Class 11 Mathematics Functions MCQs Set 06

Composite functions:

Question. If \( f(x) \) and \( g(x) \) are two functions with \( g(x) = x - \frac{1}{x} \) and \( fog(x) = x^3 - \frac{1}{x^3} \), then \( f(x) = \)
(a) \( x^3 + 3x \)
(b) \( x^2 - \frac{1}{x^2} \)
(c) \( 1 + \frac{1}{x^2} \)
(d) \( 3x^2 + \frac{3}{x^4} \)
Answer: (a) \( x^3 + 3x \)

Question. Let \( f(x) = ax + b \) and \( g(x) = cx + d \), \( a \neq 0, c \neq 0 \). Assume \( a = 1, b = 2 \). If \( (fog)(x) = (gof)(x) \) for all x, what can you say about c and d
(a) c and d both arbitrary
(b) \( c = 1, d \text{ arbitrary} \)
(c) c arbitrary, \( d = 1 \)
(d) \( c = 1, d = 1 \)
Answer: (b) \( c = 1, d \text{ arbitrary} \)

Question. If \( f(x) = \sin^2 x \) and the composite functions \( g\{f(x)\} = |\sin x| \), then the function \( g(x) = \)
(a) \( \sqrt{x - 1} \)
(b) \( \sqrt{x} \)
(c) \( \sqrt{x + 1} \)
(d) \( -\sqrt{x} \)
Answer: (b) \( \sqrt{x} \)

Question. If \( f : R \rightarrow R \) and \( g : R \rightarrow R \) are given by \( f(x) = |x| \) and \( g(x) = [x] \) for each \( x \in R \), then \( \{x \in R : g(f(x)) \le f(g(x))\} = \)
(a) \( Z \cup (-\infty, 0) \)
(b) \( (-\infty, 0) \)
(c) Z
(d) R
Answer: (d) R

Question. Let \( g : R \rightarrow R \) be given by \( g(x) = 3 + 4x \). If \( g^n(x) = gogo \dots og(x) \), and \( g^n(x) = A + Bx \) then A and B are
(a) \( 2^{n+1} - 1, 2^{n+1} \)
(b) \( 4^n - 1, 4^n \)
(c) \( 3^n, 3^n + 1 \)
(d) \( 5^n - 1, 5^n \)
Answer: (b) \( 4^n - 1, 4^n \)

MULTIPLE ANSWER QUESTIONS

Question. If a polynomial of degree ‘n’ satisfies \( f(x) = f'(x) \cdot f''(x) \forall x \in R \) then f(x) is
(a) an onto function
(b) an into function
(c) no such function is possible
(d) even function
Answer: (a) an onto function

Question. Let \( f(x) = \ln(2x-x^{2}) + \sin \left( \frac{\pi x}{2} \right) \) then
(a) graph of ‘f’ is symmetrical about the line x=1
(b) graph of ‘f’ is symmetrical about the line x=2
(c) max. value of ‘f’ is 1
(d) min.value of ‘f’ does not exist
Answer: (a, c, d)

PARAGRAPH QUESTIONS

Let \( F(x) = f(x) + g(x) \); \( G(x) = f(x) - g(x) \) and \( H(x) = \frac{f(x)}{g(x)} \) where \( f(x) = 1 - 2\sin^{2}x \) and \( g(x) = \cos(2x) \forall f : R \rightarrow [-1,1] \) and \( g : R \rightarrow [-1,1] \)
Now answer the following:

Question. Domain and range of \( H(x) \) are
(a) \( R \) and \( \{1\} \)
(b) \( R \) and \( \{0,1\} \)
(c) \( R - \left\{ (2n+1) \frac{\pi}{4} \right\} \) and \( \{1\} \) \( n \in z \)
(d) None of the options
Answer: (c) \( R - \left\{ (2n+1) \frac{\pi}{4} \right\} \) and \( \{1\} \) \( n \in z \)

Question. If \( F : R \rightarrow [-2,2] \) then \( F(x) \) is
(a) one-one
(b) onto
(c) into
(d) None of the options
Answer: (b) onto

