JEE Mathematics Differential Coefficient MCQs

Choose the most appropriate option (a, b, c or d).

Question. If \( y = (1 + x)(1 + x^2)(1 + x^4) \dots (1 + x^{2^n}) \) then \( \frac{dy}{dx} \) at \( x = 0 \) is
(a) 1
(b) -1
(c) 0
(d) none of the options
Answer: (a) 1

Question. If \( y = |\cos x| + |\sin x| \) then \( \frac{dy}{dx} \) at \( x = \frac{2\pi}{3} \) is
(a) \( \frac{1 - \sqrt{3}}{2} \)
(b) 0
(c) \( \frac{1}{2}(\sqrt{3} - 1) \)
(d) none of the options
Answer: (c) \( \frac{1}{2}(\sqrt{3} - 1) \)

Question. The differential coefficient of \( f(\log_e x) \) with respect to \( x \), where \( f(x) = \log_e x \), is
(a) \( \frac{x}{\log_e x} \)
(b) \( \frac{1}{x} \log_e x \)
(c) \( \frac{1}{x \log_e x} \)
(d) none of the options
Answer: (c) \( \frac{1}{x \log_e x} \)

Question. If \( f(x) = \cos x \cdot \cos 2x \cdot \cos 4x \cdot \cos 8x \cdot \cos 16x \) then \( f'\left( \frac{\pi}{4} \right) \) is
(a) \( \sqrt{2} \)
(b) \( \frac{1}{\sqrt{2}} \)
(c) 1
(d) none of the options
Answer: (a) \( \sqrt{2} \)

Question. If \( y = \cos^{-1}(\cos x) \) then \( \frac{dy}{dx} \) at \( x = \frac{5\pi}{4} \) is equal to
(a) 1
(b) -1
(c) \( \frac{1}{\sqrt{2}} \)
(d) none of the options
Answer: (b) -1

Question. If \( y = \sec(\tan^{-1} x) \) then \( \frac{dy}{dx} \) at \( x = 1 \) is equal to
(a) \( \frac{1}{\sqrt{2}} \)
(b) \( -\frac{1}{\sqrt{2}} \)
(c) 1
(d) none of the options
Answer: (a) \( \frac{1}{\sqrt{2}} \)

Question. If \( x^y = e^{x-y} \) then \( \frac{dy}{dx} \) at \( x = 1 \) is equal to
(a) 0
(b) -2
(c) 1
(d) none of the options
Answer: (a) 0

Question. If \( xe^{xy} - y = \sin^2 x \) then \( \frac{dy}{dx} \) at \( x = 0 \) is
(a) 0
(b) 1
(c) -1
(d) none of the options
Answer: (b) 1

Question. If \( x = e^{y+e^{y+\dots \text{ to } \infty}} \) then \( \frac{dy}{dx} \) is
(a) \( \frac{x}{1 + x} \)
(b) \( \frac{1}{x} \)
(c) \( \frac{1 - x}{x} \)
(d) none of the options
Answer: (c) \( \frac{1 - x}{x} \)

Question. If \( y = \log_2(\log_e x) \) then \( \frac{dy}{dx} \) is
(a) \( \frac{1}{x} \log_2 e \cdot \log_e e \)
(b) \( \frac{1}{x} \log_2 x \)
(c) \( \frac{1}{x} \log_e x \)
(d) none of the options
Answer: (a) \( \frac{1}{x} \log_2 e \cdot \log_e e \)

Question. If \( f'(x) = \sqrt{2x^2 - 1} \) and \( y = f(x^2) \) then \( \frac{dy}{dx} \) at \( x = 1 \) is
(a) 2
(b) 1
(c) -2
(d) none of the options
Answer: (a) 2

Question. If \( y = \cos^{-1} \left( \frac{5 \cos x - 12 \sin x}{13} \right); x \in \left( 0, \frac{\pi}{2} \right) \), then \( \frac{dy}{dx} \) is equal to
(a) 1
(b) -1
(c) 0
(d) none of the options
Answer: (a) 1

Question. If \( y = \tan^{-1} \sqrt{\frac{x+1}{x-1}} \) then \( \frac{dy}{dx} \) is equal to
(a) \( \frac{-1}{2 | x | \sqrt{x^2 - 1}} \)
(b) \( \frac{-1}{2x \sqrt{x^2 - 1}} \)
(c) \( \frac{1}{2x \sqrt{x^2 - 1}} \)
(d) none of the options
Answer: (a) \( \frac{-1}{2 | x | \sqrt{x^2 - 1}} \)

