JEE Mathematics Functions MCQs Set 01

Question. If \( f(x) = x^2 + \lambda x + \mu \) be integral function of the integral variable \( x \) then
(a) \( \lambda \) is an integer and \( \mu \) is a rational fraction
(b) \( \lambda \) and \( \mu \) are integers
(c) \( \mu \) is an integer and \( \lambda \) is a rational fraction
(d) \( \lambda \) and \( \mu \) are rational fractions
Answer: (b) \( \lambda \) and \( \mu \) are integers

Question. Let \( f(x) = ax^2 + bx + c \), where \( a, b, c \) are rational, and \( f : \mathbb{Z} \to \mathbb{Z} \) where \( \mathbb{Z} \) is the set of integers. Then \( a + b \) is
(a) a negative integer
(b) an integer
(c) nonintegral rational number
(d) none of the options
Answer: (b) an integer

Question. If \( f(x) = \cos [\pi]x + \cos [\pi x] \), where \( [y] \) is the greatest integer function of \( y \) then \( f\left(\frac{\pi}{2}\right) \) is equal to
(a) \( \cos 3 \)
(b) 0
(c) \( \cos 4 \)
(d) none of the options
Answer: (c) \( \cos 4 \)

Question. Let \( f(x) = \sin (\tan^{-1}x) \). Then \( [f(-\sqrt{3})] \), where \( [.] \) denotes the greatest integer function, is
(a) \( -\frac{\sqrt{3}}{2} \)
(b) 0
(c) -1
(d) none of the options
Answer: (c) -1

Question. If \( f(x) = \frac{x - 1}{x + 1} \) then \( f(ax) \) in term of \( f(x) \) is equal to
(a) \( \frac{f(x) + a}{1 + af(x)} \)
(b) \( \frac{(a - 1)f(x) + a + 1}{(a + 1)f(x) + a - 1} \)
(c) \( \frac{(a + 1)f(x) + a - 1}{(a - 1)f(x) + a + 1} \)
(d) none of the options
Answer: (c) \( \frac{(a + 1)f(x) + a - 1}{(a - 1)f(x) + a + 1} \)

Question. Let \( f(1) = 1 \) and \( f(n) = 2\sum_{r=1}^{n-1} f(r) \). Then \( \sum_{n=1}^{m} f(n) \) is equal to
(a) \( 3^m - 1 \)
(b) \( 3^m \)
(c) \( 3^{m-1} \)
(d) none of the options
Answer: (c) \( 3^{m-1} \)

Question. If \( f(x + 1) + f(x - 1) = 2f(x) \) and \( f(0) = 0 \) then \( f(n), n \in N \), is
(a) \( nf(1) \)
(b) \( \{f(1)\}^n \)
(c) 0
(d) none of the options
Answer: (a) \( nf(1) \)

Question. If \( af(x + 1) + bf\left(\frac{1}{x+1}\right) = x, x \neq -1, a \neq b \) then \( f(2) \) is equal to
(a) \( \frac{2a + b}{2(a^2 - b^2)} \)
(b) \( \frac{a}{a^2 - b^2} \)
(c) \( \frac{a + 2b}{a^2 - b^2} \)
(d) none of the options
Answer: (a) \( \frac{2a + b}{2(a^2 - b^2)} \)

Question. Let \( f \) be a function satisfying \( f(x + y) = f(x) + f(y) \) for all \( x, y \in R \). If \( f(1) = k \) then \( f(n), n \in N \), is equal to
(a) \( k^n \)
(b) \( nk \)
(c) \( n^k \)
(d) none of the options
Answer: (b) \( nk \)

Question. Let \( f \) be a function satisfying \( f(x + y) = f(x).f(y) \) for all \( x, y \in R \). If \( f(1) = 3 \) then \( \sum_{r=1}^{n} f(r) \) is equal to
(a) \( \frac{3}{2}(3^n - 1) \)
(b) \( \frac{3}{2}n(n + 1) \)
(c) \( 3^{n+1} - 3 \)
(d) none of the options
Answer: (a) \( \frac{3}{2}(3^n - 1) \)

Question. If \( f(x + y) = f(x) + f(y) - xy - 1 \) for all \( x, y \), and \( f(1) = 1 \) then the number of solutions of \( f(n) = n, n \in N \), is
(a) one
(b) two
(c) four
(d) none of the options
Answer: (a) one

Question. Let \( f(x) = 1 + | x |, x < -1 \)
\( [x], x \geq -1 \), where \( [.] \) denotes the greatest integer function.
Then \( f\{f(-2, 3)\} \) is equal to
(a) 4
(b) 2
(c) -3
(d) 3
Answer: (d) 3