Question. Which of the following is correct
(a) periods of f(x) ,g(x) and F(x) makes A.P with the common difference \( \frac{\pi}{3} \)
(b) period of f(x),g(x) and F(x) are same and is equal to \( 2\pi \)
(c) sum of the periods of f(x),g(x) and F(x) is \( 3\pi \)
(d) sum of the periods of f(x),g(x) and F(x) is \( 6\pi \)
Answer: (c) sum of the periods of f(x),g(x) and F(x) is \( 3\pi \)

Question. Which of the following is correct
(a) the domain of G(x) and H(x) are same
(b) the range of G(x) and H(x) are same
(c) the union of the domain of G(x) and H(x) are all real numbers
(d) None of the options
Answer: (c) the union of the domain of G(x) and H(x) are all real numbers

Question. If the solutions of \( F(x) - G(x) = 0 \) are \( x_{1}, x_{2}, x_{3} \dots x_{n} \) where \( x \in [0,5\pi] \) then
(a) \( x_{1}, x_{2}, x_{3} \dots x_{n} \) are in A.P with common difference \( \frac{\pi}{4} \)
(b) the no.of solutions of \( F(x) - G(x) = 0 \) is \( 10 \forall x \in [0,5\pi] \)
(c) the sum of all solutions of \( F(x) - G(x) = 0 \) is \( \forall x \in [0,5\pi] \) is \( 25\pi \)
(d) b,c are true
Answer: (d) b,c are true

Let \( f : N \rightarrow N \) be a function defined by f(x) = the biggest +ve integer obtained by reshuffling the digits of ‘x’. For example f(296)=962. Now answer the following.

Question. f is
(a) one-one ,onto
(b) one-one and into
(c) many-one and onto
(d) many-one and into
Answer: (d) many-one and into

Question. The biggest +ve integer which divides \( f(n) - n, \forall n \in N \) is
(a) 3
(b) 9
(c) 18
(d) 27
Answer: (b) 9

Question. The range of f is
(a) N
(b) set of +ve integers whose digits are non-increasing from left to right
(c) set of +ve whoe digits are non-decreaseing from left to right
(d) None of the options
Answer: (b) set of +ve integers whose digits are non-increasing from left to right

A function ‘f’ from a set X to Y is called onto if every \( y \in Y, \exists x \in X \) such that f(x)=y. Unless the contrary is specified, a real function is onto if it takes all real values, Otherwise, it is called on into function. Thus, if X and Y are finite sets , then ‘f’ can not be onto if Y contains more elements than ‘x’. Now answer the following.

Question. The polynomial function \( f(x) = a_{0}x^{n} + a_{1}x^{n-1} + a_{2}x^{n-2} + \dots + a_{n} \) where \( a_{0} \neq 0 \) is onto, for
(a) all positive integers n
(b) all even +ve integers ‘n’
(c) all odd +ve integers’n’
(d) no +ve interger
Answer: (c) all odd +ve integers’n’

Question. The function \( f(x) = \frac{x^{2} + 2x + c}{x^{2} + 4x + 3c} \) is onto, if
(a) \( 0 < c < 2 \)
(b) \( 0 < c < 4 \)
(c) \( \frac{-1}{2} < c < \frac{1}{2} \)
(d) \( 0 < c < 1 \)
Answer: (d) \( 0 < c < 1 \)

Question. Which of the following is not true
(a) A one-one function from the set \( \{a, b, c\} \) to \( \{\alpha, \beta, \gamma\} \) is onto also
(b) An onto function from an infinite set to a finite set can not be one-one
(c) An onto function is alwasys invertible
(d) the functions tanx and cotx are onto
Answer: (c) An onto function is alwasys invertible

Let \( f : R \rightarrow R \) be a continuous function such that \( f(x) - 2f\left( \frac{x}{2} \right) + f\left( \frac{x}{4} \right) = x^{2} \). Now answer the following

Question. \( f(3) = \)
(a) f(0)
(b) 4+f(0)
(c) 9+f(0)
(d) 16+f(0)
Answer: (d) 16+f(0)

