24 (B) 0.3 (C) 0.48 (D) 0.96

62. If A and B are two events and A φ, B φ, then

(A) P(A | B) = P(A).P(B) (B) P(A | B) =P(A B)

P(B)

∩

(C) P(A | B).P(B | A)=1 (D) P(A | B) = P(A) | P(B)

63. A and B are events such that P(A) = 0.4, P(B) = 0.3 and P(A ∪ B) = 0.5.

Then P (B A) equals

(A)2

3(B)

1

2(C)

3

10(D)

1

5

64. You are given that A and B are two events such that P(B)=3

5, P(A | B) =

1

2 and

P(A∪

B) =

4

5 , then P(A) equals

(A)3

10(B)

1

5(C)

1

2(D)

3

5

65. In Exercise 64 above, P(B | A ) is equal to

(A)1

5(B)

3

10(C)

1

2(D)

3

5

66. If P(B) =3

5, P(A | B) =

1

2 and P(A ∪ B) =

4

5, then P(A ∪ B ) + P( A ∪ B) =

(A)1

5(B)

4

5(C)

1

2(D) 1



PROBABILITY 281

67. Let P(A) = 713

, P(B) = 913

and P(A ∩ B) = 413

. Then P( A | B) is equal to

(A)6

13(B)

4

13(C)

4

9(D)

5

9

68. If A and B are such events that P(A) > 0 and P(B) ≠ 1, then P( A | B )

equals.

(A) 1 – P(A | B) (B) 1– P( A | B)

(C)1–P(A B)

P(B')

∪(D) P( A ) | P( B )

69. If A and B are two independent events with P(A) =3

5 and P(B) =

4

9, then

P( A ∩ B ) equals

(A)

4

15 (B)

8

45 (C)

1

3 (D)

2

9

70. If two events are independent, then

(A) they must be mutually exclusive

(B) the sum of their probabilities must be equal to 1

(C) (A) and (B) both are correct

(D) None of the above is correct

71. Let A and B be two events such that P(A) =

3

8 , P(B) =

5

8 and P(A ∪ B) =

3

4 .

Then P(A | B).P( A | B) is equal to

(A)2

5(B)

3

8(C)

3

20(D)

6

25

72. If the events A and B are independent, then P(A ∩ B) is equal to



282 MATHEMATICS

(A) P (A) + P (B) (B) P(A) – P(B)

(C) P (A) . P(B) (D) P(A) | P(B)

73. Two events E and F are independent. If P(E) = 0.3, P(E ∪ F) = 0.5, then

P(E | F)–P(F | E) equals

(A)2

7(B)

3

35(C)

1

70 (D)

1

7

74. A bag contains 5 red and 3 blue balls. If 3 balls are drawn at random without

replacement the probability of getting exactly one red ball is

(A)45

196(B)

135

392(C)

15

56(D)

15

29

75. Refer to Question 74 above. The probability that exactly two of the three balls

were red, the first ball being red, is

(A)1

3(B)

4

7(C)

15

28(D)

5

28

76. Three persons, A, B and C, fire at a target in turn, starting with A. Their probabilityof hitting the target are 0.4, 0.3 and 0.2 respectively. The probability of two hits

is

(A) 0.024 (B) 0.188 (C) 0.336 (D) 0.452

77. Assume that in a family, each child is equally likely to be a boy or a girl. A family

with three children is chosen at random. The probability that the eldest child is a

girl given that the family has at least one girl is

(A)

1

2 (B)

1

3 (C)

2

3 (D)

4

7

78. A die is thrown and a card is selected at random from a deck of 52 playing cards.

The probability of getting an even number on the die and a spade card is

(A)1

2(B)

1

4(C)

1

8(D)

3

4



PROBABILITY 283

79. A box contains 3 orange balls, 3 green balls and 2 blue balls. Three balls are

drawn at random from the box without replacement. The probability of drawing

2 green balls and one blue ball is

(A)3

28(B)

2

21(C)

1

28(D)

167

168

80. A flashlight has 8 batteries out of which 3 are dead. If two batteries are selected

without replacement and tested, the probability that both are dead is

(A)33

56 (B)9

64 (C)1

14 (D)3

28

81. Eight coins are tossed together. The probability of getting exactly 3 heads is

(A)1

256(B)

7

32(C)

5

32(D)

3

32

82. Two dice are thrown. If it is known that the sum of numbers on the dice was less

than 6, the probability of getting a sum 3, is

(A) 118

(B) 518

(C) 15

(D) 25

83. Which one is not a requirement of a binomial distribution?

(A) There are 2 outcomes for each trial

(B) There is a fixed number of trials

(C) The outcomes must be dependent on each other

(D) The probability of success must be the same for all the trials

84. Two cards are drawn from a well shuffled deck of 52 playing cards with

replacement. The probability, that both cards are queens, is

(A)1

13×

1

13(B)

1

13+

1

13(C)

1

13×

1

17(D)

1

13×

4

51

85. The probability of guessing correctly at least 8 out of 10 answers on a true-false

type examination is



284 MATHEMATICS

(A) 764

(B) 7128

(C) 451024

(D) 741

86. The probability that a person is not a swimmer is 0.3. The probability that out of

5 persons 4 are swimmers is

(A) 5C4 (0.7)4 (0.3) (B) 5C

1 (0.7) (0.3)4

(C) 5C4 (0.7) (0.3)4 (D) (0.7)4 (0.3)

87. The probability distribution of a discrete random variable X is given below:

X 2 3 4 5

P(X)5

k

7

k

9

k

11

k

The value of k is

(A) 8 (B) 16 (C) 32 (D) 48

88. For the following probability distribution:

X – 4 –3 –2 –1 0

P(X) 0.1 0.2 0.3 0.2 0.2

E(X) is equal to :

(A) 0 (B) –1 (C) –2 (D) –1.8

89. For the following probability distribution

X 1 2 3 4

P (X)1

10

1

5

3

10

2

5E(X2) is equal to

(A) 3 (B) 5 (C) 7 (D) 10

90. Suppose a random variable X follows the binomial distribution with parameters n

and p, where 0 < p < 1. If P( x = r ) / P( x = n – r ) is independent of n and r , then

p equals



PROBABILITY 285

(A) 12

(B) 13

(C) 15

(D) 17

91. In a college, 30% students fail in physics, 25% fail in mathematics and 10% fail

in both. One student is chosen at random. The probability that she fails in physics

if she has failed in mathematics is

(A)1

10(B)

2

5(C)

9

20(D)

1

3

92. A and B are two students. Their chances of solving a problem correctly are13

and1

4, respectively. If the probability of their making a common error is,

1

20

and they obtain the same answer, then the probability of their answer to be

correct is

(A)1

12(B)

1

40(C)

13

120(D)

10

13

93. A box has 100 pens of which 10 are defective. What is the probability that out of

a sample of 5 pens drawn one by one with replacement at most one is defective?

(A)

59

10

⎛ ⎞⎜ ⎟⎝ ⎠

(B)

41 9

2 10

⎛ ⎞⎜ ⎟⎝ ⎠

(C)

51 9

2 10

⎛ ⎞⎜ ⎟⎝ ⎠

(D)

5 49 1 9

10 2 10

⎛ ⎞ ⎛ ⎞+⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠


94. Let P(A) > 0 and P(B) > 0. Then A and B can be both mutually exclusive and

independent.

95. If A and B are independent events, then A and B are also independent.

96. If A and B are mutually exclusive events, then they will be independent also.

97. Two independent events are always mutually exclusive.

98. If A and B are two independent events then P(A and B) = P(A).P(B).



286 MATHEMATICS

99. Another name for the mean of a probability distribution is expected value.

100. If A and B′ are independent events, then P(A' ∪ B) = 1 – P (A) P(B')

101. If A and B are independent, then

P (exactly one of A, B occurs) = P(A)P(B )+P B P A

102. If A and B are two events such that P(A) > 0 and P(A) + P(B) >1, then

P(B | A) ≥ P(B)

1

P(A)

′−

103. If A, B and C are three independent events such that P(A) = P(B) = P(C) =

p, then

P (At least two of A, B, C occur) = 2 33 2 p−

Fill in the blanks in each of the following questions:

104. If A and B are two events such that

P (A | B) = p, P(A) = p, P(B) =

1

3

and P(A ∪ B)=5

9, then p = _____

105. If A and B are such that

P(A' ∪ B') =2

3 and P(A ∪ B)=

5

9,

then P(A') + P(B') = ..................

106. If X follows binomial distribution with parameters n = 5, p and

P (X = 2) = 9, P (X = 3), then p = ___________

107. Let X be a random variable taking values x1, x

2,..., x

n with probabilities

p1, p

2, ..., p

n, respectively. Then var (X) = ________

108. Let A and B be two events. If P(A | B) = P(A), then A is ___________ of B.



MCQ I

2 .1 A capacitor of 4 μ F is connected as shown in the circuit (Fig. 2.1).

The internal resistance of the battery is 0.5Ω . The amount of charge

on the capacitor plates will be

(a) 0

(b) 4 μ C

(c) 16 μ C

(d) 8 μ C

2 .2 A positively charged particle is released from rest in an uniformelectric field. The electric potential energy of the charge

(a) remains a constant because the electric field is uniform.

(b) increases because the charge moves along the electric field.

(c) decreases because the charge moves along the electric field.

(d) decreases because the charge moves opposite to the electric field.

Chapter Two

ELECTROSTATIC

POTENTIAL ANDCAPACITANCE

4 F

10

2.5V

2Fig. 2.1



11

Electrostatic Potentialand Capacitance

2 .3 Figure 2.2 shows some equipotential lines distributed in space.

A charged object is moved from point A to point B.

(a) The work done in Fig. (i) is the greatest.

(b) The work done in Fig. (ii) is least.

(c) The work done is the same in Fig. (i), Fig. (ii) and Fig. (iii).

(d) The work done in Fig. (iii) is greater than Fig. (ii)but equal tothat in Fig. (i).

2 .4 The electrostatic potential on the surface of a charged conductingsphere is 100V. Two statments are made in this regard:

S1 : At any point inside the sphere, electric intensity is zero.