Question. The domain of the function \( y = \log_{10} \log_{10} \log_{10} \dots \log_{10} x \) is
(a) \( [10^n, +\infty) \)
(b) \( (10^{n-1}, +\infty) \)
(c) \( (10^{n-2}, +\infty) \)
(d) none of the options
Answer: (d) none of the options

Question. The largest set of real values of \( x \) for which \( f(x) = \sqrt{(x+2)(5-x)} - \frac{1}{\sqrt{x^2 - 4}} \) is real function is
(a) \( [1, 2) \cup (2, 5] \)
(b) \( (2, 5] \)
(c) \( [3, 4] \)
(d) none of the options
Answer: (b) \( (2, 5] \)

Question. Let \( f(x) = (x^{12} - x^9 + x^4 - x + 1)^{-1/2} \). The domain of the function is
(a) \( (1, +\infty) \)
(b) \( (-\infty, -1) \)
(c) \( (-1, 1) \)
(d) \( (-\infty, +\infty) \)
Answer: (d) \( (-\infty, +\infty) \)

Question. The domain of the function \( f(x) = \sqrt{x - \sqrt{1 - x^2}} \) is
(a) \( \left[-1, -\frac{1}{\sqrt{2}}\right] \cup \left[\frac{1}{\sqrt{2}}, 1\right] \)
(b) \( [-1, 1] \)
(c) \( \left(-\infty, -\frac{1}{\sqrt{2}}\right] \cup \left[\frac{1}{\sqrt{2}}, +\infty\right) \)
(d) \( \left[\frac{1}{\sqrt{2}}, 1\right] \)
Answer: (d) \( \left[\frac{1}{\sqrt{2}}, 1\right] \)

Question. The domain of the function \( f(x) = \sqrt{1 - \sqrt{1 - \sqrt{1 - x^2}}} \) is
(a) \( \{x | x < 1\} \)
(b) \( \{x | x > -1\} \)
(c) \( [0, 1] \)
(d) \( [-1, 1] \)
Answer: (d) \( [-1, 1] \)

Question. The domain of the function \( f(x) = \log_{10} \log_{10} (1 + x^3) \) is
(a) \( (-1, +\infty) \)
(b) \( (0, +\infty) \)
(c) \( [0, +\infty) \)
(d) \( (-1, 0) \)
Answer: (b) \( (0, +\infty) \)

Question. The domain of the function \( f(x) = \sqrt{x^2 - [x]^2} \), where \( [x] = \) the greatest integer less than or equal to \( x \), is
(a) \( R \)
(b) \( [0, +\infty) \)
(c) \( (-\infty, 0] \)
(d) none of the options
Answer: (d) none of the options

Question. The domain of \( f(x) = \frac{1}{\sqrt{|\cos x| + \cos x}} \) is
(a) \( [-2n\pi, 2n\pi] \)
(b) \( (2n\pi, 2n + 1)\pi \)
(c) \( \left(\frac{(4n + 1)\pi}{2}, \frac{(4n + 3)\pi}{2}\right) \)
(d) \( \left(\frac{(4n - 1)\pi}{2}, \frac{(4n + 1)\pi}{2}\right) \)
Answer: (d) \( \left(\frac{(4n - 1)\pi}{2}, \frac{(4n + 1)\pi}{2}\right) \)

Question. The domain of \( f(x) = \sqrt{\log_{x^2 - 1}(x)} \) is
(a) \( (\sqrt{2}, +\infty) \)
(b) \( (0, +\infty) \)
(c) \( (1, +\infty) \)
(d) none of the options
Answer: (a) \( (\sqrt{2}, +\infty) \)

Question. The domain of the function \( f(x) = {}^{16-x}C_{2x-1} + {}^{20-3x}P_{4x-5} \), where the symbols have their usual meanings, is the set
(a) \( \{1, 2, 3, 4, 5\} \)
(b) \( \{2, 3, 4\} \)
(c) \( \{2, 3\} \)
(d) none of the options
Answer: (c) \( \{2, 3\} \)

Question. The domain of \( f(x) = \sin^{-1}\left(\frac{1 + x^2}{2x}\right) + \sqrt{1 - x^2} \) is
(a) \( \{1\} \)
(b) \( (-1, 1) \)
(c) \( \{1, -1\} \)
(d) none of the options
Answer: (c) \( \{1, -1\} \)

Question. The domain of the function \( f(x) = \sqrt{\sec^{-1}\left\{\frac{1 - |x|}{2}\right\}} \) is
(a) \( (-\infty, -3] \cup [3, +\infty) \)
(b) \( [3, +\infty) \)
(c) \( \phi \)
(d) \( R \)
Answer: (a) \( (-\infty, -3] \cup [3, +\infty) \)

