Class 11 Mathematics Functions MCQs Set 08

Real valued functions:

Question. \( f : R \to R \) is a function defined by \( f(x) = \frac{1}{2} \) then \( f\left(-\frac{2}{5}\right) = \)
(a) 2
(b) \( -\frac{1}{2} \)
(c) -2
(d) \( \frac{1}{2} \)
Answer: (d) \( \frac{1}{2} \)

Question. If \( f(x) = \sin \left( \frac{\pi}{3}[x] - x^2 \right) \) then the value of \( f\left(\sqrt{\frac{\pi}{3}}\right) \) is
(a) 1
(b) -1
(c) 0
(d) \( \frac{-3}{4} \)
Answer: (c) 0

Question. The function \( y = f(x) \) such that \( f\left(x + \frac{1}{x}\right) = x^2 + \frac{1}{x^2} \)
(a) \( 2 - x^2 \)
(b) \( x^2 - 2 \)
(c) \( x^2 + 4 \)
(d) \( 4x^2 - 2 \)
Answer: (b) \( x^2 - 2 \)

Question. If \( f = \{(a, 1), (b, -2), (c, 3)\} \), \( g = \{(a, -2), (b, 0), (c, 1)\} \) then \( f^2 + g^2 = \)
(a) \( \{(a, -1), (b, -2), (c, 4)\} \)
(b) \( \{(a, 3), (b, -2), (c, 2)\} \)
(c) \( \{(a, -4), (b, -4), (c, 9)\} \)
(d) \( \{(a, 5), (b, 4), (c, 10)\} \)
Answer: (d) \( \{(a, 5), (b, 4), (c, 10)\} \)

Question. Which of the following relations are functions
\( f : \{(2, 1), (3, 1), (4, 2)\} \)
\( g : \{(2, 2), (2, 4), (3, 3), (4, 4)\} \)
\( h = \{(1, 2), (2, 3), (3, 4), (4, 5), (5, 6)\} \)
(a) f, g
(b) g, h
(c) h, f
(d) f, g, h
Answer: (c) h, f

Even and odd functions

Question. If \( f(x) = ax^5 + bx^3 + cx + d \) is odd then
(a) \( a = 0 \)
(b) \( b = 0 \)
(c) \( c = 0 \)
(d) \( d = 0 \)
Answer: (d) \( d = 0 \)

Question. A function whose graph is symmetrical about the y axis is given by
(a) \( f(x) = \sin \left[ \log \left( x + \sqrt{x^2 + 1} \right) \right] \)
(b) \( f(x) = \frac{\sec^4 x + \text{cosec}^4 x}{x^3 + x^4 \cot x} \)
(c) \( f(x+y) = f(x) + f(y) \, \forall x, y \in R \)
(d) \( f(x) = x^2 \)
Answer: (d) \( f(x) = x^2 \)

Periodic functions :

Question. The period of \( \text{sgn} (x - [x]) \) is
(a) any real number
(b) 0
(c) 1
(d) non periodic function
Answer: (c) 1

Question. The period of \( x \cos x \) is
(a) \( 2\pi \)
(b) \( \pi \)
(c) \( \frac{\pi}{2} \)
(d) non periodic
Answer: (d) non periodic

Domain of the function:

Question. Domain of \( f(x) = \frac{x^2 + 3x + 5}{x^2 - 5x + 4} \) is
(a) \( R - \{1, 4\} \)
(b) \( \{1, 4\} \)
(c) \( (1, 4) \)
(d) \( [1, 4] \)
Answer: (a) \( R - \{1, 4\} \)

Question. Domain of \( [x] + x \) is
(a) R
(b) Z
(c) R – Z
(d) Q
Answer: (a) R

Question. Domain of \( |x - 1| \) is
(a) \( [1, \infty) \)
(b) R
(c) \( [0, \infty) \)
(d) Z
Answer: (b) R

Question. The domain of \( f(x) = \sqrt{-x^2} \) is
(a) \( (0, \infty) \)
(b) \( (-\infty, 0) \)
(c) \( \{0\} \)
(d) \( (1, \infty) \)
Answer: (c) \( \{0\} \)

