# NCERT Solutions for Class 11 Maths Maths Chapter 2

## Relations and Functions Class 11

### Exercise 2.1 : Solutions of Questions on Page Number : 33

Q1 :

If , find the values of xand y.

It is given that .

Since the ordered pairs are equal, the corresponding elements will also be equal.

Therefore, and .  x= 2 and y= 1

Q2 :

If the set A has 3 elements and the set B = {3, 4, 5}, then find the number of elements in (A x B)?

It is given that set A has 3 elements and the elements of set B are 3, 4, and 5.

Number of elements in set B = 3

Number of elements in (A x B)

= (Number of elements in A) x (Number of elements in B)

= 3 x 3 = 9

Thus, the number of elements in (A x B) is 9.

Q3 :

If G = {7, 8} and H = {5, 4, 2}, find G x H and H x G.

Q4 :

State whether each of the following statement are true or false. If the statement is false, rewrite the given statement correctly.

(i) If P = {m, n} and Q = {n, m}, then P x Q = {(m, n), (n, m)}.

(ii) If A and B are non-empty sets, then A x B is a non-empty set of ordered pairs (x, y) such that x A and y B.

(iii) If A = {1, 2}, B = {3, 4}, then A x (B Φ) = Φ.

Q5 :

If A = {-1, 1}, find A x A x A.

Q6 :

If A x B = {(a, x), (a, y), (b, x), (b, y)}. Find A and B.

Q7 :

Let A = {1, 2}, B = {1, 2, 3, 4}, C = {5, 6} and D = {5, 6, 7, 8}. Verify that

(i) A x (B C) = (A x B) (A x C)

(ii) A x C is a subset of B x D

Q8 :

Let A = {1, 2} and B = {3, 4}. Write A x B. How many subsets will A x B have? List them.

Q9 :

Let A and B be two sets such that n(A) = 3 and n (B) = 2. If (x, 1), (y, 2), (z, 1) are in A x B, find A and B, where x, y and z are distinct elements.

Q10 :

The Cartesian product A x A has 9 elements among which are found (-1, 0) and (0, 1). Find the set A and the remaining elements of A x A.

### Exercise 2.2 : Solutions of Questions on Page Number : 35

Q1 :

Let A = {1, 2, 3, ... , 14}. Define a relation R from A to A by R = {(x, y): 3x - y = 0, where x, y A}. Write down its domain, codomain and range.

Q2 :

Define a relation R on the set Nof natural numbers by R = {(x, y): y= x+ 5, xis a natural number less than 4; x, y N}. Depict this relationship using roster form. Write down the domain and the range.

Q3 :

A = {1, 2, 3, 5} and B = {4, 6, 9}. Define a relation R from A to B by R = {(x, y): the difference between xand yis odd; x A, y B}. Write R in roster form.

Q4 :

The given figure shows a relationship between the sets P and Q. write this relation

(i) in set-builder form (ii) in roster form.

What is its domain and range? Q5 :

Let A = {1, 2, 3, 4, 6}. Let R be the relation on A defined by

{(a, b): a, b A, bis exactly divisible by a}.

(i) Write R in roster form

(ii) Find the domain of R

(iii) Find the range of R.

Q6 :

Determine the domain and range of the relation R defined by R = {(x, x+ 5): x {0, 1, 2, 3, 4, 5}}.

Q7 :

Write the relation R = {(x, x3): x is a prime number less than 10} in roster form.

Q8 :

Let A = {x, y, z} and B = {1, 2}. Find the number of relations from A to B.

Q9 :

Let R be the relation on Zdefined by R = {(a, b): a, b Z, a - bis an integer}. Find the domain and range of R.

### Exercise 2.3 : Solutions of Questions on Page Number : 44

Q1 :

Which of the following relations are functions? Give reasons. If it is a function, determine its domain and range.

(i) {(2, 1), (5, 1), (8, 1), (11, 1), (14, 1), (17, 1)}

(ii) {(2, 1), (4, 2), (6, 3), (8, 4), (10, 5), (12, 6), (14, 7)}

(iii) {(1, 3), (1, 5), (2, 5)}

Q2 :

Find the domain and range of the following real function:

(i) f(x) = â€“|x| (ii) Q3 :

A function fis defined by f(x) = 2x- 5. Write down the values of

(i) f(0), (ii) f(7), (iii) f(-3)

Q4 :

The function 't' which maps temperature in degree Celsius into temperature in degree Fahrenheit is defined by .

Find (i) t(0) (ii) t(28) (iii) t(â€“10) (iv) The value of C, when t(C) = 212

Q5 :

Find the range of each of the following functions.

(i) f(x) = 2 - 3x, x R, x> 0.

(ii) f(x) = x2+ 2, x, is a real number.

(iii) f(x) = x, xis a real number

### Exercise Miscellaneous : Solutions of Questions on Page Number : 46

Q1 :

The relation f is defined by The relation gis defined by Show that f is a function and g is not a function.

Q2 :

If f(x) = x2, find .

Q3 :

Find the domain of the function Q4 :

Find the domain and the range of the real function fdefined by .

Q5 :

Find the domain and the range of the real function fdefined by f(x) = |x- 1|.

Q6 :

Let be a function from Rinto R. Determine the range of f.

Q7 :

Let f, g: R Ã¢â€ ’ R be defined, respectively by f(x) = x + 1, g(x) = 2x â€“ 3. Find f+ g, fâ€“ gand .

Q8 :

Let f = {(1, 1), (2, 3), (0, -1), (-1, -3)} be a function from Zto Zdefined by f(x) = ax+ b, for some integers a, b. Determine a, b.

Q9 :

Let R be a relation from N to N defined by R = {(a, b): a, b N and a = b2}. Are the following true?

(i) (a, a) R, for all a N
(ii) (a, b) R, implies (b, a) R

(iii) (a, b) R, (b, c) R implies (a, c) R.

Q10 :

Let A = {1, 2, 3, 4}, B = {1, 5, 9, 11, 15, 16} and f = {(1, 5), (2, 9), (3, 1), (4, 5), (2, 11)}. Are the following true?

(i) fis a relation from A to B (ii) fis a function from A to B.

Q11 :

Let fbe the subset of Z x Zdefined by f = {(ab, a+ b): a, b Z}. Is fa function from Zto Z: justify your answer.

Q12 :

Let A = {9, 10, 11, 12, 13} and let f: A Nbe defined by f(n) = the highest prime factor of n. Find the range of f.