S2 : At any point inside the sphere, the electrostatic potential is

100V. Which of the following is a correct statement?

(a) S1 is true but S

2is false.

(b) Both S1 & S

2 are false.

(c) S1 is true, S2 is also true and S1 is the cause of S2.(d) S

1 is true, S

2 is also true but the statements are independant.

2 .5 Equipotentials at a great distance from a collection of charges whosetotal sum is not zero are approximately

(a) spheres.

(b) planes.

(c) paraboloids

(d) ellipsoids.

Fig. 2.2

(i) (ii) (iii)

10V

A B

30V

10V 20V 40V

Fig III

50V

A B

10V 20V 40V 30V

Fig I

50V

A

30V 40V

20V

Fig II

50V

B



12

Exemplar Problems–Physics

2 .6 A parallel plate capacitor is made of two dielectric blocks in series.

One of the blocks has thickness d 1 and dielectric constant k 1 and

the other has thickness d 2 and dielectric constant k

2 as shown in

Fig. 2.3. This arrangement can be thought as a dielectric slab of thickness d (= d

1+d

2) and effective dielectric constant k . The k is

(a)k d k d k d k d k k d d k k

b c d d d k k k d k d k k

1 1 2 2 1 1 2 2 1 2 1 2 1 2

1 2 1 2 1 1 2 2 1 2

( ) 2( ) ( ) ( )

( )

+ + +

+ + + +

MCQ II

2 .7 Consider a uniform electric field in the z direction. The potential isa constant

(a) in all space.

(b) for any x for a given z .

(c) for any y for a given z .

(d) on the x -y plane for a given z.

2 .8 Equipotential surfaces

(a) are closer in regions of large electric fields compared to regionsof lower electric fields.

(b) will be more crowded near sharp edges of a conductor.

(c) will be more crowded near regions of large charge densities.

(d) will always be equally spaced.

2.9 The work done to move a charge along an equipotential from A to B

(a) cannot be defined as

B

A

d – .∫ lE

(b) must be defined as

B

A

d – .∫ lE

(c) is zero.(d) can have a non-zero value.

2 .10 In a region of constant potential

(a) the electric field is uniform

(b) the electric field is zero

(c) there can be no charge inside the region.

(d) the electric field shall necessarily change if a charge is placedoutside the region.

2 .11 In the circuit shown in Fig. 2.4. initially key K 1 is closed and key

d 1

d 1

k 1

k 2

Fig. 2.3



13


K 2 is open. Then K

1 is opened and K

2 is closed (order is important).

[Take Q1′ and Q2′ as charges on C1 and C2 and V 1 and V

2 as voltage

respectively.]

Then

(a) charge on C 1 gets redistributed such that V

1 = V

2

(b) charge on C 1 gets redistributed such that Q

1′ =Q

2′

(c) charge on C 1 gets redistributed such that C

1V

1 + C

2V

2 = C

1 E

(d) charge on C 1 gets redistributed such that Q

1′ +Q

2′ = Q

2 .12 If a conductor has a potential 0V ≠ and there are no charges

anywhere else outside, then

(a) there must be charges on the surface or inside itself.

(b) there cannot be any charge in the body of the conductor.

(c) there must be charges only on the surface.

(d) there must be charges inside the surface.

2 .13 A parallel plate capacitor is connected to a battery as shown inFig. 2.5. Consider two situations:

A: Key K is kept closed and plates of capacitors are moved apart using insulating handle.

B: Key K is opened and plates of capacitors are moved apart usinginsulating handle.

Choose the correct option(s).

(a) In A : Q remains same but C changes.

(b) In B : V remains same but C changes.

(c) In A : V remains same and hence Q changes.

(d) In B : Q remains same and hence V changes.

VSA

2 .14 Consider two conducting spheres of radii R 1and R

2 with R

1> R

2 . If

the two are at the same potential, the larger sphere has more chargethan the smaller sphere. State whether the charge density of the

( ) ( )K 2

C1

E

K 1

C2

Fig. 2.4

Fig. 2.5

( )

E

C

E

K



14


Q

b a

θ

Fig. 2.6

smaller sphere is more or less than that of the larger one.

2 .15 Do free electrons travel to region of higher potential or lower potential?

2 .16 Can there be a potential difference between two adjacent conductors carrying the same charge?

2 .17 Can the potential function have a maximum or minimum in freespace?

2 .18 A test charge q is made to move in the electric field of a point chargeQ along two different closed paths (Fig. 2.6). First path has sectionsalong and perpendicular to lines of electric field. Second path is a rectangular loop of the same area as the first loop. How does the

work done compare in the two cases?

SA

2 .19 Prove that a closed equipotential surface with no charge withinitself must enclose an equipotential volume.

2 .20 A capacitor has some dielectric between its plates, and the capacitor is connected to a DC source. The battery is now disconnected and

then the dielectric is removed. State whether the capacitance, theenergy stored in it, electric field, charge stored and the voltage willincrease, decrease or remain constant.

2 .21 Prove that, if an insulated, uncharged conductor is placed near a charged conductor and no other conductors are present, theuncharged body must be intermediate in potential between that of the charged body and that of infinity.

2 .22 Calculate potential energy of a point charge –q placed along theaxis due to a charge +Q uniformly distributed along a ring of radiusR . Sketch P.E. as a function of axial distance z from the centre of the ring. Looking at graph, can you see what would happen if -q isdisplaced slightly from the centre of the ring (along the axis)?

2 .23 Calculate potential on the axis of a ring due to charge Q uniformly distributed along the ring of radius R .

LA

2 .24 Find the equation of the equipotentials for an infinite cylinder of radius r

0, carrying charge of linear density λ.



15


2 .25 Two point charges of magnitude +q and -q are placed at

(-d /2, 0,0) and (d /2, 0,0), respectively. Find the equation of theequipoential surface where the potential is zero.

2 .26 A parallel plate capacitor is filled by a dielectric whose relativepermittivity varies with the applied voltage (U ) as ε = αU whereα = 2V –1 .A similar capacitor with no dielectric is charged toU

0 = 78 V. It is then connected to the uncharged capacitor with

the dielectric. Find the final voltage on the capacitors.

2 . 2 7 A capacitor is made of two circular plates of radius R each,separated by a distance d <<R . The capacitor is connected to a constant voltage. A thin conducting disc of radius r <<R andthickness t <<r is placed at a centre of the bottom plate. Findthe minimum voltage required to lift the disc if the mass of thedisc is m .

2 .28 (a) In a quark model of elementary particles, a neutron is made of one up quarks [charge (2/3) e ] and two down quarks [charges

–(1/3) e ]. Assume that they have a triangle configuration withside length of the order of 10 –15 m. Calculate electrostaticpotential energy of neutron and compare it with its mass 939MeV.

(b) Repeat above exercise for a proton which is made of two up

and one down quark.

2 .29 Two metal spheres, one of radius R and the other of radius 2R , both have same surface charge density σ . They are brought incontact and separated. What will be new surface charge densitieson them?

2 .30 In the circuit shown in Fig. 2.7, initially K 1

is closed and K 2 is open. What are the charges

on each capacitors. Then K

1 was opened and K

2 was closed

(order is important), What will be the chargeon each capacitor now? [C = 1μF]

2 .31 Calculate potential on the axis of a disc of radius R due to a charge Q uniformly distributed on its surface.

2 .32 Two charges q 1 and q

2 are placed at (0, 0, d ) and (0, 0, –d )

respectively. Find locus of points where the potential a zero.

2 .33 Two charges – q each are separated by distance 2d . A third charge+ q is kept at mid point O. Find potential energy of + q as a functionof small distance x from O due to – q charges. Sketch P.E. v/s x andconvince yourself that the charge at O is in an unstable equilibrium.

Fig. 2.7

K 2K 1

C =3C2

C =6C1

C =3C3

C2E = 9V




16

MCQ I

3 .1 Consider a current carrying wire (current I ) in the shape of a circle.Note that as the current progresses along the wire, the direction of j (current density) changes in an exact manner, while the current I remain unaffected. The agent that is essential ly responsible for is

(a) source of emf.

(b) electric field produced by charges accumulated on the surface

of wire.(c) the charges just behind a given segment of wire which push

them just the right way by repulsion.

(d) the charges ahead.

3 .2 Two batteries of emf ε 1 and ε

2 (ε

2> ε

1) and internal

resistances r 1 and r

2 respectively are connected in parallel

as shown in Fig 3.1.

(a) The equivalent emf ε eq

of the two cells is between ε 1

and ε 2, i.e. ε

1<

ε

eq<

ε

2.

Chapter Three

CURRENT

ELECTRICITY

B

r 2

r 1

A

1

Fig 3.1



Current Electricity

17

(b) The equivalent emf ε eq

is smaller than ε 1.

(c) The ε eq

is given by ε eq

= ε 1+

ε

2 always.

(d) ε eq

is independent of internal resistances r 1and r

2.

3 .3 A resistance R is to be measured using a meter bridge. Student chooses the standard resistance S to be 100Ω. He finds the nullpoint at l

1 = 2.9 cm. He is told to attempt to improve the accuracy.

Which of the following is a useful way?

(a) He should measure l 1 more accurately.

(b) He should change S to 1000Ω and repeat the experiment.(c) He should change S to 3Ω and repeat the experiment.(d) He should give up hope of a more accurate measurement with

a meter bridge.

3 .4 Two cells of emf’s approximately 5V and 10V are to be accurately compared using a potentiometer of length 400cm.

(a) The battery that runs the potentiometer should have voltage of 8V.

(b) The battery of potentiometer can have a voltage of 15V and R

adjusted so that the potential drop across the wire slightly exceeds 10V.

(c) The first portion of 50 cm of wire itself should have a potentialdrop of 10V.

(d) Potentiometer is usually used for comparing resistances and

not voltages.

3 .5 A metal rod of length 10 cm and a rectangular cross-section of

1cm ×1

2 cm is connected to a battery across opposite faces. The

resistance will be

(a) maximum when the battery is connected across 1 cm ×1

2 cm

faces.

(b) maximum when the battery is connected across 10 cm × 1 cmfaces.