Question. The domain of \( f(x) = \frac{1}{|x| + x} \) is
(a) \( (-\infty, 0) \)
(b) \( (0, \infty) \)
(c) \( (-\infty, 1) \)
(d) (-2, -1)
Answer: (b) \( (0, \infty) \)

Question. The domain of \( f(x) = \sqrt{1 - |x|} \) is
(a) [-1, 1]
(b) (-1, 1)
(c) (0, 1)
(d) R
Answer: (a) [-1, 1]

Question. The domain of \( f(x) = \log \{ (x - 3)(6 - x) \} \) is
(a) \( (3, \infty) \)
(b) \( (3, 6) \)
(c) \( (0, \infty) \)
(d) \( (-\infty, \infty) \)
Answer: (b) \( (3, 6) \)

Question. The domain of \( f(x) = \cot \frac{x}{3} \) is
(a) \( (-\infty, \infty) \)
(b) \( R - \{n\pi : n \in Z\} \)
(c) \( R - \{3n\pi : n \in Z\} \)
(d) \( (0, \infty) \)
Answer: (c) \( R - \{3n\pi : n \in Z\} \)

Question. The domain of \( f(x) = \text{Tan}^{-1}(5x) \) is
(a) \( (-\infty, \infty) \)
(b) \( (0, \infty) \)
(c) \( (-\infty, 0) \)
(d) \( \left(-\frac{1}{5}, \frac{1}{5}\right) \)
Answer: (a) \( (-\infty, \infty) \)

Question. The domain of \( f(x) = \sqrt[3]{x} \cot x \) is
(a) R
(b) \( R - \{n\pi : n \in Z\} \)
(c) \( R - \left\{ (2n+1)\frac{\pi}{2} : n \in Z \right\} \)
(d) \( (0, \infty) \)
Answer: (b) \( R - \{n\pi : n \in Z\} \)

Question. Domain of \( f(x) = \frac{|x| - x}{2x} \) is
(a) R
(b) \( R - \{0\} \)
(c) Z
(d) N
Answer: (b) \( R - \{0\} \)

Question. Domain of \( \frac{3^x}{x+1} \) is
(a) R
(b) \( R - \{-1\} \)
(c) \( (1, \infty) \)
(d) \( (-\infty, 1) \)
Answer: (b) \( R - \{-1\} \)

Range of the function:

Question. Range of \( \frac{|x - 4|}{x - 4} \) is
(a) \( R - \{4\} \)
(b) R
(c) \( \{-1, 1\} \)
(d) \( R - \{-1, 1\} \)
Answer: (c) \( \{-1, 1\} \)

Question. If x is positive, the values of \( f(x) = -3 \cos \sqrt{3 + x + x^2} \) lie in the intervel
(a) [-1, 3]
(b) [-3, 3]
(c) [0, 3]
(d) [-3, 0]
Answer: (b) [-3, 3]

Question. The range of \( f(x) = \frac{\sin \pi [x^2 - 1]}{x^4 + 1} \) is, where [.] is greatest integer function
(a) R
(b) [-1, 1]
(c) {0, 1}
(d) {0}
Answer: (d) {0}

Question. The range of \( f(x) = |x - 2| + |x - 12| \) is
(a) \( [2, \infty) \)
(b) \( (12, \infty) \)
(c) \( [10, \infty) \)
(d) \( [14, \infty) \)
Answer: (c) \( [10, \infty) \)

Question. The range of \( f(x) = 3x^2 + 7x + 10 \) is
(a) \( \left[ \frac{70}{3}, \infty \right) \)
(b) \( \left[ \frac{71}{12}, \infty \right) \)
(c) \( [0, \infty) \)
(d) \( \left(-\infty, \frac{70}{3}\right) \)
Answer: (b) \( \left[ \frac{71}{12}, \infty \right) \)

Question. The range of the function \( f(x) = \frac{2+x}{2-x}, x \neq 2 \) is
(a) R
(b) \( R - \{-1\} \)
(c) \( R - \{1\} \)
(d) \( R - \{2\} \)
Answer: (b) \( R - \{-1\} \)

Question. The range of \( f(x) = x - [x] \) is
(a) \( x = \{1, 2, 3...\} \)
(b) \( x \ge 0 \)
(c) \( x < 1 \)
(d) \( 0 \le x < 1 \)
Answer: (d) \( 0 \le x < 1 \)