Question. The function \( f(x) = \sqrt{e^{\cos^{-1}(\log_4 x^2)}} \) is real valued. It is defined if
(a) \( x \in \left[\frac{1}{2}, 2\right] \)
(b) \( x \in \left[-2, -\frac{1}{2}\right] \cup \left[\frac{1}{2}, 2\right] \)
(c) \( x \in \left[-2, -\frac{1}{2}\right] \)
(d) none of the options
Answer: (b) \( x \in \left[-2, -\frac{1}{2}\right] \cup \left[\frac{1}{2}, 2\right] \)

Question. The domain of the real-valued function \( f(x) = \log_e | \log_e x | \) is
(a) \( (1, +\infty) \)
(b) \( (0, +\infty) \)
(c) \( (e, +\infty) \)
(d) none of the options
Answer: (d) none of the options

Question. If \( [.] \) denotes the greatest integer function then the domain of the real valued function \( \log_{[x + 1/2]} | x^2 - x - 2 | \) is
(a) \( \left[\frac{3}{2}, +\infty\right) \)
(b) \( \left[\frac{3}{2}, 2\right) \cup (2, +\infty) \)
(c) \( (e, +\infty) \)
(d) none of the options
Answer: (b) \( \left[\frac{3}{2}, 2\right) \cup (2, +\infty) \)

Question. The domain of the function \( f(x) = \log_e (x - [x]) \), where \( [.] \) denotes the greatest integer function, is
(a) \( R \)
(b) \( R - Z \)
(c) \( (0, +\infty) \)
(d) none of the options
Answer: (b) \( R - Z \)

Question. The domain of the function \( f(x) = \sin^{-1} (x + [x]) \), where \( [.] \) denote the greatest integer function, is
(a) \( [0, 1) \)
(b) \( [-1, 1] \)
(c) \( (-1, 0) \)
(d) none of the options
Answer: (a) \( [0, 1) \)

Question. Let \( f(x) = \log_{x^2} 25 \) and \( g(x) = \log_x 5 \) then \( f(x) = g(x) \) holds for \( x \) belonging to
(a) \( R \)
(b) \( (0, 1) \cup (1, +\infty) \)
(c) \( \phi \)
(d) none of the options
Answer: (b) \( (0, 1) \cup (1, +\infty) \)

Question. Let \( f(x) = \sin^2 \frac{x}{2} + \cos^2 \frac{x}{2} \) and \( g(x) = \sec^2 x - \tan^2 x \). The two functions are equal over the set
(a) \( \phi \)
(b) \( R \)
(c) \( R - \{x | x = (2n + 1)\frac{\pi}{2}, n \in Z\} \)
(d) none of the options
Answer: (c) \( R - \{x | x = (2n + 1)\frac{\pi}{2}, n \in Z\} \)

Question. The range of the function \( f(x) = x^2 + \frac{1}{x^2 + 1} \) is
(a) \( [1, +\infty) \)
(b) \( [2, +\infty) \)
(c) \( [\frac{3}{2}, +\infty) \)
(d) none of the options
Answer: (a) \( [1, +\infty) \)

Question. Let \( f(x) = \cos^{-1}\left(\frac{x^2}{1 + x^2}\right) \). The range of \( f \) is
(a) \( [0, \frac{\pi}{2}] \)
(b) \( [-\frac{\pi}{2}, \frac{\pi}{2}] \)
(c) \( [-\frac{\pi}{2}, 0] \)
(d) none of the options
Answer: (d) none of the options

Question. The range of the real-valued function \( f(x) = \sqrt{9 - x^2} \) is
(a) \( [0, 3] \)
(b) \( [-3, 3] \)
(c) \( [-3, 0] \)
(d) none of the options
Answer: (a) \( [0, 3] \)

Question. The range of the function \( f(x) = | x - 1 | + | x - 2 |, -1 \leq x \leq 3, \) is
(a) \( [1, 3] \)
(b) \( [1, 5] \)
(c) \( [3, 5] \)
(d) none of the options
Answer: (b) \( [1, 5] \)

Question. The range of the function \( y = \log_3 (5 + 4x - x^2) \) is
(a) \( (0, 2] \)
(b) \( (-\infty, 2] \)
(c) \( (0, 9] \)
(d) none of the options
Answer: (b) \( (-\infty, 2] \)

Question. Let \( f : \{x, y, z\} \to \{a, b, c\} \) be a one-one function and only one of the conditions (i) \( f(x) \neq b \), (ii) \( f(y) = b \), (iii) \( f(z) \neq a \) is true then the function \( f \) is given by the set
(a) \( \{(x, a), (y, b), (z, c)\} \)
(b) \( \{(x, a), (y, c), (z, b)\} \)
(c) \( \{(x, b), (y, a), (z, c)\} \)
(d) \( \{(x, c), (y, b), (z, a)\} \)
Answer: (c) \( \{(x, b), (y, a), (z, c)\} \)

Choose the correct options. One or more options may be correct.