Question. If \( x^y \cdot y^x = 16 \) then \( \frac{dy}{dx} \) at \( (2, 2) \) is
(a) 1
(b) -1
(c) 0
(d) none of the options
Answer: (b) -1

Question. If \( y = x^{1/x} \), the value of \( \frac{dy}{dx} \) at \( x = e \) is
(a) 1
(b) 0
(c) -1
(d) none of the options
Answer: (b) 0

Question. The derivative of \( \tan^{-1} \frac{2x}{1 - x^2} \) with respect to \( \sin^{-1} \frac{2x}{1 + x^2} \) is
(a) \( \frac{1}{1 + x^2} \)
(b) \( \frac{1}{1 - x^2} \)
(c) 0
(d) 1
Answer: (d) 1

Question. Let the function \( y = f(x) \) be given by \( x = t^5 - 5t^3 - 20t + 7 \) and \( y = 4t^3 - 3t^2 - 18t + 3 \), where \( t \in (-2, 2) \). Then \( f'(x) \) at \( t = 1 \) is
(a) \( \frac{5}{2} \)
(b) \( \frac{2}{5} \)
(c) \( \frac{7}{5} \)
(d) none of the options
Answer: (b) \( \frac{2}{5} \)

Question. If \( f(x) = \cos \left[ \frac{\pi}{2} [x] - x^3 \right], 1 < x < 2 \), and \( [x] = \) the greatest integer \( \le x \), then \( f'\left( \sqrt[3]{\frac{\pi}{2}} \right) \) is equal to
(a) 0
(b) \( 3 \left( \frac{\pi}{2} \right)^{2/3} \)
(c) \( -3 \left( \frac{\pi}{2} \right)^{3/2} \)
(d) none of the options
Answer: (a) 0

Question. If \( y = \sin x^\circ \) and \( z = \cos x \) then \( \frac{dy}{dz} \) is equal to
(a) \( -\text{cosec } x \cdot \cos x \)
(b) \( \frac{\pi}{180} \text{cosec } \frac{\pi x}{180} \cdot \cos x \)
(c) \( -\frac{\pi}{180} \text{cosec } x \cdot \cos \frac{\pi x}{180} \)
(d) none of the options
Answer: (c) \( -\frac{\pi}{180} \text{cosec } x \cdot \cos \frac{\pi x}{180} \)

Question. The derivative of \( \tan^{-1} \frac{\sqrt{1 + x^2} - 1}{x} \) with respect to \( \tan^{-1} x \) is
(a) \( \frac{\sqrt{1 + x^2} - 1}{x^2} \)
(b) 1
(c) \( \frac{1}{1 + x^2} \)
(d) none of the options
Answer: (d) none of the options

Question. The differential coefficient of \( \text{cosec}^{-1} \frac{1}{2x^2 - 1} \) with respect to \( \sqrt{1 - x^2} \) at \( x = \frac{1}{2} \) is
(a) -4
(b) 4
(c) -1
(d) none of the options
Answer: (a) -4

Question. If the prime sign(') represents differentiation w.r.t. x and \( f'(x) = \sin x + \sin 4x \cdot \cos x \) then \( f'\left( 2x^2 + \frac{\pi}{2} \right) \) at \( x = \sqrt{\frac{\pi}{2}} \) is equal to
(a) 0
(b) -1
(c) \( -2\sqrt{2\pi} \)
(d) none of the options
Answer: (c) \( -2\sqrt{2\pi} \)

Question. If \( u = f(x^3), v = g(x^2), f'(x) = \cos x \) and \( g'(x) = \sin x \) then \( \frac{du}{dv} \) is
(a) \( \frac{3}{2} x \cdot \cos x^3 \cdot \text{cosec } x^2 \)
(b) \( \frac{2}{3} \sin x^3 \cdot \sec x^2 \)
(c) \( \tan x \)
(d) none of the options
Answer: (a) \( \frac{3}{2} x \cdot \cos x^3 \cdot \text{cosec } x^2 \)

Question. If \( y = f\left( \frac{2x - 1}{x^2 + 1} \right) \) and \( f'(x) = \sin x^2 \) then \( \frac{dy}{dx} \) is
(a) \( \sin \left( \frac{2x - 1}{x^2 + 1} \right)^2 \)
(b) \( \frac{2(1 + x - x^2)}{(x^2 + 1)^2} \sin \left( \frac{2x - 1}{x^2 + 1} \right)^2 \)
(c) \( \frac{2(2x - 1)}{x^2 + 1} \sin \left( \frac{2x - 1}{x^2 + 1} \right)^2 \)
(d) none of the options
Answer: (b) \( \frac{2(1 + x - x^2)}{(x^2 + 1)^2} \sin \left( \frac{2x - 1}{x^2 + 1} \right)^2 \)