(c) maximum when the battery is connected across 10 cm × 12

cm faces.

(d) same irrespective of the three faces.

3 .6 Which of the following characteristics of electrons determines thecurrent in a conductor?

(a) Drift velocity alone.(b) Thermal velocity alone.(c) Both drift velocity and thermal velocity.(d) Neither drift nor thermal velocity.




18

MCQ II3 .7 Kirchhoff ’s junction rule is a reflection of

(a) conservation of current density vector.

(b) conservation of charge.

(c) the fact that the momentum with which a charged particleapproaches a junction is unchanged (as a vector) as thecharged particle leaves the junction.

(d) the fact that there is no accumulation of charges at a junction.

3 .8 Consider a simple circuit shown in Fig 3.2. stands for a

variable resistance R ′. R ′ can vary from R 0 to infinity. r is internal

resistance of the battery (r <<R <<R 0).

(a) Potential drop across AB is nearly constant as R ′ is varied.(b) Current through R ′ is nearly a constant as R ′ is varied.(c) Current I depends sensitively on R ′.

(d)V

I r R

≥+

always.

3 .9 Temperature dependence of resistivity ρ (T) of semiconductors,insulators and metals is significantly based on the followingfactors:

(a) number of charge carriers can change with temperature T .(b) time interval between two successive collisions can depend

on T .(c) length of material can be a function of T .(d) mass of carriers is a function of T .

3 .10 The measurement of an unknown resistance R is to be carriedout using Wheatstones bridge (see Fig. 3.25 of NCERT Book).

Two students perform an experiment in two ways. The first students takes R

2 = 10Ω and R

1 = 5Ω. The other student takes R

2

= 1000Ω and R 1 = 500Ω. In the standard arm, both take R

3 = 5Ω.

Both find 23

1

10R R R R

= = Ω within errors.

(a) The errors of measurement of the two students are the same.(b) Errors of measurement do depend on the accuracy with which

R 2 and R

1 can be measured.

(c) If the student uses large values of R 2 and R

1 , the currents

through the arms will be feeble. This will make determinationof null point accurately more difficult.

(d) Wheatstone bridge is a very accurate instrument and has noerrors of measurement.

V

R

I

A

R

B

r

Fig 3.2



Current Electricity

19

R S

( )

A C

D

G

B

l 1 100 l 1

3 . 1 1 In a meter bridge the point D is a neutral point

(Fig 3.3).

(a) The meter bridge can have no other neutralpoint for this set of resistances.

(b) When the jockey contacts a point on meter wireleft of D, current flows to B from the wire.

(c) When the jockey contacts a point on the meter wire to the right of D, current flows from B tothe wire through galvanometer.

(d) When R is increased, the neutral point shifts to

left.

VSA

3 . 1 2 Is the motion of a charge across junction momentum conserving? Why or why not?

3 . 1 3 The relaxation time τ is nearly independent of applied E field whereas it changes significantly with temperature T . First fact is(in part) responsible for Ohm’s law whereas the second fact leadsto variation of ρ with temperature. Elaborate why?

3 . 1 4 What are the advantages of the null-point method in a Wheatstone bridge? What additional measurements would be required tocalculate R

unknown by any other method?

3 . 1 5 What is the advantage of using thick metallic strips to join wiresin a potentiometer?

3 . 1 6 For wiring in the home, one uses Cu wires or Al wires. What considerations are involved in this?

3 . 1 7 Why are alloys used for making standard resistance coils?

3 . 1 8 Power P is to be delivered to a device via transmissioncables having resistance R

C . If V is the voltage across R

and I the current through it, find the power wastedand how can it be reduced.

3 .19 AB is a potentiometer wire (Fig 3.4). If the value of R isincreased, in which direction will the balance point Jshift?

Fig 3.3

( )

G

E

J

R

B A

Fig 3.4




20

3 . 2 0 While doing an experiment with potentiometer (Fig 3.5) it was

found that the deflection is one sided and (i) the deflectiondecreased while moving from one end A of the wire to the end B;(ii) the deflection increased. while the jockey was moved towardsthe end B.

Fig 3.5

E1 E2

A B

Fig 3.6

(i) Which terminal +or –ve of the cell E 1 , is connected at

X in case (i) and how is E 1 related to E ?

(ii) Which terminal of the cellE 1 is connected at X in case (ii)?

3 .21 A cell of emf E and internal resistance r is connected across anexternal resistance R . Plot a graph showing the variation of P.D.

across R, verses R .

SA

3 .22 First a set of n equal resistors of R each are connected in series toa battery of emf E and internal resistanceR . A current I is observedto flow. Then the n resistors are connected in parallel to the same

battery. It is observed that the current is increased 10 times. What is ‘n ’?

3 .23 Let there be n resistors R 1 ............R

n with R

max = max (R

1......... R

n)

and R min = min {R 1 ..... R n}. Show that when they are connected inparallel, the resultant resistance R P< R

min and when they are

connected in series, the resultant resistance R S > R

max . Interpret

the result physically.

3 .24 The circuit in Fig 3.6 shows twocells connected in opposition toeach other. Cell E

1 is of emf 6V

and internal resistance 2Ω; thecell E

2 is of emf 4V and internal

resistance 8Ω. Find thepotential difference between the

points A and B.

3 .25 Two cells of same emf E but internal resistance r

1 and r

2 are

connected in series to anexternal resistor R (Fig 3.7).

What should be the value of R so that the potential differenceacross the terminals of the first cell becomes zero.

Fig. 3.7

( )

G

E

B A E

1

X y

E EB

R



Current Electricity

21

3 .26 Two conductors are made of the same material and have the same

length. Conductor A is a solid wire of diameter 1mm. Conductor Bis a hollow tube of outer diameter 2mm and inner diameter 1mm.Find the ratio of resistanceR

A to R

B.

3 .27 Suppose there is a circuit consisting of only resistances and batteries. Suppose one is to double (or increase it to n -times) all voltages and all resistances. Show that currents are unaltered.Do this for circuit of Example 3.7 in the NCERT Text Book for Class XII.

LA

3 .28 Two cells of voltage 10V and 2V and internal resistances 10Ωand 5Ω respectively, are connected in parallel with the positiveend of 10V battery connected to negative pole of 2V battery (Fig 3.8). Find the effective voltage and effective resistance of thecombination.

3 .29 A room has AC run for 5 hours a day at a voltage of 220V. The wiring of the room consists of Cu of 1 mm radius and a length of

10 m. Power consumption per day is 10 commercial units. What fraction of it goes in the joule heating in wires? What would happenif the wiring is made of aluminium of the same dimensions?

[ρcu

= 1.7 × –810 m Ω, ρ

Al= 2.7 × 10 –8 Ωm]

3 .30 In an experiment with a potentiometer, V B = 10V. R is adjusted to

be 50Ω (Fig. 3.9). A student wanting to measure voltage E 1 of a

battery (approx. 8V) finds no null point possible. He thendiminishes R to 10Ω and is able to locate the null point on thelast (4th) segment of the potentiometer. Find the resistance of thepotentiometer wire and potential drop per unit length across the

wire in the second case.

3 .31 (a) Consider circuit in Fig 3.10. How much energy is absorbed by electrons from the initial state of no current (ignore thermalmotion) to the state of drift velocity?

(b) Electrons give up energy at the rate of RI 2 per second to thethermal energy. What time scale would one associate withenergy in problem (a)? n = no of electron/volume = 1029/m3,length of circuit = 10 cm, cross-section = A = (1mm)2

Fig 3.8

B

( )

A G

N1

CK 1

R

E1E2

1

2

3

Fig 3.9

R= 6

I

V = 6V

I

Fig 3.10

2V

I 2

I

R

1010V

I 1




MCQ I

4 .1 Two charged particles traverse identical helical paths in a completely

opposite sense in a uniform magnetic field B = B0 k .

(a) They have equal z-components of momenta.

(b) They must have equal charges.

(c) They necessarily represent a particle-antiparticle pair.

(d) The charge to mass ratio satisfy :1 2

0e e

m m

⎛ ⎞ ⎛ ⎞+ =⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

.

4 .2 Biot-Savart law indicates that the moving electrons (velocity v )produce a magnetic field B such that

(a) B ⊥ v.

(b) B ||v.

(c) it obeys inverse cube law.

(d) it is along the line joining the electron and point of observation.

Chapter Four

MOVING CHARGES

AND MAGNETISM



Moving Charges and Magnetism

23

4 .3 A current carrying circular loop of radius

R is placed in the

x -y

plane with centre at the origin. Half of the loop with x > 0 is now bent so that it now lies in the y -z plane.

(a) The magnitude of magnetic moment now diminishes.

(b) The magnetic moment does not change.

(c) The magnitude of B at (0.0.z ), z >>R increases.

(d) The magnitude of B at (0.0.z ), z >>R is unchanged.

4 .4 An electron is projected with uniform velocity along the axis of a current carrying long solenoid. Which of the following is true?

(a) The electron will be accelerated along the axis.

(b) The electron path will be circular about the axis.

(c) The electron will experience a force at 45° to the axis and henceexecute a helical path.

(d) The electron will continue to move with uniform velocity alongthe axis of the solenoid.

4 .5 In a cyclotron, a charged particle

(a) undergoes acceleration all the time.

(b) speeds up between the dees because of the magnetic field.

(c) speeds up in a dee.

(d) slows down within a dee and speeds up between dees.

4 .6 A circular current loop of magnetic moment M is in an arbitrary orientation in an external magnetic field B. The work done to rotatethe loop by 30° about an axis perpendicular to its plane is

(a) MB.

(b) 32

MB .

(c)2

MB .

(d) zero.

MCQ II

4 .7 The gyro-magnetic ratio of an electron in an H-atom, according toBohr model, is

(a) independent of which orbit it is in.

(b) negative.

(c) positive.

(d) increases with the quantum number n .

23




24

4 .8 Consider a wire carrying a steady current, I placed in a uniform

magnetic field B perpendicular to its length. Consider the chargesinside the wire. It is known that magnetic forces do no work. Thisimplies that,

(a) motion of charges inside the conductor is unaffected by B sincethey do not absorb energy.

