Class 11 Mathematics Functions MCQs Set 05

SINGLE ANSWER QUESTIONS

Question. Let \( f : R \rightarrow R \) defined by \( f(x) = \frac{e^{x^{2}} - e^{-x^{2}}}{e^{x^{2}} + e^{-x^{2}}} \) then f(x) is
(a) one-one but not onto
(b) neither one-one nor onto
(c) many-one but onto
(d) one-one but not onto
Answer: (b) neither one-one nor onto

Question. If the graph of a function f(x) is symmetrical about the line x=a then
(a) \( f(a+x) = f(a-x) \)
(b) \( f(a+x) = f(x-a) \)
(c) \( f(x) = f(-x) \)
(d) \( f(x) = -f(-x) \)
Answer: (a) \( f(a+x) = f(a-x) \)

Question. If \( f(x) = f(2a-x) \) then the graph of f(x) is symmetric about the line
(a) y=a
(b) x=2a
(c) x=a
(d) x=-a
Answer: (c) x=a

Question. There are exactly two linear functions which map from [-1,1] onto [0,2] they are
(a) \( y = x+1, y = x-1 \)
(b) \( y = x+1, y = -x+1 \)
(c) \( y = 2x-1, y = x-4 \)
(d) \( y = x, y = 3x \)
Answer: (b) \( y = x+1, y = -x+1 \)

Question. \( f : (-\infty,-1] \rightarrow (0,e^{5}] \) defined by \( f(x) = e^{x^{3}-3x+2} \) is
(a) one-one and into
(b) one-one and onto
(c) many-one and into
(d) many-one and onto
Answer: (a) one-one and into

Question. Let \( f(x) = x + 2|x+1| + 2|x-1| \). If \( f(x) = k \) has exactly one real solution, then k equals
(a) 3
(b) 0
(c) 1
(d) 2
Answer: (b) 0

Question. Let \( f(1)=1 \) and \( f(n) = 2 \sum_{r=1}^{n-1} f(r) \) then \( \sum_{n=1}^{m} f(n) = \)
(a) \( 3^{m}-1 \)
(b) \( 3^{m} \)
(c) \( 3^{m-1} \)
(d) \( 3^{m-2} \)
Answer: (c) \( 3^{m-1} \)

Question. The domain of \( f(x) = \log_{10} \left\{ 1 - \log_{10}(x^{2} - 5x + 16) \right\} \) is
(a) (2,3)
(b) \( (0,\infty) \)
(c) [1,3]
(d) [2,3]
Answer: (a) (2,3)

Question. If [.] denotes G.I.F then the domain of \( f(x) = \cos^{-1}(x+[x]) \) is
(a) (0,1)
(b) [0,1)
(c) [0,1]
(d) [-1,1]
Answer: (b) [0,1)

Question. The domain of \( f(x) = \cot^{-1} \left( \frac{x}{\sqrt{x^{2} - [x^{2}]}} \right) \) (where [.] is G.I.F)
(a) R
(b) \( R - \{0\} \)
(c) \( R - \left\{ \pm \sqrt{n}, n \in Z \text{ and } n \ge 0 \right\} \)
(d) None of the options
Answer: (c) \( R - \left\{ \pm \sqrt{n}, n \in Z \text{ and } n \ge 0 \right\} \)

Question. The domain of \( f(x) = \frac{1}{\ln [\cos^{-1} x]} \) ( where [.] G.I.F ) is
(a) [0,1]
(b) \( [-1,\cos 2] \)
(c) \( [-1,\cos 3) \cup (\cos 3,\cos 4) \)
(d) \( [-1,\cos 3) \cup (\cos 3,\cos 2) \)
Answer: (b) \( [-1,\cos 2] \)

Question. If \( f(x) = \cos^{-1}(x-x^{2}) + \sqrt{ 1 - \frac{1}{|x|} } + \frac{1}{ \sqrt{ [ x^{2} - 1 ] } } \) then domain of \( f(x) \) ( where [.] G.I.F) is
(a) \( \left( \sqrt{2}, \frac{1+\sqrt{5}}{2} \right] \)
(b) \( \left( -\sqrt{2}, \frac{1-\sqrt{5}}{2} \right] \)
(c) \( \left[ \sqrt{2}, \frac{1+\sqrt{5}}{2} \right] \)
(d) \( \left[ \sqrt{7}, \frac{\sqrt{7}}{2} \right] \)
Answer: (c) \( \left[ \sqrt{2}, \frac{1+\sqrt{5}}{2} \right] \)

