NCERT Exemplar solutions for Mathematics [English] Class 11 chapter 2 - Relations and Functions [Latest edition]

Let A = {1, 2, 3, 4} and B = {5, 7, 9}. Determine A × B 

Let A = {1, 2, 3, 4} and B = {5, 7, 9}. Determine B × A

Let A = {1, 2, 3, 4} and B = {5, 7, 9}. Determine is A × B = B × A?

Let A = {1, 2, 3, 4} and B = {5, 7, 9}. Determine is n (A × B) = n (B × A)?

Find x and y if: (4x + 3, y) = (3x + 5, – 2)

Find x and y if: (x – y, x + y) = (6, 10)

If A = {2, 4, 6, 9} and B = {4, 6, 18, 27, 54}, a ∈ A, b ∈ B, find the set of ordered pairs such that 'a' is factor of 'b' and a < b.

Find the domain and range of the relation R given by R = {(x, y) : y = `x + 6/x`; where x, y ∈ N and x < 6}.

Is the following relation a function? Justify your answer

R₁ = `{(2, 3), (1/2, 0), (2, 7), (-4, 6)}`

Is the following relation a function? Justify your answer

R₂ = {(x, |x |) | x is a real number}

Find the domain for which the functions f(x) = 2x² – 1 and g(x) = 1 – 3x are equal.

Find the domain of the following function.

f(x) = `x/(x^2 + 3x + 2)`

Find the domain of the following function.

f(x) = [x] + x

Find the range of the following functions given by `|x - 4|/(x - 4)`

Find the range of the following functions given by `sqrt(16 - x^2)`

Redefine the function which is given by f(x) = `|x - 1| + |1 + x|, -2 ≤ x ≤ 2`

Find the domain of the function f given by f(x) = `1/sqrt([x]^2 - [x] - 6)`

The domain of the function f defined by f(x) = `1/sqrt(x - |x|)` is ______.

If f(x) = `x^3 - 1/x^3`, then `f(x) + f(1/x)` is equal to ______.

Let A and B be any two sets such that n(B) = p, n(A) = q then the total number of functions f : A → B is equal to ______.

Let f and g be two functions given by f = {(2, 4), (5, 6), (8, – 1), (10, – 3)} g = {(2, 5), (7, 1), (8, 4), (10, 13), (11, – 5)} then. Domain of f + g is ______.

Let A = {–1, 2, 3} and B = {1, 3}. Determine A × B

Let A = {–1, 2, 3} and B = {1, 3}. Determine B × A

Let A = {–1, 2, 3} and B = {1, 3}. Determine B × B

Let A = {–1, 2, 3} and B = {1, 3}. Determine A × A

If P = {x : x < 3, x ∈ N}, Q = {x : x ≤ 2, x ∈ W}. Find (P ∪ Q) × (P ∩ Q), where W is the set of whole numbers.

A = {x : x ∈ W, x < 2} B = {x : x ∈ N, 1 < x < 5} C = {3, 5} find A × (B ∩ C)

If A = {x : x ∈ W, x < 2} B = {x : x ∈ N, 1 < x < 5} C = {3, 5} find A × (B ∪ C)

In the following find a and b 

(2a + b, a – b) = (8, 3)

In the following find a and b 

`(a/4, a - 2b)` = (0, 6 + b)

Given A = {1, 2, 3, 4, 5}, S = {(x, y) : x ∈ A, y ∈ A}. Find the ordered pairs which satisfy the conditions given below:

x + y = 5

Given A = {1, 2, 3, 4, 5}, S = {(x, y) : x ∈ A, y ∈ A}. Find the ordered pairs which satisfy the conditions given below:

x + y < 5

Given A = {1, 2, 3, 4, 5}, S = {(x, y) : x ∈ A, y ∈ A}. Find the ordered pairs which satisfy the conditions given below:

x + y > 8

Given R = {(x, y) : x, y ∈ W, x² + y² = 25}. Find the domain and Range of R.

If R₁ = {(x, y) | y = 2x + 7, where x ∈ R and – 5 ≤ x ≤ 5} is a relation. Then find the domain and Range of R₁.

If R₂ = {(x, y) | x and y are integers and x² + y² = 64} is a relation. Then find R₂.

If R₃ = {(x, x) | x is a real number} is a relation. Then find domain and range of R₃.

Is the given relation a function? Give reasons for your answer.

h = {(4, 6), (3, 9), (– 11, 6), (3, 11)}

Is the given relation a function? Give reasons for your answer.

f = {(x, x) | x is a real number}

Is the given relation a function? Give reasons for your answer.

g = `"n", 1/"n" |"n"` is a positive integer

Is the given relation a function? Give reasons for your answer.

s = {(n, n²) | n is a positive integer}

Is the given relation a function? Give reasons for your answer.

t = {(x, 3) | x is a real number

If f and g are real functions defined by f(x) = x² + 7 and g(x) = 3x + 5, find the following:

f(3) + g(– 5)

If f and g are real functions defined by f(x) = x² + 7 and g(x) = 3x + 5, find the following:

`f(1/2) xx g(14)`

If f and g are real functions defined by f(x) = x² + 7 and g(x) = 3x + 5, find the following:

f(– 2) + g(– 1)

If f and g are real functions defined by f(x) = x² + 7 and g(x) = 3x + 5, find the following:

f(t) – f(– 2)

If f and g are real functions defined by f(x) = x² + 7 and g(x) = 3x + 5, find the following:

`(f(t) - f(5))/(t - 5)`, if t ≠ 5

Let f and g be real functions defined by f(x) = 2x + 1 and g(x) = 4x – 7. For what real numbers x, f(x) = g(x)?