(b) some charges inside the wire move to the surface as a result of B.

(c) if the wire moves under the influence of B, no work is done by the force.

(d) if the wire moves under the influence of B, no work is done by

the magnetic force on the ions, assumed fixed within the wire.

4 .9 Two identical current carrying coaxial loops, carry current I in anopposite sense. A simple amperian loop passes through both of them once. Calling the loop as C ,

(a) 02c

I μ =∫ B.dl m Ñ .

(b) the value ofc

∫ B.dl Ñ is independent of sense of C.

(c) there may be a point on C where B and dl are perpendicular.(d) B vanishes everywhere on C.

4 .10 A cubical region of space is filled with some uniform electric andmagnetic fields. An electron enters the cube across one of its faces

with velocity v and a positron enters via opposite face with velocity - v. At this instant,

(a) the electric forces on both the particles cause identicalaccelerations.

(b) the magnetic forces on both the particles cause equalaccelerations.

(c) both particles gain or loose energy at the same rate.

(d) the motion of the centre of mass (CM) is determined by B alone.

4 .11 A charged particle would continue to move with a constant velocity in a region wherein,

(a) E = 0, 0.≠B

(b) 0, 0.≠ ≠E B

(c) 0, 0.≠ =E B

(d) E = 0, B = 0.




25

VSA4 .12 Verify that the cyclotron frequency ω = eB /m has the correct

dimensions of [T] –1.

4 .13 Show that a force that does no work must be a velocity dependent force.

4 .14 The magnetic force depends on v which depends on the inertialframe of reference. Does then the magnetic force differ from inertialframe to frame? Is it reasonable that the net acceleration has a

different value in different frames of reference?

4 .15 Describe the motion of a charged particle in a cyclotron if thefrequency of the radio frequency (rf) field were doubled.

4 .16 Two long wires carrying current I 1 and I

2 are arranged as shown in

Fig. 4.1. The one carrying current I 1 is along is the x -axis. The

other carrying current I 2 is along a line parallel to the y -axis given

by x = 0 and z = d . Find the force exerted at O2 because of the wire

along the x -axis.

SA

4 .17 A current carrying loop consists of 3 identical quarter circles of radius R , lying in the positive quadrants of the x-y, y -z and z-x planes with their centres at the origin, joined together. Find thedirection and magnitude of B at the origin.

4 .18 A charged particle of charge e and mass m is moving in an electricfield E and magnetic field B. Construct dimensionless quantitiesand quantities of dimension [T ] –1.

4 .19 An electron enters with a velocity v = v 0i into a cubical region (faces

parallel to coordinate planes) in which there are uniform electricand magnetic fields. The orbit of the electron is found to spiraldown inside the cube in plane parallel to the x-y plane. Suggest a configuration of fields E and B that can lead to it.

4 .20 Do magnetic forces obey Newton’s third law. Verify for two current

elements d l 1 = d l i located at the origin and d l

2 = d l j located at

(0, R , 0). Both carry current I .

O1

O2

z

I 2

y

d

I 1

x Fig. 4.1




26

GR 1 R 2 R 3

2V 20V 200V

4 .21 A mult irange vo ltmeter can be constructed by us ing a

galvanometer circuit as shown in Fig. 4.2. We want to construct a voltmeter that can measure 2V, 20V and 200V using a galvanometer of resistance 10Ω and that produces maximumdeflection for current of 1 mA. Find R

1, R

2 and R

3 that have to be

used.

4 .22 A long straight wire carrying current of 25A rests on a table asshown in Fig. 4.3. Another wire PQ of length 1m, mass 2.5 gcarries the same current but in the opposite direction. The wirePQ is free to slide up and down. To what height will PQ rise?

LA

4 .23 A 100 turn rectangular coil ABCD (in XY plane) is hung from onearm of a balance (Fig. 4.4). A mass 500g is added to the other armto balance the weight of the coil. A current 4.9 A passes throughthe coil and a constant magnetic field of 0.2 T acting inward (in xz

plane) is switched on such that only arm CD of length 1 cm lies inthe field. How much additional mass ‘m ’ must be added to regainthe balance?

4 .24 A rectangular conducting loop consists of two wires on two oppositesides of length l joined together by rods of length d . The wires areeach of the same material but with cross-sections differing by a factor of 2. The thicker wire has a resistance R and the rods are of low resistance, which in turn are connected to a constant voltagesource V

0. The loop is placed in uniform a magnetic field B at 45°

to its plane. Find τττττ, the torque exerted by the magnetic field on theloop about an axis through the centres of rods.

4 .25 An electron and a positron are released from (0, 0, 0) and

(0, 0, 1.5R ) respectively, in a uniform magnetic field B = 0ˆB i , each

with an equal momentum of magnitude p = e BR. Under what conditions on the direction of momentum will the orbits be non-intersecting circles?

4 .26 A uniform conducting wire of length 12a and resistance R is woundup as a current carrying coil in the shape of (i) an equilateraltriangle of side a ; (ii) a square of sides a and, (iii) a regular hexagonof sides a . The coil is connected to a voltage source V

0. Find the

magnetic moment of the coils in each case.

Fig. 4.2

Fig. 4.3

X X X

X X X

B A

D C X

X

X

X X

Fig. 4.4

P Q

h




27

4 .27 Consider a circular current-carrying loop of radius R in the x-y

plane with centre at origin. Consider the line intergral

( ) –

.L

L

Lℑ = ∫ B dl taken along z-axis.

(a) Show thatℑ (L) monotonically increases with L.

(b) Use an appropriate Amperian loop to show that ( ) I 0μ ℑ =∞ ,

where I is the current in the wire.

(c) Verify directly the above result.

(d) Suppose we replace the circular coil by a square coil of sides R

carrying the same current I . What can you say about

( )Lℑ and ( )ℑ ∞ ?

4 .28 A multirange current meter can be constructed by using a galvanometer circuit as shown in Fig. 4.5. We want a current meter that can measure 10mA, 100mA and 1A using a galvanometer of resistance 10Ω and that prduces maximum deflection for current of 1mA. Find S

1, S

2 and S

3 that have to be used

4 .29 Five long wires A, B, C, D and E, each carrying current I arearranged to form edges of a pentagonal prism as shown in Fig.

4.6. Each carries current out of the plane of paper.(a) What will be magnetic induction at a point on the axis O? Axis

is at a distance R from each wire.

(b) What will be the field if current in one of the wires (say A) isswitched off?

(c) What if current in one of the wire (say) A is reversed?

A

B

C

E

D

O

R

G

S3

S1 S2

10mA 100mA 1A

Fig. 4.5

Fig. 4.6




28

MCQ I

5 .1 A toroid of n turns, mean radius R and cross-sectional radius a

carries current I . It is placed on a horizontal table taken asx -y plane. Its magnetic moment m

(a) is non-zero and points in the z -direction by symmetry.

(b) points along the axis of the tortoid ( m =m ).

(c) is zero, otherwise there would be a field falling as 31r

at large

distances outside the toroid.

(d) is pointing radially outwards.

5 .2 The magnetic field of Earth can be modelled by that of a point dipoleplaced at the centre of the Earth. The dipole axis makes an angle of 11.3° with the axis of Earth. At Mumbai, declination is nearly zero. Then,

(a) the declination varies between 11.3° W to 11.3° E.

(b) the least declination is 0°.

Chapter Five

MAGNETISM AND

MATTER



Magnetism and Matter

29

(c) the plane defined by dipole axis and Earth axis passes through

Greenwich.

(d) declination averaged over Earth must be always negative.

5 .3 In a permanent magnet at room temperature

(a) magnetic moment of each molecule is zero.

(b) the individual molecules have non-zero magnetic moment which are all perfectly aligned.

(c) domains are partially aligned.

(d) domains are all perfectly aligned.

5 .4 Consider the two idealized systems: (i) a parallel plate capacitor

with large plates and small separation and (ii) a long solenoid of length L >> R , radius of cross-section. In (i) E is ideally treated asa constant between plates and zero outside. In (ii) magnetic field isconstant inside the solenoid and zero outside. These idealisedassumptions, however, contradict fundamental laws as below:

(a) case (i) contradicts Gauss’s law for electrostatic fields.

(b) case (ii) contradicts Gauss’s law for magnetic fields.

(c) case (i) agrees with .d 0=∫ E l .

(d) case (ii) contradicts en I .d =∫ H l

5 .5 A paramagnetic sample shows a net magnetisation of 8 Am –1 whenplaced in an external magnetic field of 0.6T at a temperature of 4K. When the same sample is placed in an external magnetic fieldof 0.2T at a temperature of 16K, the magnetisation will be

(a) –132

Am3

(b) –12

m3

(c) –16 Am

(d) –12.4 Am .

MCQ II

5 .6 S is the surface of a lump of magnetic material.

(a) Lines of B are necessarily continuous across S .

(b) Some lines of B must be discontinuous across S .

(c) Lines of H are necessarily continuous across S .

(d) Lines of H cannot all be continuous across S .




30

5 .7 The primary origin(s) of magnetism lies in

(a) atomic currents.(b) Pauli exclusion principle.(c) polar nature of molecules.(d) intrinsic spin of electron.

5 .8 A long solenoid has 1000 turns per metre and carries a current of

1 A. It has a soft iron core of r 1000μ = . The core is heated beyond

the Curie temperature, T c .

(a) The H field in the solenoid is (nearly) unchanged but the B field

decreases drastically.

(b) The H and B fields in the solenoid are nearly unchanged.(c) The magnetisation in the core reverses direction.

(d) The magnetisation in the core diminishes by a factor of about 108.

5 .9 Essential difference between electrostatic shielding by a conductingshell and magnetostatic shielding is due to

(a) electrostatic field lines can end on charges and conductors havefree charges.

(b) lines of B can also end but conductors cannot end them.

(c) lines of B cannot end on any material and perfect shielding is

not possible.(d) shells of high permeability materials can be used to divert lines

of B from the interior region.

5 .10 Let the magnetic field on earth be modelled by that of a point magnetic dipole at the centre of earth. The angle of dip at a point onthe geographical equator

(a) is always zero.

(b) can be zero at specific points.