Question. The range of \( f(x) = \sin^{-1} \left( \sqrt{x^{2}+x+1} \right) \) is
(a) \( \left[ 0, \frac{\pi}{2} \right] \)
(b) \( \left[ 0, \frac{\pi}{3} \right] \)
(c) \( \left[ \frac{\pi}{3}, \frac{\pi}{2} \right] \)
(d) \( \left[ \frac{\pi}{6}, \frac{\pi}{2} \right] \)
Answer: (c) \( \left[ \frac{\pi}{3}, \frac{\pi}{2} \right] \)

Question. The function \( f(x) = |px - q| + r|x|, x \in (-\infty,\infty) \) where p>0,q>0,r>0 assumes its minimum value only at one point if
(a) \( p \neq q \)
(b) \( r \neq q \)
(c) \( r \neq p \)
(d) \( p = q = r \)
Answer: (c) \( r \neq p \)

Question. Let f,g,h be real -valued functions defined on the interval [0,1] by \( f(x) = e^{x^{2}} + e^{-x^{2}} ; g(x) = x \cdot e^{x^{2}} + e^{-x^{2}} \) and \( h(x) = x^{2} e^{x^{2}} + e^{-x^{2}} \). If a,b,c denote respectively the absolute max.values of f,g,h on [0,1] then
(a) \( a = b \text{ and } c \neq b \)
(b) \( a = c \text{ and } a \neq b \)
(c) \( a \neq b \text{ and } c \neq b \)
(d) a=b=c
Answer: (d) a=b=c

Question. The range of \( f(x) = \cos^{-1} \left( \frac{\sqrt{2x^{2}+1}}{x^{2}+1} \right) \) is
(a) \( \left[ -\frac{\pi}{2}, \frac{\pi}{2} \right] \)
(b) \( [0,\pi] \)
(c) \( \left[ 0,\frac{\pi}{2} \right) \)
(d) \( (0,\pi) \)
Answer: (c) \( \left[ 0,\frac{\pi}{2} \right) \)

Question. Let ‘f’ be an injective mapping with domain \( \{x, y, z\} \) and range \( \{1,2,3\} \) such that exactly one of the following statements is correct and the remaining are false
\( f(x) = 1; f(y) \neq 1; f(z) \neq 2 \) then \( f^{-1}(1) = \) [IIT 1982]
(a) x
(b) y
(c) z
(d) None of the options
Answer: (b) y

Question. If \( f(x) = -1 + |x - 2| ; 0 \le x \le 4 \)
\( g(x) = 2 - |x| ; -1 \le x \le 3 \) then \( (fog)(x) = \)
(a) \( \begin{cases} -1 - x; & -1 \le x \le 0 \\ x - 1; & 0 < x \le 2 \end{cases} \)
(b) \( \begin{cases} -1 + x; & 0 \le x \le 1 \\ x - 1; & 1 < x \le 2 \end{cases} \)
(c) \( -1 - x ; -1 \le x \le 2 \)
(d) does not exist
Answer: (a) \( \begin{cases} -1 - x; & -1 \le x \le 0 \\ x - 1; & 0 < x \le 2 \end{cases} \)

Question. If \( f : [1,\infty) \rightarrow [2,\infty) \) is given by \( f(x) = x + \frac{1}{x} \) then \( f^{-1}(x) = \)
(a) \( \frac{x \pm \sqrt{x^{2}-3}}{4} \)
(b) \( \frac{x \pm \sqrt{x^{2}-4}}{2} \)
(c) \( \frac{x + \sqrt{x^{2}-4}}{2} \)
(d) \( \frac{x - \sqrt{x^{2}-4}}{2} \)
Answer: (c) \( \frac{x + \sqrt{x^{2}-4}}{2} \)