Question. The equation \( f(x) - x - f(0) = 0 \) have exactly
(a) no solution
(b) one solution
(c) two solution
(d) infinite solutions
Answer: (c) two solution

Question. \( f'(0) = \)
(a) 0
(b) 1
(c) \( f(0) \)
(d) \( -f(0) \)
Answer: (a) 0

If \( (f(x))^{2} f\left( \frac{1-x}{1+x} \right) = 64x, x \neq 0 \) then

Question. \( f(x) = \)
(a) \( 4x^{\frac{2}{3}} \left( \frac{1+x}{1-x} \right)^{\frac{1}{3}} \)
(b) \( x^{\frac{1}{3}} \left( \frac{1-x}{1+x} \right)^{\frac{1}{3}} \)
(c) \( x^{\frac{2}{3}} \left( \frac{1-x}{1+x} \right)^{\frac{1}{3}} \)
(d) \( 16x^{1/3} \left( \frac{1+x}{1-x} \right)^{1/3} \)
Answer: (a) \( 4x^{\frac{2}{3}} \left( \frac{1+x}{1-x} \right)^{\frac{1}{3}} \)

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

Question. The value of \( f\left( \frac{9}{7} \right) \) is
(a) \( 8\left( \frac{7}{9} \right)^{2/3} \)
(b) \( 4\left( \frac{9}{7} \right)^{1/3} \)
(c) \( -8\left( \frac{9}{7} \right)^{2/3} \)
(d) \( -4\left( \frac{9}{7} \right)^{1/3} \)
Answer: (c) \( -8\left( \frac{9}{7} \right)^{2/3} \)

Based upon each paragraph, three multiple choice questions have to be answered. Each question has four choices a,b,c and d, out of which only one is correct.
Consider the functions \( f(x) = \begin{cases} x+1, & x \le 1 \\ 2x+1, & 1 < x \le 2 \end{cases} \) and \( g(x) = \begin{cases} x^{2}, & -1 \le x < 2 \\ x+2, & 2 \le x \le 3 \end{cases} \)

Question. The domain of the function \( f(g(x)) \) is
(a) \( \left[ 0, \sqrt{2} \right] \)
(b) \( [-1,2] \)
(c) \( \left[ -1, \sqrt{2} \right] \)
(d) None of the options
Answer: (c) \( \left[ -1, \sqrt{2} \right] \)

Question. The range of the function \( f(g(x)) \) is
(a) \( [1,5] \)
(b) \( [2,3] \)
(c) \( [1,2] \cup (3,5] \)
(d) None of the options
Answer: (c) \( [1,2] \cup (3,5] \)

Question. The number of roots of the equation \( f(g(x)) = 2 \) is
(a) 1
(b) 2
(c) 4
(d) None of the options
Answer: (b) 2

Let \( f(x) = \sin x - x \cos x, \forall x \in R \). Now answer the following.

Question. The least +ve value of ‘x’ for which f(x)=0 lies in the quadrant
(a) \( Q_{1} \)
(b) \( Q_{2} \)
(c) \( Q_{3} \)
(d) \( Q_{4} \)
Answer: (c) \( Q_{3} \)

Question. The set of all the values of \( x \in (0,2\pi) \) for which \( f(x) > 0 \) is
(a) \( (0,\pi) \)
(b) \( (\pi, 2\pi) \)
(c) \( \left( \frac{\pi}{2}, \frac{3\pi}{2} \right) \)
(d) None of the options
Answer: (d) None of the options

Question. If \( \alpha \) is the least +ve value for which \( \tan\alpha = \alpha \) then the area bounded by \( y = f(x) \), \( X - \text{axis} \), \( x=0 \) and \( x=2\pi \) is
(a) 4
(b) \( 4(1 - \cos\alpha) \)
(c) \( 4(1 + \cos\alpha) \)
(d) \( 4 - 2(2 + \alpha^{2}) \cos\alpha \)
Answer: (d) \( 4 - 2(2 + \alpha^{2}) \cos\alpha \)