Question. If \( e^x = \frac{\sqrt{1+t} - \sqrt{1-t}}{\sqrt{1+t} + \sqrt{1-t}} \) and \( \tan \frac{y}{2} = \sqrt{\frac{1-t}{1+t}} \) then \( \frac{dy}{dx} \) at \( t = \frac{1}{2} \) is
(a) \( -\frac{1}{2} \)
(b) \( \frac{1}{2} \)
(c) 0
(d) none of the options
Answer: (a) \( -\frac{1}{2} \)

Question. If \( t(1 + x^2) = x \) and \( x^2 + t^2 = y \) then \( \frac{dy}{dx} \) at \( x = 2 \) is
(a) \( \frac{88}{125} \)
(b) \( \frac{488}{125} \)
(c) 1
(d) none of the options
Answer: (b) \( \frac{488}{125} \)

Question. Let \( f(x) \) be a polynomial function of the second degree. If \( f(1) = f(-1) \) and \( a_1, a_2, a_3 \) are in AP then \( f'(a_1), f'(a_2), f'(a_3) \) are in
(a) AP
(b) GP
(c) HP
(d) none of the options
Answer: (a) AP

Question. If \( P(x) \) is a polynomial such that \( P(x^2 + 1) = \{P(x)\}^2 + 1 \) and \( P(0) = 0 \) then \( P'(0) \) is equal to
(a) 1
(b) 0
(c) -1
(d) none of the options
Answer: (a) 1

Question. If \( 5f(x) + 3f\left( \frac{1}{x} \right) = x + 2 \) and \( y = xf(x) \) then \( \left( \frac{dy}{dx} \right)_{x=1} \) is equal to
(a) 14
(b) \( \frac{7}{8} \)
(c) 1
(d) none of the options
Answer: (b) \( \frac{7}{8} \)

Question. If for all \( x, y \) the function \( f \) is defined by \( f(x) + f(y) + f(x) \cdot f(y) = 1 \) and \( f(x) > 0 \) then
(a) \( f'(x) \) does not exist
(b) \( f'(x) = 0 \) for all \( x \)
(c) \( f'(0) < f'(1) \)
(d) none of the options
Answer: (b) \( f'(x) = 0 \) for all \( x \)

Question. There exists a function \( f(x) \) satisfying \( f(0) = 1, f'(0) = -1, f(x) > 0 \) for all \( x \) and
(a) \( f'(x) < 0 \) for all \( x \)
(b) \( -1 < f''(x) < 0 \) for all \( x \)
(c) \( -2 \le f''(x) \le -1 \) for all \( x \)
(d) \( f''(x) \le -2 \) for all \( x \)
Answer: (a) \( f'(x) < 0 \) for all \( x \)

Question. If \( g \) is the inverse function of \( f \) and \( f'(x) = \sin x \) then \( g'(x) \) is
(a) \( \text{cosec } \{g(x)\} \)
(b) \( \sin \{g(x)\} \)
(c) \( \frac{1}{\sin \{g(x)\}} \)
(d) none of the options
Answer: (a) \( \text{cosec } \{g(x)\} \)

Question. Let \( f(x) \) be a polynomial function of degree 2 and \( f(x) > 0 \) for all \( x \in \mathbb{R} \). If \( g(x) = f(x) + f'(x) + f''(x) \) then for any \( x \)
(a) \( g(x) < 0 \)
(b) \( g(x) > 0 \)
(c) \( g(x) = 0 \)
(d) \( g(x) \ge 0 \)
Answer: (b) \( g(x) > 0 \)

Question. If \( y^2 = P(x) = \) a polynomial of degree 3 then \( 2 \frac{d}{dx} \left( y^3 \frac{d^2 y}{dx^2} \right) \) equals
(a) \( P'''(x) + P'(x) \)
(b) \( P''(x) \cdot P'''(x) \)
(c) \( P(x) \cdot P'''(x) \)
(d) none of the options
Answer: (c) \( P(x) \cdot P'''(x) \)

Question. Let \( f(x) = \begin{vmatrix} x^3 & \sin x & \cos x \\ 6 & -1 & 0 \\ p & p^2 & p^3 \end{vmatrix} \), where \( p \) is a constant. Then \( \frac{d^3}{dx^3} \{f(x)\} \) at \( x = 0 \) is
(a) \( p \)
(b) \( p + p^2 \)
(c) \( p + p^3 \)
(d) independent of \( p \)
Answer: (d) independent of \( p \)