Question. The period of the real - valued function satisfying \( f(x) + f(x+4) = f(x+2) + f(x+6) \) is
(a) 10
(b) 8
(c) 12
(d) 6
Answer: (b) 8

Question. Let f(x) be a periodic function with period 3 and \( f\left( -\frac{2}{3} \right) = 7 \) and \( g(x) = \int_{0}^{x} f(t+n)dt \) where \( n=3K, K \in N \) then \( g'\left( \frac{7}{3} \right) = \)
(a) \( \frac{-2}{3} \)
(b) 7
(c) -7
(d) \( \frac{7}{3} \)
Answer: (b) 7

Question. If \( f(x) \) is an even function and satisfies the relation \( x^{2} f(x) - 2 f\left( \frac{1}{x} \right) = g(x) \) where \( g(x) \) is an odd function, then the value of \( f(5) \) is
(a) 0
(b) \( \frac{37}{55} \)
(c) 4
(d) 5
Answer: (a) 0

Question. Consider a real valued function f(x) satisfying \( 2f(xy) = (f(x))^{y} + (f(y))^{x} \) \( \forall x,y \in R \) and \( f(1) = a \) where \( a \neq 1 \) then \( (a-1) \sum_{i=1}^{n} f(i) = \)
(a) \( a^{n+1} + a \)
(b) \( a^{n+1} \)
(c) \( a^{n+1} - a \)
(d) \( a^{n+2} - a \)
Answer: (c) \( a^{n+1} - a \)

Question. If 'p' and 'q' are +ve integers, f is a function defined for +ve numbers and attains only +ve values such that \( f(x, f(y)) = x^{p} y^{q} \) then \( p^{2} = \)
(a) 2q
(b) q
(c) 3q
(d) 4q
Answer: (b) q

Question. If the function ‘f’ satisfies the relation \( f(x+y) + f(x-y) = 2f(x)f(y) \) \( \forall x,y \in R \) and \( f(0) \neq 0 \) then f(x) is an
(a) even function
(b) odd function
(c) Neither even nor odd
(d) can not decide
Answer: (a) even function

Question. A real valued function f(x) satisfies the function \( f(x-y) = f(x) f(y) - f(a-x) f(a+y) \) where ‘a’ is a given constant and \( f(0)=1 \), then the graph of the function is symmetrical about [IIT 2005]
(a) point \( (2a,0) \)
(b) point \( (a,0) \)
(c) line \( x=2a \)
(d) line \( x=a \)
Answer: (b) point \( (a,0) \)

Question. The function ‘f’ satisfies the functional equation \( 3f(x) + 2f\left( \frac{x+59}{x-1} \right) = 10x + 30 \) for all real \( x \neq 1 \), then the value of \( f(7) \) is
(a) 8
(b) 4
(c) -8
(d) 11
Answer: (b) 4

Question. Let ‘f’ be a real valued function defined for all \( x \in R \) such that for some fixed a>0, \( f(x+a) = \frac{1}{2} + \sqrt{ f(x) - (f(x))^{2} } \) for all 'x' then the period of \( f(x) \) is
(a) \( \frac{a}{4} \)
(b) \( \frac{a}{3} \)
(c) 2a
(d) None of the options
Answer: (c) 2a

Question. Let \( f(x,y) \) be a periodic function satisfying \( f(x,y) = f(2x+2y, 2y-2x) \) \( \forall x,y \in R \) defined \( g(x) = f(2^{x},0) \) then the period of \( g(x) \) is
(a) 4
(b) 6
(c) 8
(d) 12
Answer: (d) 12

Question. If \( f(x) = \min \left( x^{2}, x, \text{sgn} (x^{2}+4x+5) \right) \) then the value of \( f(2) \) is equal to
(a) 2
(b) 0
(c) 1
(d) 4
Answer: (c) 1

Question. If \( f(x) = \sin^{2}x + \sin^{2} \left( \frac{\pi}{3} + x \right) + \cos(x) \cos \left( x + \frac{\pi}{3} \right) \) and \( g\left( \frac{5}{4} \right) = 1 \) then the graph of \( y = g(f(x)) \) is
(a) a circle
(b) a straight line
(c) a parabola
(d) a pair of straight lines
Answer: (b) a straight line

