MCQ (Single Correct Answer)

1

If $$x = {e^t}\sin t$$, $$y = {e^t}\cos t$$ then $${{{d^2}y} \over {d{x^2}}}$$ at x = $$\pi$$ is

WB JEE 2008
2

The value of $${{dy} \over {dx}}$$ at $$x = {\pi \over 2}$$, where y is given by $$y = {x^{\sin x}} + \sqrt x $$ is

WB JEE 2008
3

Select the correct statement from (a), (b), (c), (d). The function $$f(x) = x{e^{1 - x}}$$

WB JEE 2008
4

The function $$f(x) = {e^{ax}} + {e^{ - ax}},a > 0$$ is monotonically increasing for

WB JEE 2008
5

The second order derivative of a sin3t with respect to a cos3t at $$t = {\pi \over 4}$$ is

WB JEE 2009
6

If $$y = {\tan ^{ - 1}}\sqrt {{{1 - \sin x} \over {1 + \sin x}}} $$, then the value of $${{dy} \over {dx}}$$ at $$x = {\pi \over 6}$$ is

WB JEE 2009
7

If x2 + y2 = 4, then $$y{{dy} \over {dx}} + x = $$

WB JEE 2011
8

If $$y = {A \over x} + B{x^2}$$, then $${x^2}{{{d^2}y} \over {d{x^2}}}$$ =

WB JEE 2011
9

Let $$f(x) = ta{n^{ - 1}}x$$. Then $$f'(x) + f''(x)=0$$, when x is equal to

WB JEE 2011
10

If $$y = {\tan ^{ - 1}}{{\sqrt {1 + {x^2}} - 1} \over x}$$, then y'(1) =

WB JEE 2011
11

If $$y = 2{x^3} - 2{x^2} + 3x - 5$$, then for x = 2 and $$\Delta$$x = 0.1 the value of $$\Delta$$y is

WB JEE 2011
12

The approximate value of $$\root 5 \of {33} $$ correct to 4 decimal places is

WB JEE 2011
13

Let $$f(x) = {x^3}{e^{ - 3x}},\,x > 0$$. Then the maximum value of f(x) is

WB JEE 2011
14

If ' $f$ ' is the inverse function of ' $g$ ' and $g^{\prime}(x)=\frac{1}{1+x^n}$, then the value of $f^{\prime}(x)$ is

WB JEE 2025
15

Let $f(x)$ be a second degree polynomial. If $f(1)=f(-1)$ and $p, q, r$ are in A.P., then $f^{\prime}(p), f^{\prime}(q), f^{\prime}(r)$ are

WB JEE 2025
16

If $$\mathrm{U}_{\mathrm{n}}(\mathrm{n}=1,2)$$ denotes the $$\mathrm{n}^{\text {th }}$$ derivative $$(\mathrm{n}=1,2)$$ of $$\mathrm{U}(x)=\frac{\mathrm{L} x+\mathrm{M}}{x^2-2 \mathrm{~B} x+\mathrm{C}}$$ (L, M, B, C are constants), then $$\mathrm{PU}_2+\mathrm{QU}_1+\mathrm{RU}=0$$, holds for

WB JEE 2024
17

$$ \text { If } y=\tan ^{-1}\left[\frac{\log _e\left(\frac{e}{x^2}\right)}{\log _e\left(e x^2\right)}\right]+\tan ^{-1}\left[\frac{3+2 \log _e x}{1-6 \cdot \log _e x}\right] \text {, then } \frac{d^2 y}{d x^2}= $$

WB JEE 2024
18

Suppose $$f:R \to R$$ be given by $$f(x) = \left\{ \matrix{ 1,\,\,\,\,\,\,\,\,\,\,\mathrm{if}\,x = 1 \hfill \cr {e^{({x^{10}} - 1)}} + {(x - 1)^2}\sin {1 \over {x - 1}},\,\mathrm{if}\,x \ne 1 \hfill \cr} \right.$$

then

WB JEE 2023
19

Let $${\cos ^{ - 1}}\left( {{y \over b}} \right) = {\log _e}{\left( {{x \over n}} \right)^n}$$, then $$A{y_2} + B{y_1} + Cy = 0$$ is possible for, where $${y_2} = {{{d^2}y} \over {d{x^2}}},{y_1} = {{dy} \over {dx}}$$

WB JEE 2023
20

The function $$y = {e^{kx}}$$ satisfies $$\left( {{{{d^2}y} \over {d{x^2}}} + {{dy} \over {dx}}} \right)\left( {{{dy} \over {dx}} - y} \right) = y{{dy} \over {dx}}$$. It is valid for

WB JEE 2023
21

If $$y = {\log ^n}x$$, where $${\log ^n}$$ means $${\log _e}{\log _e}{\log _e}\,...$$ (repeated n times), then $$x\log x{\log ^2}x{\log ^3}x\,.....\,{\log ^{n - 1}}x{\log ^n}x{{dy} \over {dx}}$$ is equal to

WB JEE 2023
22

If $$x = \sin \theta $$ and $$y = \sin k\theta $$, then $$(1 - {x^2}){y_2} - x{y_1} - \alpha y = 0$$, for $$\alpha=$$

WB JEE 2023
23

If $$y = {e^{{{\tan }^{ - 1}}x}}$$, then

WB JEE 2022
24
Let $$g(x) = \int\limits_x^{2x} {{{f(t)} \over t}dt} $$ where x > 0 and f be continuous function and f(2x) = f(x), then
WB JEE 2021
25
A bulb is placed at the centre of a circular track of radius 10 m. A vertical wall is erected touching the track at a point P. A man is running along the track with a speed of 10 m/sec. Starting from P the speed with which his shadow is running along the wall when he is at an angular distance of 60$$^\circ$$ from P is
WB JEE 2021
26
Let f(x) > 0 for all x and f'(x) exists for all x. If f is the inverse function of h and $${h'(x) = {1 \over {1 + \log x}}}$$. Then, f'(x) will be
WB JEE 2019
27
Let f(x) be a derivable function, f'(x) > f(x) and f(0) = 0. Then,
WB JEE 2019
28
Let $$f(x) = {x^4} - 4{x^3} + 4{x^2} + c,\,c \in R$$. Then
WB JEE 2019
29
Let $${f_1}(x) = {e^x}$$, $${f_2}(x) = {e^{{f_1}(x)}}$$, ......, $${f_{n + 1}}(x) = {e^{{f_n}(x)}}$$ for all n $$ \ge $$ 1. Then for any fixed n, $${d \over {dx}}{f_n}(x)$$ is
WB JEE 2018
30
The equation x log x = 3 $$-$$ x
WB JEE 2018
31
If $$f(x) = {\log _5}{\log _3}x$$, then f'(e) is equal to
WB JEE 2017

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