(c) can be positive or negative.

(d) is bounded.

VSA

5 .11 A proton has spin and magnetic moment just like an electron. Why then its effect is neglected in magnetism of materials?

5 .12 A permanent magnet in the shape of a thin cylinder of length 10 cmhas M = 106 A/m. Calculate the magnetisation current I

M .

5 .13 Explain quantitatively the order of magnitude difference between thediamagnetic susceptibility of N

2 (~5 × 10 –9) (at STP) and Cu (~10 –5).



Magnetism and Matter

31

5 .14 From molecular view point, discuss the temperature dependence

of susceptibility for diamagnetism, paramagnetism andferromagnetism.

5 .15 A ball of superconducting material is dipped in liquid nitrogen andplaced near a bar magnet. (i) In which direction will it move?(ii) What will be the direction of it’s magnetic moment?

SA

5 .16 Verify the Gauss’s law for magnetic field of a point dipole of dipolemoment m at the origin for the surface which is a sphere of radius R .

5 .17 Three identical bar magnets are rivetted together at centre in thesame plane as shown in Fig. 5.1. This system is placed at rest in a slowly varying magnetic field. It is found that the system of magnetsdoes not show any motion. The north-south poles of one magnet isshown in the Fig. 5.1. Determine the poles of the remaining two.

5 .18 Suppose we want to verify the analogy between electrostatic andmagnetostatic by an explicit experiment. Consider the motion of (i) electric dipole p in an electrostatic field E and (ii) magnetic dipole

m in a magnetic field B. Write down a set of conditions on E, B, p,

m so that the two motions are verified to be identical. (Assume

identical initial conditions.)

5 .19 A bar magnet of magnetic momentm and moment of inertia I (about centre, perpendicular to length) is cut into two equal pieces,perpendicular to length. Let T be the period of oscillations of theoriginal magnet about an axis through the mid point, perpendicular to length, in a magnetic field B. What would be the similar period T ′

for each piece?

5 .20 Use (i) the Ampere’s law for H and (ii) continuity of lines of B, toconclude that inside a bar magnet, (a) lines of H run from the N poleto S pole, while (b) lines of B must run from the S pole to N pole.

LA

5.21 Verify the Ampere’s law for magnetic field of a point dipole of dipole

moment m =m k . Take C as the closed curve running clockwise

along (i) the z -axis from z = a > 0 to z = R ; (ii) along the quarter circleof radius R and centre at the origin, in the first quadrant of x -z plane; (iii) along the x -axis from x = R to x = a , and (iv) along thequarter circle of radius a and centre at the origin in the first quadrant of x -z plane.

60°

60°

? ?

??

N

S

Fig. 5.1




32

5 .22 What are the dimensions of χ , the magnetic susceptibility? Consider

an H-atom. Guess an expression for χ , upto a constant by constructing a quantity of dimensions of χ , out of parameters of

the atom: e , m , v , R and 0μ . Here, m is the electronic mass, v is

electronic velocity, R is Bohr radius. Estimate the number so

obtained and compare with the value of –5~10 χ for many

solid materials.

5 .23 Assume the dipole model for earth’s magnetic fieldB which is given

by B V

= vertical component of magnetic field = 03

2 cos

4

m

r

μ θ

π

B H = Horizontal component of magnetic field =

0

34

s in m

r

μ θ

π

θ = 90° – lattitude as measured from magnetic equator.

Find loci of points for which (i) B is minimum; (ii) dip angle is zero;

and (iii) dip angle is ± 45°.

5 .24 Consider the plane S formed by the dipole axis and the axis of earth.Let P be point on the magnetic equator and in S . Let Q be the point of intersection of the geographical and magnetic equators. Obtainthe declination and dip angles at P and Q .

5 .25 There are two current carrying planar coils made each from identical wires of length L. C

1

is circular (radius R ) and C2

is square (side a ). They are so constructed that they have same frequency of oscillation when they are placed in the same uniform B and carry the samecurrent. Find a in terms of R .



MCQ I

9 .1 A ray of light incident at an angle θ on a refracting face of a prismemerges from the other face normally. If the angle of the prism is 5°and the prism is made of a material of refractive index 1.5, the angleof incidence is

(a) 7.5°.

(b) 5°.

(c) 15°.

(d) 2.5°.

9 .2 A short pulse of white light is incident from air to a glass slab at normal incidence. After travelling through the slab, the first colour to emerge is

(a) blue.

(b) green.

(c) violet.

(d) red.

Chapter Nine

RAY OPTICS AND

OPTICALINSTRUMENTS



Ray Optics and Optical Instruments

55

9 .3 An object approaches a convergent lens from the left of the lens

with a uniform speed 5 m/s and stops at the focus. The image

(a) moves away from the lens with an uniform speed 5 m/s.

(b) moves away from the lens with an uniform accleration.

(c) moves away from the lens with a non-uniform acceleration.

(d) moves towards the lens with a non-uniform acceleration.

9 .4 A passenger in an aeroplane shall

(a) never see a rainbow.

(b) may see a primary and a secondary rainbow as concentric circles.

(c) may see a primary and a secondary rainbow as concentric arcs.

(d) shall never see a secondary rainbow.

9 .5 You are given four sources of light each one providing a light of a single colour – red, blue, green and yellow. Suppose the angle of refraction for a beam of yellow light corresponding to a particular angle of incidence at the interface of two media is 90°. Which of thefollowing statements is correct if the source of yellow light is replaced

with that of other lights without changing the angle of incidence?

(a) The beam of red light would undergo total internal reflection.

(b) The beam of red light would bend towards normal while it getsrefracted through the second medium.

(c) The beam of blue light would undergo total internal reflection.(d) The beam of green light would bend away from the normal as it

gets refracted through the second medium.

9 .6 The radius of curvature of the curved surface of a plano-convex lens is 20 cm. If the refractive index of the material of the lens be1.5, it will

(a) act as a convex lens only for the objects that lie on its curvedside.

(b) act as a concave lens for the objects that lie on its curved side.

(c) act as a convex lens irrespective of the side on which the object

lies.(d) act as a concave lens irrespective of side on which the object lies.

9 .7 The phenomena involved in the reflection of radiowaves by ionosphere is similar to

(a) reflection of light by a plane mirror.

(b) total internal reflection of light in air during a mirage.

(c) dispersion of light by water molecules during the formation of a rainbow.

(d) scattering of light by the particles of air.




56

1

2

C

P3

F

Q 49 .8 The direction of ray of light incident on a concave mirror is shown

by PQ while directions in which the ray would travel after reflectionis shown by four rays marked 1, 2, 3 and 4(Fig 9.1). Which of the four rays correctly shows the direction of reflected ray?

(a) 1

(b) 2

(c) 3

(d) 4

9 .9 The optical density of turpentine is higher than that of water while

its mass density is lower. Fig 9.2. shows a layer of turpentinefloating over water in a container. For which one of the four raysincident on turpentine in Fig 9.2, the path shown is correct?

(a) 1

(b) 2

(c) 3

(d) 4

1 2 3 4

Air

Turpentine

Water

9 .10 A car is moving with at a constant speed of 60 km h –1 on a straight road. Looking at the rear view mirror, the driver finds that the car following him is at a distance of 100 m and is approaching with a speed of 5 km h –1. In order to keep track of the car in the rear, thedriver begins to glance alternatively at the rear and side mirror of

his car after every 2 s till the other car overtakes. If the two cars were maintaining their speeds, which of the following statement (s) is/are correct?

(a) The speed of the car in the rear is 65 km h –1.

(b) In the side mirror the car in the rear would appear to approach with a speed of 5 km h –1 to the driver of the leading car.

(c) In the rear view mirror the speed of the approaching car wouldappear to decrease as the distance between the cars decreases.

(d) In the side mirror, the speed of the approaching car wouldappear to increase as the distance between the cars decreases.

Fig. 9.1

Fig. 9.2




57

9 .11 There are certain material developed in laboratories which have a

negative refractive index (Fig. 9.3). A ray incident from air (medium 1)into such a medium (medium 2) shall follow a path given by

i

r

1

2

i

r

1

2

1

2

(a) (b)

(d)(c)

i r 1

2

MCQ II

9 .12 Consider an extended object immersed in water contained in a plane trough. When seen from close to the edge of the trough theobject looks distorted because

(a) the apparent depth of the points close to the edge are nearer the surface of the water compared to the points away from theedge.

(b) the angle subtended by the image of the object at the eye issmaller than the actual angle subtended by the object in air.

(c) some of the points of the object far away from the edge may not be visible because of total internal reflection.

(d) water in a trough acts as a lens and magnifies the object.

9 .13 A rectangular block of glass ABCD has a refractive index 1.6. A pin is placed midway on the face AB (Fig. 9.4). When observed

from the face AD, the pin shall

(a) appear to be near A.

(b) appear to be near D.

(c) appear to be at the centre of AD.

(d) not be seen at all.

9 .14 Between the primary and secondary rainbows, there is a dark band known as Alexandar’s dark band. This is because

(a) light scattered into this region interfere destructively.(b) there is no light scattered into this region.

A B

D C

Fig. 9.3

Fig. 9.4




58

(c) light is absorbed in this region.

(d) angle made at the eye by the scattered rays with respect to theincident light of the sun lies between approximately 42° and50°.

9 . 1 5 A magnifying glass is used, as the object to be viewed can be brought closer to the eye than the normal near point. Thisresults in

(a) a larger angle to be subtended by the object at the eye andhence viewed in greater detail.

(b) the formation of a virtual erect image.

(c) increase in the field of view.

(d) infinite magnification at the near point.

9 .16 An astronomical refractive telescope has an objective of focal length20m and an eyepiece of focal length 2cm.

(a) The length of the telescope tube is 20.02m.(b) The magnification is 1000.(c) The image formed is inverted.(d) An objective of a larger aperture will increase the brightness

and reduce chromatic aberration of the image.

VSA

9 .17 Will the focal length of a lens for red light be more, same or lessthan that for blue light?

9 .18 The near vision of an average person is 25cm. To view an object with an angular magnification of 10, what should be the power of the microscope?