Question. Let \( f(x) = x^2, 0 < x < 2; \quad 2x - 3, 2 \leq x < 3; \quad x + 2, x \geq 3. \) Then
(a) \( f\{f(f(f(\frac{3}{2})))\} = f(\frac{3}{2}) \)
(b) \( 1 + f\{f(f(\frac{5}{2}))\} = f(\frac{5}{2}) \)
(c) \( f\{f(1)\} = f(1) = 1 \)
(d) none of the options
Answer: (a) \( f\{f(f(f(\frac{3}{2})))\} = f(\frac{3}{2}) \), (b) \( 1 + f\{f(f(\frac{5}{2}))\} = f(\frac{5}{2}) \), (c) \( f\{f(1)\} = f(1) = 1 \)

Question. If \( f(x) = \cos^2 x + \cos^2 \left(x + \frac{\pi}{3}\right) - \cos x . \cos \left(x + \frac{\pi}{3}\right) \) then
(a) \( f(x) \) is an even function
(b) \( f\left(\frac{\pi}{8}\right) = f\left(\frac{\pi}{4}\right) \)
(c) \( f(x) \) is a constant function
(d) \( f(x) \) is not periodic function
Answer: (a) \( f(x) \) is an even function, (b) \( f\left(\frac{\pi}{8}\right) = f\left(\frac{\pi}{4}\right) \), (c) \( f(x) \) is a constant function

Question. If one of the roots of \( x^2 + f(a) . x + a = 0 \) is equal to the third power of the other for real \( a \) then
(a) the domain of the real-valued function \( f \) is the set of non-negative real numbers
(b) \( f(x) = -x^{1/4}(1 + x^{1/2}) \)
(c) \( f(x) = x^{1/4} + x^{3/4} \)
(d) none of the options
Answer: (a) the domain of the real-valued function \( f \) is the set of non-negative real numbers, (b) \( f(x) = -x^{1/4}(1 + x^{1/2}) \)

Question. If \( f \) is an even function defined on the interval (-5, 5) then a value of \( x \) satisfying the equation \( f(x) = f\left(\frac{x + 1}{x + 2}\right) \) is
(a) \( \frac{-1 + \sqrt{5}}{2} \)
(b) \( \frac{-2 + \sqrt{5}}{2} \)
(c) \( \frac{-1 - \sqrt{5}}{2} \)
(d) \( \frac{-3 - \sqrt{5}}{2} \)
Answer: (a) \( \frac{-1 + \sqrt{5}}{2} \), (b) \( \frac{-2 + \sqrt{5}}{2} \), (c) \( \frac{-1 - \sqrt{5}}{2} \), (d) \( \frac{-3 - \sqrt{5}}{2} \)

Question. Let \( f(x) = [x] = \) the greatest integer less than or equal to \( x \) and \( g(x) = x - [x] \). Then for any two real numbers \( x \) and \( y \)
(a) \( f(x + y) = f(x) + f(y) \)
(b) \( g(x + y) = g(x) + g(y) \)
(c) \( f(x + y) = f(x) + f\{y + g(x)\} \)
(d) none of the options
Answer: (c) \( f(x + y) = f(x) + f\{y + g(x)\} \)

Question. Let \( x \in N \) and let \( x \) be a perfect square. Let \( f(x) = \) the quotient when \( x \) is divided by 5 and \( g(x) = \) the remainder when \( x \) is divided by 5. Then \( \sqrt{x} = f(x) + g(x) \) holds for \( x \) equal to
(a) 0
(b) 16
(c) 25
(d) none of the options
Answer: (b) 16, (c) 25

Question. If \( f(x) = 27x^3 + \frac{1}{x^3} \) and \( \alpha, \beta \) are the roots of \( 3x + \frac{1}{x} = 2 \) then
(a) \( f(\alpha) = f(\beta) \)
(b) \( f(\alpha) = 10 \)
(c) \( f(\beta) = -10 \)
(d) none of the options
Answer: (a) \( f(\alpha) = f(\beta) \), (c) \( f(\beta) = -10 \)

Question. If \( f(x) = \sin^{-1}(\sin x) \) then
(a) \( f(x) = \pi - x, 0 \leq x \leq \frac{\pi}{2} \)
(b) \( f(x) = \pi - x, \frac{\pi}{2} \leq x \leq \pi \)
(c) \( f(x) = x, 0 \leq x \leq \pi \)
(d) \( f(x) = -x, -\frac{\pi}{2} \leq x \leq 0 \)
Answer: (b) \( f(x) = \pi - x, \frac{\pi}{2} \leq x \leq \pi \)