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# NCERT solutions Mathematics Class 11 chapter 2 Relations and Functions

## Chapter 2 - Relations and Functions

#### Pages 33 - 34

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

Q 2 | Page 33

if (x/3 + 1, y - 2/3) = (5/3, 1/3) find the values of x and y.

Q 2.1 | Page 33

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

Q 3 | Page 33

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

If P = {mn} and Q = {nm}, then P × Q = {(mn), (nm)}.

Q 4.1 | Page 33

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

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

Q 4.2 | Page 33

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

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

Q 4.3 | Page 33

If A = {–1, 1}, find A × A × A.

Q 5 | Page 33

If A × B = {(ax), (ay), (bx), (by)}. Find A and B.

Q 6 | Page 33

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

Q 7.1 | Page 33

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

Q 7.2 | Page 33

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

Q 8 | Page 33

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 × B, find A and B, where xy and z are distinct elements.

Q 9 | Page 33

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

Q 10 | Page 34

#### Pages 35 - 36

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

Q 1 | Page 35

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

Q 2 | Page 36

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

Q 3 | Page 36

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?

Q 4 | Page 36

Let A = {1, 2, 3, 4, 6}. Let R be the relation on A defined by {(ab): ab ∈ A, b is exactly divisible by a}.

(i) Write R in roster form

(ii) Find the domain of R

(iii) Find the range of R.

Q 5 | Page 36

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

Q 6 | Page 36

Write the relation R = {(xx3): is a prime number less than 10} in roster form.

Q 7 | Page 36

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

Q 8 | Page 36

Let R be the relation on Z defined by R = {(ab): ab ∈ Z– b is an integer}. Find the domain and range of R.

Q 9 | Page 36

#### Page 44

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)}

Q 1 | Page 44

Find the domain and range of the given real function:

f(x) = –|x|

Q 2.1 | Page 44

Find the domain and range of the following real function:

f(x) = sqrt(9 - x^2)

Q 2.2 | Page 44

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

(i) f(0), (ii) f(7), (iii) f(–3)

Q 3 | Page 44

The function ‘t’ which maps temperature in degree Celsius into temperature in degree Fahrenheit is defined by t(C) = "9C"/5 + 32

Find

(i) t(0)

(ii) t(28)

(iii) t(–10)

(iv) The value of C, when t(C) = 212.

Q 4 | Page 44

Find the range of each of the following functions.

f(x) = 2 – 3xx ∈ Rx > 0.

Q 5.1 | Page 44

Find the range of each of the following functions.

f(x) = xx is a real number

Q 5.2 | Page 44

Find the range of each of the following functions.

f(x) = x2 + 2, x, is a real number.

Q 5.2 | Page 44

#### Pages 46 - 47

The relation f is defined by f(x) = {(x^2,0<=x<=3),(3x,3<=x<=10):}

The relation g is defined by  g(x) = {(x^2, 0 <= x <= 2),(3x,2<= x <= 10):}

Show that f is a function and g is not a function.

Q 1 | Page 46

If f(x) = x2, find (f(1.1) - f(1))/((1.1 - 1))

Q 2 | Page 46

Find the domain of the function  f(x) = (x^2 + 2x + 1)/(x^2 - 8x + 12)

Q 3 | Page 46

Find the domain and the range of the real function f defined by f(x)=sqrt((x-1))

Q 4 | Page 46

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

Q 5 | Page 46

Let f = {(x, x^2/(1+x^2)):x ∈ R} be a function from R into R. Determine the range of f.

Q 6 | Page 46

Let fgR → R be defined, respectively by f(x) = + 1, g(x) = 2x – 3. Find f + gf – g and f/g

Q 7 | Page 46

Let = {(1, 1), (2, 3), (0, –1), (–1, –3)} be a function from Z to Z defined by f(x) = ax + b, for some integers ab. Determine ab.

Q 8 | Page 46

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

(i) (aa) ∈ R, for all a ∈ N

(ii) (ab) ∈ R, implies (ba) ∈ R

(iii) (ab) ∈ R, (bc) ∈ R implies (ac) ∈ R.

Q 9 | Page 46

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

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

Q 10 | Page 46

Let f be the subset of Z × Z defined by = {(aba + b): ab ∈ Z}. Is f a function from Z to Z: justify your answer.

Q 11 | Page 47

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

Q 12 | Page 47

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