9 .19 An unsymmetrical double convex thin lens forms the image of a point object on its axis. Will the position of the image change if the lens is reversed?

9 .20 Three immiscible liquids of densities d 1 > d 2 > d 3 and refractiveindices μ

1 > μ

2 > μ

3 are put in a beaker. The height of each liquid

column is3

h . A dot is made at the bottom of the beaker. For near

normal vision, find the apparent depth of the dot.

9 .21 For a glass prism (μ = 3 ) the angle of minimum deviation is

equal to the angle of the prism. Find the angle of the prism.




59

SA9 .22 A short object of length L is placed along the principal axis of a

concave mirror away from focus. The object distance is u . If themirror has a focal length f , what will be the length of the image?

You may take L <<|v - f |.

9 .23 A circular disc of radius ‘R ’ is placed co-axially and horizontally inside an opaque hemispherical bowl of radius ‘a ’ (Fig. 9.5). Thefar edge of the disc is just visible when viewed from the edge of the

bowl. The bowl is filled with transparent liquid of refractive index μ and the near edge of the disc becomes just visible. How far below

the top of the bowl is the disc placed?

9 .24 A thin convex lens of focal length 25 cm is cut into two pieces0.5 cm above the principal axis. The top part is placed at (0,0) andan object placed at (–50 cm, 0). Find the coordinates of the image.

9 .25 In many experimental set-ups the source and screen are fixed at a distance say D and the lens is movable. Show that there are twopositions for the lens for which an image is formed on the screen.Find the distance between these points and the ratio of the imagesizes for these two points.

9 .26 A jar of height h is filled with a transparent liquid of refractiveindex μ (Fig. 9.6). At the centre of the jar on the bottom surface isa dot. Find the minimum diameter of a disc, such that when placedon the top surface symmetrically about the centre, the dot isinvisible.

O

9 0 -

A BC

i

d

Ma a

R R

Fig. 9.5

Fig. 9.6

d

O

h i i

9 .27 A myopic adult has a far point at 0.1 m. His power of accomodationis 4 diopters. (i) What power lenses are required to see distant




60

objects? (ii) What is his near point without glasses? (iii) What is

his near point with glasses? (Take the image distance from thelens of the eye to the retina to be 2 cm.)

LA

9 .28 Show that for a material with refractive index μ ≥ 2 , light incident

at any angle shall be guided along a length perpendicular to theincident face.

9 .29 The mixture a pure liquid and a solution in a long vertical column(i.e, horizontal dimensions << vertical dimensions) producesdiffusion of solute particles and hence a refractive index gradient along the vertical dimension. A ray of light entering the column at right angles to the vertical is deviated from its original path. Findthe deviation in travelling a horizontal distance d << h , the height of the column.

9 .30 If light passes near a massive object, the gravitational interactioncauses a bending of the ray. This can be thought of as happeningdue to a change in the effective refrative index of the mediumgiven by

n (r ) = 1 + 2 GM /rc 2

where r is the distance of the point of consideration from the centreof the mass of the massive body, G is the universal gravitationalconstant, M the mass of the body and c the speed of light in

vacuum. Considering a spherical object find the deviation of theray from the original path as it grazes the object.

9 .31 An infinitely long cylinder of radius R is made of an unusual exoticmaterial with refractive index –1 (Fig. 9.7). The cylinder is placed

between two planes whose normals are along the y direction. Thecenter of the cylinder O lies along the y -axis. A narrow laser beam

is directed along the y direction from the lower plate. The laser source is at a horizontal distance x from the diameter in they direction. Find the range of x such that light emitted from thelower plane does not reach the upper plane.

9 .32 (i) Consider a thin lens placed between a source (S) and anobserver (O) (Fig. 9.8). Let the thickness of the lens vary as

2

0( ) – α

= b

w b w , where b is the verticle distance from the pole.

w 0is a constant. Using Fermat’s principle i.e. the time of transit

y

x

i R

x

o

i

r

Fig. 9.7




61

for a ray between the source and observer is an

extremum, find the condition that all paraxial raysstarting from the source will converge at a point Oon the axis. Find the focal length.

(ii) A gravitational lens may be assumed to have a varying width of the form

21( ) k

w b k l n b

⎛ ⎞= ⎜ ⎟⎝ ⎠

b min

< b < b max

21

min

k k ln

b

⎛ ⎞= ⎜ ⎟

⎝ ⎠

b < b min

Show that an observer will see an image of a point object as a ring about the center of the lens with an angular radius

( ) 1 –1 u k n

v

u v β =

+.

P 1

S u v O

b

Fig. 9.8



MCQ I

10 .1 Consider a light beam incident from air to a glass slab at Brewster’sangle as shown in Fig. 10.1.

A polaroid is placed in the path of the emergent ray at point Pand rotated about an axis passing through the centreand perpendicular to the plane of the polaroid.

(a) For a particular orientation there shall be darknessas observed through the polaoid.

(b) The intensity of light as seen through the polaroidshall be independent of the rotation.

(c) The intensity of light as seen through the Polaroidshall go through a minimum but not zero for twoorientations of the polaroid.

(d) The intensity of light as seen through the polaroid shallgo through a minimum for four orientations of thepolaroid.

10 .2 Consider sunlight incident on a slit of width 104 A. The imageseen through the slit shall

Chapter Ten

WAVE OPTICS

PFig. 10.1



Wave Optics

63

(a) be a fine sharp slit white in colour at the center.

(b) a bright slit white at the center diffusing to zero intensities at the edges.

(c) a bright slit white at the center diffusing to regions of different colours.

(d) only be a diffused slit white in colour.

10 .3 Consider a ray of light incident from air onto a slab of glass(refractive index n ) of width d , at an angle θ . The phase difference

between the ray reflected by the top surface of the glass and the bottom surface is

(a)1/2

2

2

4 11 – sin θ π ⎛ ⎞ + π⎜ ⎟λ ⎝ ⎠

d

n

(b)

1/22

2

4 11 – sin

d

n θ

π ⎛ ⎞⎜ ⎟λ ⎝ ⎠

(c)

1/22

2

4 11 – sin

2

d

n θ

π π⎛ ⎞ +⎜ ⎟λ ⎝ ⎠

(d)

1/22

2

4 121 – sin

d

n θ

π ⎛ ⎞ + π⎜ ⎟λ ⎝ ⎠.

10 .4 In a Young’s double slit experiment, the source is white light. Oneof the holes is covered by a red filter and another by a blue filter.In this case

(a) there shall be alternate interference patterns of red and blue.

(b) there shall be an interference pattern for red distinct from that for blue.

(c) there shall be no interference fringes.

(d) there shall be an interference pattern for red mixing with onefor blue.

10 .5 Figure 10.2 shows a standard two slit arrangement with slits S

1, S

2. P

1, P

2 are the two

minima points on either side of P (Fig. 10.2).

At P2 on the screen, there is a hole and behind P

2

is a second 2- slit arrangement with slits S3, S

4

and a second screen behind them.

(a) There would be no interference pattern on thesecond screen but it would be lighted.

(b) The second screen would be totally dark.

Screen

P1

P2

S3

S4

S1

S2 Second

Screen

SP

Fig. 10.2




64

(c) There would be a single bright point on the second screen.

(d) There would be a regular two slit pattern on the second screen.

MCQ II

10 .6 Two source S1and S

2 of intensity I

1 and I

2 are placed in front of a

screen [Fig. 10.3 (a)]. The patteren of intensity distribution seenin the central portion is given by Fig. 10.3 (b).

In this case which of the following statements are true.

(a) S 1 and S

2 have the same intensities.

(b) S 1 and S

2 have a constant phase difference.

(c) S 1 and S 2 have the same phase.(d) S

1 and S

2 have the same wavelength.

10 .7 Consider sunlight incident on a pinhole of width 103 A. The imageof the pinhole seen on a screen shall be

(a) a sharp white ring.

(b) different from a geometrical image.

(c) a diffused central spot, white in colour.

(d) diffused coloured region around a sharp central white spot.

10 .8 Consider the diffraction patern for a small pinhole. As the size of

the hole is increased(a) the size decreases.

(b) the intensity increases.

(c) the size increases.

(d) the intensity decreases.

10 .9 For light diverging from a point source

(a) the wavefront is spherical.

(b) the intensity decreases in proportion to the distance squared.

(c) the wavefront is parabolic.

(d) the intensity at the wavefront does not depend on the distance.

VSA

10 .10 Is Huygen’s principle valid for longitudunal sound waves?

10.11 Consider a point at the focal point of a convergent lens. Another convergent lens of short focal length is placed on the other side.

What is the nature of the wavefronts emerging from the final image?

10 .12 What is the shape of the wavefront on earth for sunlight?

x x = 0

Fig. 10.3 (b)

x

S1

S2

Fig. 10.3 (a)



Wave Optics

65

10 .13 Why is the diffraction of sound waves more evident in daily

experience than that of light wave?

10 .14 The human eye has an approximate angular resolution of

–45.8 10= ×φ rad and a typical photoprinter prints a minimum

of 300 dpi (dots per inch, 1 inch = 2.54 cm). At what minimaldistance z should a printed page be held so that one does not seethe individual dots.

10 .15 A polariod (I) is placed in front of a monochromatic source. Another polatiod (II) is placed in front of this polaroid (I) androtated till no light passes. A third polaroid (III) is now placed in

between (I) and (II). In this case, will light emerge from (II). Explain.

SA

10 .16 Can reflection result in plane polarised light if the light is incident on the interface from the side with higher refractive index?

10 .17 For the same objective, find the ratio of the least separation between two points to be distinguished by a microscope for light

of 5000 A o

and electrons accelerated through 100V used as the

illuminating substance.

10 .18 Consider a two slit interference arrangements (Fig. 10.4) suchthat the distance of the screen from the slits is half the distance

between the slits. Obtain the value of D in terms of λ suchthat the first minima on the screen falls at a distance D from thecentre O.