Question. Range of \( \sqrt{9 - x^2} \) is
(a) [0, 3]
(b) [–3, 3]
(c) [–3, 0]
(d) R
Answer: (a) [0, 3]

Question. Range of \( f(x) = e^x \) is
(a) \( (0, \infty) \)
(b) \( [0, \infty) \)
(c) \( (-\infty, \infty) \)
(d) \( [e, \infty) \)
Answer: (a) \( (0, \infty) \)

Types of functions

Question. If \( f : R \to S \), defined by \( f(x) = \sin x - \sqrt{3} \cos x + 1 \), is onto, then the interval of 'S' is
(a) [0, 3]
(b) [-1, 3]
(c) [0, 1]
(d) [-1, 1]
Answer: (b) [-1, 3]

Question. If \( f : [1, \infty) \to B \) defined by \( f(x) = x^2 - 2x + 6 \) is a surjection then B =
(a) \( [1, \infty) \)
(b) \( [5, \infty) \)
(c) \( [6, \infty) \)
(d) \( [2, \infty) \)
Answer: (b) \( [5, \infty) \)

Question. \( f : (-\infty, \infty) \to (-\infty, \infty) \) is defined by \( f(x) = ax + b, a, b \in R \quad (a \neq 0) \) then f is
(a) injective but not surjective
(b) surjective but not injective
(c) bijective
(d) neither injective nor surjective
Answer: (c) bijective

Question. \( f : Z \to Z \) and \( f(x) = x^2 \) then \( f \) is 
(a) bijection
(b) injection
(c) surijection
(d) not bijection
Answer: (d) not bijection

Question. The function \( f : R \to R \) defined by \( f(x) = \sin x \) is
(a) Neither one one nor onto
(b) onto
(c) one-one
(d) many one
Answer: (a) Neither one one nor onto

Question. \( f : Z \to Z \) defined as \( f(x) = [x] \) then f is
(a) not a function
(b) many-to-one function
(c) into function
(d) identity function
Answer: (d) identity function

Question. \( f : Q \to Q \) is defined by \( f(x) = 15x + 7 \) is
(a) injective only
(b) surjective only
(c) bijective
(d) neither injective nor surjective
Answer: (c) bijective

Question. \( f : (0, \infty) \to [0, \infty) \) defined by \( f(x) = x^2 \) is
(a) one-one but not onto
(b) onto but not one-one
(c) bijective
(d) neither one-one nor onto
Answer: (a) one-one but not onto

Question. Let \( f : [0, \infty) \to [0, 2] \) be defined by \( f(x) = \frac{2x}{1+x} \) then \( f \) is
(a) one-one but not onto
(b) onto but not one-one
(c) both one-one and onto
(d) neither one-one nor onto
Answer: (a) one-one but not onto

Number of functions:

Question. The total number of functions from A to itself is 256, then n(A) =
(a) 2
(b) 3
(c) 4
(d) 5
Answer: (c) 4

Question. If B = {1, 2, 3} and A = {4, 5, 6, 7, 8} then the number of surjections from A to B is
(a) 81
(b) 64
(c) 48
(d) 150
Answer: (d) 150

Question. The number of one-one functions that can be defined from A = {1, 2, 3} to B = {a, e, i, o, u} is
(a) \( 3^5 \)
(b) \( 5^3 \)
(c) \( ^5P_3 \)
(d) 5!
Answer: (c) \( ^5P_3 \)

Question. The number of possible many to one functions from A = {6, 36} to B = {1, 2, 3, 4, 5} is
(a) 32
(b) 25
(c) 5
(d) 20
Answer: (d) 20

Question. If n (A) = 4 and n(B) = 6, then the number of surjections from A to B is
(a) \( 4^6 \)
(b) \( 6^4 \)
(c) 0
(d) 24
Answer: (c) 0

Question. The number of bijections from the set A to itself when A contains 106 elements is
(a) 106
(b) \( 106^2 \)
(c) 106!
(d) \( 2^{106} \)
Answer: (c) 106!

Question. The number of non-surjective mappings that can be defined from A = {1, 4, 9, 16} to B = {2, 8, 16, 32, 64} is
(a) 1024
(b) 20
(c) 505
(d) 625
Answer: (d) 625