Let f and g be real functions defined by f(x) = 2x + 1 and g(x) = 4x – 7. For what real numbers x, f(x) < g(x)?

If f and g are two real valued functions defined as f(x) = 2x + 1, g(x) = x² + 1, then find f + g

If f and g are two real valued functions defined as f(x) = 2x + 1, g(x) = x² + 1, then find f – g

If f and g are two real valued functions defined as f(x) = 2x + 1, g(x) = x² + 1, then find fg

If f and g are two real valued functions defined as f(x) = 2x + 1, g(x) = x² + 1, then find `f/g`

Express the following functions as set of ordered pairs and determine their range.
f : X → R, f(x) = x³ + 1, where X = {–1, 0, 3, 9, 7}

Find the values of x for which the functions f(x) = 3x² – 1 and g(x) = 3 + x are equal.

Is g = {(1, 1), (2, 3), (3, 5), (4, 7)} a function? Justify. If this is described by the relation, g(x) = αx + β, then what values should be assigned to α and β?

Find the domain of the following functions given by f(x) = `1/sqrt(1 - cos x)`

Find the domain of the following functions given by f(x) = `1/sqrt(x + |x|)`

Find the domain of the following functions given by f(x) = x|x|

Find the domain of the following functions given by f(x) = `(x^3 - x + 3)/(x^2 - 1)`

Find the domain of the following function given by:

f(x) = `(3x)/(2x - 8)`

Find the range of the following functions given by f(x) = `3/(2 - x^2)`

Find the range of the following functions given by f(x) = 1 – |x – 2| 

Find the range of the following functions given by f(x) = |x − 3|

Find the range of the following functions given by f(x) = 1 + 3 cos2x

(Hint: –1 ≤ cos 2x ≤ 1 ⇒ –3 ≤ 3 cos 2x ≤ 3 ⇒ –2 ≤ 1 + 3cos 2x ≤ 4)

Redefine the function f(x) = x − 2 + 2 + x , – 3 ≤ x ≤ 3

If f(x) = `(x - 1)/(x + 1)`, then show that `f(1/x)` = – f(x)

If f(x) = `(x - 1)/(x + 1)`, then show that `f(- 1/x) = (-1)/(f(x))`

Let f(x) = `sqrt(x)` and g(x) = x be two functions defined in the domain R⁺ ∪ {0}. Find (f + g)(x)

Let f(x) = `sqrt(x)` and g(x) = x be two functions defined in the domain R⁺ ∪ {0}. Find (f – g)(x)

Let f(x) = `sqrt(x)` and g(x) = x be two functions defined in the domain R⁺ ∪ {0}. Find (fg)(x)

Let f(x) = `sqrt(x)` and g(x) = x be two functions defined in the domain R⁺ ∪ {0}. Find `(f/g)(x)`

Find the domain and range of the function f(x) = `1/sqrt(x - 5)`

If f(x) = y = `(ax - b)/(cx - a)`, then prove that f(y) = x.

Let n(A) = m, and n(B) = n. Then the total number of non-empty relations that can be defined from A to B is ______.

If [x]² – 5[x] + 6 = 0, where [ . ] denote the greatest integer function, then ______.

Range of f(x) = `1/(1 - 2 cosx)` is ______.

Let f(x) = `sqrt(1 + x^2)`, then ______.

Domain of `sqrt(a^2 - x^2)  (a > 0)` is ______.

If f(x) = ax + b, where a and b are integers, f(–1) = – 5 and f(3) = 3, then a and b are equal to ______.

The domain of the function f defined by f(x) = `sqrt(4 - x) + 1/sqrt(x^2 - 1)` is equal to ______.

The domain and range of the real function f defined by f(x) = `(4 - x)/(x - 4)` is given by ______.

The domain and range of real function f defined by f(x) = `sqrt(x - 1)` is given by ______.

The domain of the function f given by f(x) = `(x^2 + 2x + 1)/(x^2 - x - 6)` is ______.

The domain and range of the function f given by f(x) = 2 – |x – 5| is ______.

The domain for which the functions defined by f(x) = 3x² – 1 and g(x) = 3 + x are equal is ______.

Let f and g be two real functions given by
f = {(0, 1), (2, 0), (3, – 4), (4, 2), (5, 1)}
g = {(1, 0), (2, 2), (3, – 1), (4, 4), (5, 3)}
then the domain of f . g is given by ______.

Let f = {(2, 4), (5, 6), (8, – 1), (10, – 3)}
g = {(2, 5), (7, 1), (8, 4), (10, 13), (11, 5)}
be two real functions. Then Match the following :

State True or False for the following statement.

The ordered pair (5, 2) belongs to the relation R = {(x, y) : y = x – 5, x, y ∈ Z}.

State True or False for the following statement.

If P = {1, 2}, then P × P × P = {(1, 1, 1), (2, 2, 2), (1, 2, 2), (2, 1, 1)}

State True or False for the following statement.

If A = {1, 2, 3}, B = {3, 4} and C = {4, 5, 6}, then (A × B) ∪ (A × C) = {(1, 3), (1, 4), (1, 5), (1, 6), (2, 3), (2, 4), (2, 5), (2, 6), (3, 3), (3, 4), (3, 5), (3, 6)}.

State True or False for the following statement.

If (x – 2, y + 5) = `(-2, 1/3)` are two equal ordered pairs, then x = 4, y = `(-14)/3`

State True or False for the following statement.

If A × B = {(a, x), (a, y), (b, x), (b, y)}, then A = {a, b}, B = {x, y}