S1

P

S

S2

1

2

Fig. 10.4

Fig. 10.5

S1 T

1

P

O

T 2

S

Source

C

S2

Screen

OP = x

CO = D

S1C = CS

2 = D

LA

10 .19 Figure 10.5 shown a two slit arrangement with a source whichemits unpolarised light. P is a polariser with axis whose directionis not given. If I

0 is the intensity of the principal maxima when no

polariser is present, calculate in the present case, the intensity of the principal maxima as well as of the first minima.




66

R 1 A B C

D

/2

/2 /2

R 2

10 .20

Screen

P1

S1

S2

A C

O

L

d =

/ 4

AC = CO = D , S1C = S

2C = d << D

A small transparent slab containing material of 1.5μ = is placed

along AS2(Fig.10.6). What will be the distance from O of the

principal maxima and of the first minima on either side of theprincipal maxima obtained in the absence of the glass slab. .

10 .21 Four identical monochromatic sources A,B,C,D as shown in the(Fig.10.7) produce waves of the same wavelength λ and arecoherent. Two receiver R

1 and R

2 are at great but equal distaces

from B.

(i) Which of the two receivers picks up the larger signal?(ii) Which of the two receivers picks up the larger signal when Bis turned off?

(iii) Which of the two receivers picks up the larger signal when Dis turned off?

(iv) Which of the two receivers can distinguish which of the sourcesB or D has been turned off?

Fig. 10.6

Fig. 10.7

R 1B = d = R

2 B

AB = BC = BD = λ/2



Wave Optics

67

10 .22 The optical properties of a medium are governed by the relative

permitivity ( )r

ε and relative permeability (μ r). The refractive index

is defined asr r

n .μ ε = For ordinary material r ε > 0 and μ

r> 0 and

the positive sign is taken for the square root. In 1964, a Russian

scientist V. Veselago postulated the existence of material with r ε

< 0 and μr < 0. Since then such ‘metamaterials’ have been

produced in the laboratories and their optical properties studied.

For such materialsr r

n – = μ ε . As light enters a medium of such

refractive index the phases travel away from the direction of propagation.

(i) According to the description above show that if rays of light enter such a medium from air (refractive index =1) at an angleθ in 2nd quadrant, them the refracted beam is in the 3rd quadrant.

(ii) Prove that Snell’s law holds for such a medium.

10 .23 To ensure almost 100 per cent transmittivity, photographic lensesare often coated with a thin layer of dielectric material. The refractiveindex of this material is intermediated between that of air and glass(which makes the optical element of the lens). A typically useddielectric film is MgF

2 (n = 1.38). What should the thickness of the

film be so that at the center of the visible speetrum (5500 Ao

) there

is maximum transmission.



Atoms

75

MCQ I

12 .1 Taking the Bohr radius as a 0 = 53pm, the radius of Li++ ion in itsground state, on the basis of Bohr’s model, will be about

(a) 53 pm(b) 27 pm(c) 18 pm(d) 13 pm

12 .2 The binding energy of a H-atom, considering an electron moving

around a fixed nuclei (proton), is4

2 2 20

– 8me

B n h ε

= . (m = electron

mass).

If one decides to work in a frame of reference where the electron isat rest, the proton would be moving arround it. By similar arguments, the binding energy would be

4

2 2 20

– 8Me

B n h

=ε

(M = proton mass)

Chapter Twelve

ATOMS




76

This last expression is not correct because

(a) n would not be integral.(b) Bohr-quantisation applies only to electron(c) the frame in which the electron is at rest is not inertial.(d) the motion of the proton would not be in circular orbits, even

approximately.

12 .3 The simple Bohr model cannot be directly applied to calculate theenergy levels of an atom with many electrons. This is because

(a) of the electrons not being subject to a central force.(b) of the electrons colliding with each other (c) of screening effects

(d) the force between the nucleus and an electron will no longer be given by Coulomb’s law.

12 .4 For the ground state, the electron in the H-atom has an angular momentum = h , according to the simple Bohr model. Angular momentum is a vector and hence there will be infinitely many orbits with the vector pointing in all possible directions. Inactuality, this is not true,

(a) because Bohr model gives incorrect values of angular momentum.

(b) because only one of these would have a minimum energy.

(c) angular momentum must be in the direction of spin of electron.(d) because electrons go around only in horizontal orbits.

12 .5 O2 molecule consists of two oxygen atoms. In the molecule, nuclear force between the nuclei of the two atoms

(a) is not important because nuclear forces are short-ranged.(b) is as important as electrostatic force for binding the two atoms.(c) cancels the repulsive electrostatic force between the nuclei.(d) is not important because oxygen nucleus have equal number

of neutrons and protons.

12 .6 Two H atoms in the ground state collide inelastically. The

maximum amount by which their combined kinetic energy isreduced is

(a) 10.20 eV (b) 20.40 eV (c) 13.6 eV (d) 27.2 eV

12 .7 A set of atoms in an excited state decays.

(a) in general to any of the states with lower energy.(b) into a lower state only when excited by an external electric

field.



Atoms

77

(c) all together simultaneously into a lower state.

(d) to emit photons only when they collide.

MCQ II

12 .8 An ionised H-molecule consists of an electron and two protons. The protons are separated by a small distance of the order of angstrom. In the ground state,

(a) the electron would not move in circular orbits.(b) the energy would be (2)4 times that of a H-atom.(c) the electrons, orbit would go arround the protons.

(d) the molecule will soon decay in a proton and a H-atom.

12 .9 Consider aiming a beam of free electrons towards free protons. When they scatter, an electron and a proton cannot combine toproduce a H-atom,

(a) because of energy conservation.(b) without simultaneously releasing energy in the from of

radiation.(c) because of momentum conservation.(d) because of angular momentum conservation.

12 .10 The Bohr model for the spectra of a H-atom

(a) will not be applicable to hydrogen in the molecular from.(b) will not be applicable as it is for a He-atom.(c) is valid only at room temperature.(d) predicts continuous as well as discrete spectral lines.

12 .11 The Balmer series for the H-atom can be observed

(a) if we measure the frequencies of light emitted when an excitedatom falls to the ground state.

(b) if we measure the frequencies of light emitted due totransitions between excited states and the first excited state.(c) in any transition in a H-atom.(d) as a sequence of frequencies with the higher frequencies

getting closely packed.

12 .12 Let4

2 2 20

18n

me E

n h ε

−= be the energy of the n th level of H-atom. If all

the H-atoms are in the ground state and radiation of frequency (E 2-E 1)/h falls on it,

(a) it will not be absorbed at all




78

(b) some of atoms will move to the first excited state.

(c) all atoms will be excited to the n = 2 state.(d) no atoms will make a transition to the n = 3 state.

12 .13 The simple Bohr modle is not applicable to He4 atom because

(a) He4 is an inert gas.(b) He4 has neutrons in the nucleus.(c) He4 has one more electron.(d) electrons are not subject to central forces.

VSA

12 .14 The mass of a H-atom is less than the sum of the masses of a proton and electron. Why is this?

12 .15 Imagine removing one electron from He4 and He3. Their energy levels, as worked out on the basis of Bohr model will be very close. Explain why.

12 .16 When an electron falls from a higher energy to a lower energy level, the difference in the energies appears in the form of electromagnetic radiation. Why cannot it be emitted as other forms of energy?

12 .17 Would the Bohr formula for the H-atom remain unchanged if

proton had a charge (+4/3)e and electron a charge ( )3/4 e − ,

where e = 1.6 × 10 –19C. Give reasons for your answer.

12 .18 Consider two different hydrogen atoms. The electron in each atomis in an excited state. Is it possible for the electrons to havedifferent energies but the same orbital angular momentumaccording to the Bohr model?

SA

12 .19 Positronium is just like a H-atom with the proton replaced by the positively charged anti-particle of the electron (called thepositron which is as massive as the electron). What would be theground state energy of positronium?

12 .20 Assume that there is no repulsive force between the electrons inan atom but the force between positive and negative charges isgiven by Coulomb’s law as usual. Under such circumstances,calculate the ground state energy of a He-atom.



Atoms

79

12 .21 Using Bohr model, calculate the electric current created by the

electron when the H-atom is in the ground state.

12 .22 Show that the first few frequencies of light that is emitted whenelectrons fall to the n th level from levels higher than n , areapproximate harmonics (i.e. in the ratio 1 : 2: 3...) when n >>1.

12 .23 What is the minimum energy that must be given to a H atom in

ground state so that it can emit an H γ line in Balmer series. If the

angular momentum of the system is conserved, what would be

the angular momentum of such H γ photon?

LA

12 .24 The first four spectral lines in the Lyman serics of a H-atom areλ = 1218 Å, 1028Å, 974.3 Å and 951.4Å. If instead of Hydrogen,

we consider Deuterium, calculate the shift in the wavelength of these lines.

12 .25 Deutrium was discovered in 1932 by Harold Urey by measuringthe small change in wavelength for a particular transition in 1Hand 2H. This is because, the wavelength of transition depend to a certain extent on the nuclear mass. If nuclear motion is taken

into account then the electrons and nucleus revolve around their common centre of mass. Such a system is equivalent to a singleparticle with a reduced mass μ , revolving around the nucleus at a distance equal to the electron-nucleus separation. Hereμ = m

e M /(m

e +M ) where M is the nuclear mass and m

e is the

electronic mass. Estimate the percentage difference in wavelengthfor the 1st line of the Lyman series in 1H and 2H. (Mass of 1Hnucleus is 1.6725 × 10 –27 kg, Mass of 2H nucleus is3.3374 × 10 –27 kg, Mass of electron = 9.109 × 10 –31 kg.)

12 .26 If a proton had a radius R and the charge was uniformly

distributed, calculate using Bohr theory, the ground state energy of a H-atom when (i) R = 0.1Å, and (ii) R = 10 Å.

12 .27 In the Auger process an atom makes a transition to a lower state without emitting a photon. The excess energy is transferred toan outer electron which may be ejected by the atom. (This iscalled an Auger electron). Assuming the nucleus to be massive,calculate the kinetic energy of an n = 4 Auger electron emitted

by Chromium by absorbing the energy from a n = 2 to n = 1transition.