Question. If \( y = \sin 2x \) then \( \frac{d^6 y}{dx^6} \) at \( x = \frac{\pi}{2} \) is equal to
(a) -64
(b) 0
(c) 64
(d) none of the options
Answer: (b) 0

Question. \( x = t \cos t, y = t + \sin t \) then \( \frac{d^2 x}{dy^2} \) at \( t = \frac{\pi}{2} \) is equal to
(a) \( \frac{\pi + 4}{2} \)
(b) \( -\frac{\pi + 4}{2} \)
(c) -2
(d) none of the options
Answer: (b) \( -\frac{\pi + 4}{2} \)

Question. If \( y = at^2, x = 2at \), where \( a \) is a constant, then \( \frac{d^2 y}{dx^2} \) at \( x = \frac{1}{2} \) is
(a) \( \frac{1}{2a} \)
(b) 1
(c) \( 2a \)
(d) none of the options
Answer: (a) \( \frac{1}{2a} \)

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

Question. Let \( f(x) = (ax + b) \cos x + (cx + d) \sin x \) and \( f'(x) = x \cos x \) be an identity in \( x \). Then
(a) \( a = 5 \)
(b) \( b = 1 \)
(c) \( c = 1 \)
(d) \( d = -5 \)
Answer: (b) \( b = 1 \), (c) \( c = 1 \)

Question. Let \( f(x) = 2 \tan^{-1} x + \sin^{-1} \frac{2x}{1 + x^2} \). Then
(a) \( f'(2) = f'(3) \)
(b) \( f'(2) = 0 \)
(c) \( f'\left( \frac{1}{2} \right) = \frac{16}{5} \)
(d) \( f'\left( \frac{1}{2} \right) = 0 \)
Answer: (a) \( f'(2) = f'(3) \), (b) \( f'(2) = 0 \), (c) \( f'\left( \frac{1}{2} \right) = \frac{16}{5} \)

Question. If \( f(x) = x^3 + x^2 f'(1) + x f''(2) + f'''(3) \) for all \( x \in \mathbb{R} \) then
(a) \( f(0) + f(2) = f(1) \)
(b) \( f(0) + f(3) = 0 \)
(c) \( f(1) + f(3) = f(2) \)
(d) none of the options
Answer: (a) \( f(0) + f(2) = f(1) \), (b) \( f(0) + f(3) = 0 \), (c) \( f(1) + f(3) = f(2) \)

Question. If \( f(x - y), f(x) \cdot f(y) \) and \( f(x + y) \) are in AP for all \( x, y \), and \( f(0) \neq 0 \), then
(a) \( f(2) = f(-2) \)
(b) \( f(3) + f(-3) = 0 \)
(c) \( f'(2) + f'(-2) = 0 \)
(d) \( f'(3) = f'(-3) \)
Answer: (a) \( f(2) = f(-2) \), (c) \( f'(2) + f'(-2) = 0 \)

Question. Let \( f(-x) = f(x) \). Then \( f'(x) \) must be
(a) an even function
(b) an odd function
(c) a periodic function
(d) neither even nor odd
Answer: (b) an odd function

Question. If \( f_n(x) = e^{f_{n-1}(x)} \) for all \( n \in N \) and \( f_0(x) = x \) then \( \frac{d}{dx} \{f_n(x)\} \) is equal to
(a) \( f_n(x) \cdot \frac{d}{dx} \{f_{n-1}(x)\} \)
(b) \( f_n(x) \cdot f_{n-1}(x) \)
(c) \( f_n(x) \cdot f_{n-1}(x) \cdot \dots \cdot f_2(x) \cdot f_1(x) \)
(d) none of the options
Answer: (a) \( f_n(x) \cdot \frac{d}{dx} \{f_{n-1}(x)\} \), (c) \( f_n(x) \cdot f_{n-1}(x) \cdot \dots \cdot f_2(x) \cdot f_1(x) \)

Question. Let \( f(x) = x^2 + x g'(1) + g''(2) \) and \( g(x) = f(1) \cdot x^2 + x f'(x) + f''(x) \) then
(a) \( f'(1) + f'(2) = 0 \)
(b) \( g'(2) = g'(1) \)
(c) \( g''(2) + f''(3) = 0 \)
(d) none of the options
Answer: (a) \( f'(1) + f'(2) = 0 \), (b) \( g'(2) = g'(1) \), (c) \( g''(2) + f''(3) = 0 \)

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