Question. Let \( f_{1}(n) = 1 + \frac{1}{2} + \frac{1}{3} + \dots + \frac{1}{n} \) then \( f_{1}(1) + f_{1}(2) + f_{1}(3) + \dots + f_{1}(n) = \)
(a) \( n.f_{1}(n) - 1 \)
(b) \( (n+1)f_{1}(n) + n \)
(c) \( (n+1)f_{1}(n) - n \)
(d) \( n.f_{1}(n) + n \)
Answer: (c) \( (n+1)f_{1}(n) - n \)

MULTIPLE ANSWER QUESTIONS

Question. If \( f(x) = x|x|; \ 0 \le x < 1 \) and \( f(x) = 2x; \ x \ge 1 \) then its
(a) even extension is \( \begin{cases} x^{2}; & -1 \le x \le 0 \\ -2x; & -\infty < x \le -1 \end{cases} \)
(b) odd extension is \( \begin{cases} -x^{2}; & -1 \le x \le 0 \\ 2x; & -\infty < x \le -1 \end{cases} \)
(c) even extension is \( \begin{cases} x^{2}; & 0 \le x \le 1 \\ -2x; & 1 < x < \infty \end{cases} \)
(d) does not exist
Answer: (a, b)

Question. If \( f(x) = \sqrt{3|x| - x - 2} \) and \( g(x) = \sin x \) then the domain of \( (fog)(x) = \)
(a) \( \left\{ 2m\pi + \frac{\pi}{2} \right\}; m \in z \)
(b) \( \left[ 2m\pi + \frac{7\pi}{6}, 2m\pi + \frac{11\pi}{6} \right] m \in Z \)
(c) \( \left\{ 2m\pi + \frac{\pi}{3} \right\}, m \in z \)
(d) \( \phi \)
Answer: (a, b)

Question. The domain of \( f(x) = \frac{1}{\sqrt{[|x|-1]-5}} \) ( where [.] G.I.F) is
(a) \( [-7,7] \)
(b) \( (-\infty,7] \)
(c) \( (-\infty,-7] \)
(d) \( [7,\infty) \)
Answer: (c, d)

Question. For the function \( f(x) \) satisfying \( 2f(\sin x) + f(\cos x) = x, \forall x \in R \)
(a) Domain is [0,1]
(b) range is \( \left[ \frac{-2\pi}{3}, \frac{\pi}{3} \right] \)
(c) Domain is [-1,1]
(d) range is \( \left[ \frac{-\pi}{2}, \frac{\pi}{2} \right] \)
Answer: (b, c)

Question. Let \( f(x) = \sin x, g(x) = \ln|x| \) If the ranges of fog and gof are \( R_{1} \) and \( R_{2} \) respectively then [IIT 1994 ]
(a) \( R_{1} = \{U : -1 \le U \le 1\} \)
(b) \( R_{2} = \{V : -\infty < V \le 0\} \)
(c) \( R_{1} = \{U : 0 \le U \le 1\} \)
(d) None of the options
Answer: (a, b)

Question. The polynomial p(x) is such that for any polynomial q(x) we have \( p(q(x)) = q(p(x)) \) then \( p(x) \) is
(a) even
(b) odd
(c) of even degree
(d) of odd degree
Answer: (b, d)

Question. Let \( f(x) = \max \{ 1+\sin x, 1, 1-\cos x \} \); \( x \in [0,2\pi] \) and \( g(x) = \max \{ 1, |x-1| \} \); \( x \in R \) then
(a) \( g(f(0)) = 1 \)
(b) \( g(f(1)) = 1 \)
(c) \( f(g(1)) = 1 \)
(d) \( f(g(0)) = \sin 1 \)
Answer: (a, b)

Question. Let \( f(x) = \sin x + \cos \left( \left( \sqrt{4-a^{2}} \right) x \right) \). Then the integral values of ‘a’ for which f(x) is a periodic function, are given by
(a) 1
(b) 2
(c) -2
(d) 0
Answer: (b, c, d)