80

12 .28 The inverse square law in electrostatics is( )

2

20

| |4 .

e r πε

=F for the

force between an electron and a proton. The1r

⎛ ⎞⎜ ⎟⎝ ⎠

dependence of

| |F can be understood in quantum theory as being due to thefact that the ‘particle’ of light (photon) is massless. If photons hada mass m

p , force would be modified to

( )( )

e e p r

r r r

2

2 20

1| | . –

4λ

λ πε

⎡ ⎤= ×+⎢ ⎥⎣ ⎦

F where p m c /λ = h and h

2π =h .

Estimate the change in the ground state energy of a H-atom if m p were 10 –6 times the mass of an electron.

12 .29 The Bohr model for the H-atom relies on the Coulomb’s law of electrostatics. Coulomb’s law has not directly been verified for

very short distances of the order of angstroms. SupposingCoulomb’s law between two opposite charge + q

1 , – q

2 is modified

to

( )1 2

020

1| | ,

4q q

r R r πε

= ≥F

1 2 002

0 0

1,

4q q R

r R R r

ε

πε

⎛ ⎞= ≤⎜ ⎟

⎝ ⎠

Calculate in such a case, the ground state energy of a H-atom, if ε = 0.1, R

0 = 1Å.



MCQ I

13 .1 Suppose we consider a large number of containers eachcontaining initially 10000 atoms of a radioactive material with a half life of 1 year. After 1 year,

(a) all the containers will have 5000 atoms of the material.

(b) all the containers will contain the same number of atoms of the material but that number will only be approximately 5000.

(c) the containers will in general have different numbers of theatoms of the material but their average will be close to 5000.

(d) none of the containers can have more than 5000 atoms.

13 .2 The gravitational force between a H-atom and another particle of mass m will be given by Newton’s law:

2

.M m F G

r = , where r is in km and

(a) M = m proton

+ m electron

.

(b) M = m proton

+ m electron

2

B

c

− (B = 13.6 eV).

Chapter Thirteen

NUCLEI




82

(c) M is not related to the mass of the hydrogen atom.

(d) M = m proton

+ m electron

2

| |V

c − (| |V = magnitude of the potential

energy of electron in the H-atom).

13 .3 When a nucleus in an atom undergoes a radioactive decay, theelectronic energy levels of the atom

(a) do not change for any type of radioactivity .

(b) change for α and β radioactivity but not for γ -radioactivity.

(c) change for α-radioactivity but not for others.

(d) change for β-radioactivity but not for others.

13 .4 M x andM y denote the atomic masses of the parent and the daughter nuclei respectively in a radioactive decay. The Q -value for a β –

decay is Q 1 and that for a β + decay is Q

2. If m

e denotes the mass of

an electron, then which of the following statements is correct?

(a) Q 1 = (M

x – M

y ) c 2 and Q

2 = (M

x – M

y – 2m

e)c2

(b) Q 1 = (M

x – M

y ) c 2 and Q

2 = (M

x – M

y )c 2

(c) Q 1 = (M

x – M

y – 2m

e) c 2 and Q

2 = (M

x – M

y +2 m

e)c 2

(d) Q 1 = (M

x – M

y+ 2m

e) c 2 and Q

2 = (M

x – M

y +2 m

e)c 2

13 .5 Tritium is an isotope of hydrogen whose nucleus Triton contains

2 neutrons and 1 proton. Free neutrons decay into ep ν + + . If

one of the neutrons in Triton decays, it would transform into He3

nucleus. This does not happen. This is because

(a) Triton energy is less than that of a He3 nucleus.

(b) the electron created in the beta decay process cannot remainin the nucleus.

(c) both the neutrons in triton have to decay simultaneously resulting in a nucleus with 3 protons, which is not a He3

nucleus.(d) because free neutrons decay due to external perturbations

which is absent in a triton nucleus.

1 3 . 6 .Heavy stable nucle have more neutrons than protons. This is because of the fact that

(a) neutrons are heavier than protons.

(b) electrostatic force between protons are repulsive.

(c) neutrons decay into protons through beta decay.

(d) nuclear forces between neutrons are weaker than that betweenprotons.

13 .7 In a nuclear reactor, moderators slow down the neutrons whichcome out in a fission process. The moderator used have light nuclei. Heavy nuclei will not serve the purpose because



Nuclei

83

(a) they will break up.

(b) elastic collision of neutrons with heavy nuclei will not slow them down.

(c) the net weight of the reactor would be unbearably high.

(d) substances with heavy nuclei do not occur in liquid or gaseousstate at room temperature.

MCQ II

13 .8 Fusion processes, like combining two deuterons to form a Henucleus are impossible at ordinary temperatures and pressure.

The reasons for this can be traced to the fact:

(a) nuclear forces have short range.

(b) nuclei are positively charged.

(c) the original nuclei must be completely ionized before fusioncan take place.

(d) the original nuclei must first break up before combining witheach other.

13 .9 Samples of two radioactive nuclides A and B are taken. λ A and λ

B

are the disintegration constants of A and B respectively. In whichof the following cases, the two samples can simultaneously havethe same decay rate at any time?

(a) Initial rate of decay of A is twice the initial rate of decay of Band λ

A= λ

B.

(b) Initial rate of decay of A is twice the initial rate of decay of Band λ

A> λ

B.

(c) Initial rate of decay of B is twice the initial rate of decay of A and λ

A> λ

B.

(d) Initial rate of decay of B is same as the rate of decay of A at t = 2h and λ

B< λ

A .

13 .10 The variation of decay rate of two radioactive samples A and B with time is shown in Fig. 13.1.

Which of the following statements are true?

(a) Decay constant of A is greater than that of B, hence A always decays faster than B.

(b) Decay constant of B is greater than that of A but itsdecay rate is always smaller than that of A.

(c) Decay constant of A is greater than that of B but it doesnot always decay faster than B.

(d) Decay constant of B is smaller than that of A but still itsdecay rate becomes equal to that of A at a later instant.

A

B

t

dN

d t

Fig. 13.1




84

VSA

13 .11 32He and 3

1He nuclei have the same mass number. Do they have

the same binding energy?

13 .12 Draw a graph showing the variation of decay rate with number of active nuclei.

13 .13 Which sample, A or B shown in Fig. 13.2 has shorter mean-life?

13 .14 Which one of the following cannot emit radiation and why?

Excited nucleus, excited electron.

13 .15 In pair annihilation, an electron and a positron destroy each other to produce gamma radiation. How is the momentum conserved?

SA

13 .16 Why do stable nuclei never have more protons than neutrons?

13 .17 Consider a radioactive nucleus A which decays to a stable nucleusC through the following sequence:

A → B → C

Here B is an intermediate nuclei which is also radioactive.Considering that there are N

0 atoms of A initially, plot the graph

showing the variation of number of atoms of A and B versus time.

13 .18 A piece of wood from the ruins of an ancient building was foundto have a 14C activity of 12 disintegrations per minute per gramof its carbon content. The 14C activity of the living wood is 16disintegrations per minute per gram. How long ago did the tree,from which the wooden sample came, die? Given half-life of 14C is5760 years.

13 .19 Are the nucleons fundamental particles, or do they consist of still smaller parts? One way to find out is to probe a nucleon just as Rutherford probed an atom. What should be the kinetic energy of an electron for it to be able to probe a nucleon? Assume thediameter of a nucleon to be approximately 10 –15 m.

13 .20 A nuclide 1 is said to be the mirror isobar of nuclide 2 if Z 1 =N

2

and Z 2 =N

1 . (a) What nuclide is a mirror isobar of 23

11 Na ? (b) Which

nuclide out of the two mirror isobars have greater binding energy and why?

A

B

t

dN

d t

Fig. 13.2



Nuclei

85

LA13 .21 Sometimes a radioactive nucleus decays into a nucleus which

itself is radioactive. An example is :

half-life half-life38 38 38

=2.48h =0.62hSulphur Cl Ar (stable) ⎯⎯⎯⎯→ ⎯⎯⎯⎯→ .

Assume that we start with 1000 38S nuclei at time t = 0. Thenumber of 38Cl is of count zero at t = 0 and will again be zero at

t = ∞ . At what value of t , would the number of counts be a

maximum?

13 .22 Deuteron is a bound state of a neutron and a proton with a

binding energy B = 2.2 MeV. A γ -ray of energy E is aimed at a deuteron nucleus to try to break it into a (neutron + proton)such that the n and p move in the direction of the incident γ -ray.If E = B , show that this cannot happen. Hence calculate how much bigger than B must E be for such a process to happen.

13 .23 The deuteron is bound by nuclear forces just as H-atom is madeup of p and e bound by electrostatic forces. If we consider theforce between neutron and proton in deuteron as given in the

form of a Coulomb potential but with an effective charge e ′ :

2

0

1

4

e

F r πε

′

=

estimate the value of (e’ / e ) given that the binding energy of a deuteron is 2.2 MeV.

13 .24 Before the neutrino hypothesis, the beta decay process wasthrought to be the transition,

n p e → +

If this was true, show that if the neutron was at rest, the protonand electron would emerge with fixed energies and calculate

them.Experimentally, the electron energy was found to have a large range.

13 .25 The activity R of an unknown radioactive nuclide is measuredat hourly intervals. The results found are tabulated as follows:

t (h) 0 1 2 3 4

R (MBq) 100 35.36 12.51 4.42 1.56




(i) Plot the graph of R versus t and calculate half-life from the

graph.

(ii) Plot the graph of0

lnR

R

⎛ ⎞⎜ ⎟⎝ ⎠

versus t and obtain the value of

half-life from the graph.

13 .26 Nuclei with magic no. of proton Z = 2, 8, 20, 28, 50, 52 andmagic no. of neutrons N = 2, 8, 20, 28, 50, 82 and 126 are foundto be very stable. (i) Verify this by calculating the protonseparation energy S

p for 120Sn (Z = 50) and 121Sb = (Z = 51).

The proton separation energy for a nuclide is the minimum

energy required to separate the least tightly bound proton froma nucleus of that nuclide. It is given by

Sp = (M

Z–1 ,N+ M

H – M

Z,N ) c 2 .

Given 119In = 118.9058u, 12 0Sn = 119.902199u,121Sb = 120.903824u, 1H = 1.0078252u.

(ii) What does the existance of magic number indicate?

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