MCQ (Single Correct Answer)

1

If $y=\frac{\cos x}{1+\sin x}$, then

(a) $\frac{d y}{d x}=\frac{-1}{1+\sin x}$

(b) $\frac{d y}{d x}=\frac{1}{1+\sin x}$

(c) $\frac{\mathrm{dy}}{\mathrm{dx}}=-\frac{1}{2} \sec ^2\left(\frac{\pi}{4}-\frac{\mathrm{x}}{2}\right)$

(d) $\frac{\mathrm{dy}}{\mathrm{dx}}=\frac{1}{2} \sec ^2\left(\frac{\pi}{4}-\frac{\mathrm{x}}{2}\right)$

KCET 2025
2

If $\mathrm{y}=\mathrm{a} \sin ^3 \mathrm{t}, \mathrm{x}=\mathrm{a} \cos ^3 \mathrm{t}$, then $\frac{\mathrm{dy}}{\mathrm{dx}}$ at $\mathrm{t}=\frac{3 \pi}{4}$ is

KCET 2025
3

The derivative of $\sin \mathrm{x}$ with respect to $\log \mathrm{x}$ is

KCET 2025
4

If $y=2 x^{3 x}$, then $d y / d x$ at $x=1$ is

KCET 2024
5

$\frac{d}{d x}\left[\cos ^2\left(\cot ^{-1} \sqrt{\frac{2+x}{2-x}}\right)\right]$ is

KCET 2024
6

If $$y=a \sin x+b \cos x$$, then $$y^2+\left(\frac{d y}{d x}\right)^2$$ is a

KCET 2023
7

If $$f(x)=1+n x+\frac{n(n-1)}{2} x^2+\frac{n(n-1)(n-2)}{6} x^3+\ldots+x^n$$, then $$f^n(1)$$ is equal to :

KCET 2023
8

If $$f(x)$$ and $$g(x)$$ are two functions with $$g(x)=x-\frac{1}{x}$$ and $$f \circ g(x)=x^3-\frac{1}{x^3}$$, then $$f^{\prime}(x)$$ is equals to

KCET 2023
9

If $$y=\left(1+x^2\right) \tan ^{-1} x-x$$, then $$\frac{d y}{d x}$$ is

KCET 2022
10

If $$x=e^\theta \sin \theta, y=e^\theta \cos \theta$$ where $$\theta$$ is a parameter, then $$\frac{d y}{d x}$$ at $$(1,1)$$ is equal to

KCET 2022
11

If $$y=e^{\sqrt{x \sqrt{x} \sqrt{x}}...,} x >1$$, then $$\frac{d^2 y}{d x^2}$$ at $$x=\log _e 3$$ is

KCET 2022
12

If $$f(1)=1, f^{\prime}(l)=3$$, then the derivative of $$f(f(f(x)))+(f(x))^2$$ at $$x=1$$ is

KCET 2022
13

If $$y=x^{\sin x}+(\sin x)^x$$, then $$\frac{d y}{d x}$$ at $$x=\frac{\pi}{2}$$ is

KCET 2022
14

If $$e^y+x y=e$$ the ordered pair $$\left(\frac{d y}{d x}, \frac{d^2 y}{d x^2}\right)$$ at $$x=0$$ is equal to

KCET 2022
15

If $$a$$ and $$b$$ are fixed non-zero constants, then the derivative of $$\frac{a}{x^4}-\frac{b}{x^2}+\cos x$$ is $$m a+n b-p$$, where

KCET 2021
16

If $$y=\left(\cos x^2\right)^2$$, then $$\frac{d y}{d x}$$ is equal to

KCET 2021
17

For constant $$a, \frac{d}{d x}\left(x^x+x^a+a^x+a^a\right)$$ is

KCET 2021
18

Consider the following statements

Statement 1 : If $$y=\log _{10} x+\log _e x$$, then $$\frac{d y}{d x}=\frac{\log _{10} e}{x}+\frac{1}{x}$$

Statement 2 : If $$\frac{d}{d x}\left(\log _{10} x\right)=\frac{\log x}{\log 10}$$ and $$\frac{d}{d x}\left(\log _e x\right)=\frac{\log x}{\log e}$$

KCET 2021
19

If the parametric equation of curve is given by $$x=\cos \theta+\log \tan \frac{\theta}{2}$$ and $$y=\sin \theta$$, then the points for which $$\frac{d y}{d x}=0$$ are given by

KCET 2021
20

If $$y=(x-1)^2(x-2)^3(x-3)^5$$, then $$\frac{d y}{d x}$$ at $$x=4$$ is equal to

KCET 2021
21

If $$2^x+2^y=2^{x+y}$$, then $$\frac{d y}{d x}$$ is

KCET 2020
22

If $$y=2 x^{n+1}+\frac{3}{x^n}$$, then $$x^2 \frac{d^{2 y}}{d x^2}$$ is

KCET 2020
23

If $$(x e)^y=e^y$$, then $$\frac{d y}{d x}$$ is

KCET 2020
24

If $$[x]$$ represents the greatest integer function and $$f(x)=x-[x]-\cos x$$, then $$f^{\prime}\left(\frac{\pi}{2}\right)=$$

KCET 2019
25

If $$x=a \sec ^2 \theta$$ & $$y=a \tan ^2 \theta$$, then $$\frac{d^2 y}{d x^2}=$$

KCET 2019
26

$$\sqrt[3]{y} \sqrt{x}=\sqrt[6]{(x+y)^5}$$, then $$\frac{d y}{d x}=$$

KCET 2019
27
If $\cos y=x \cos (a+y)$ with $\cos a \neq \pm 1$, then $\frac{d y}{d x}$ is equal to
KCET 2018
28
If $f(x)=|\cos x-\sin x|$, then $f^{\prime}\left(\frac{\pi}{6}\right)$ is equal to
KCET 2018
29
$$ \text { If } y=\sqrt{x+\sqrt{x+\sqrt{x+\ldots \infty}}} \text {, then } \frac{d y}{d x} \text { is equal } $$ to
KCET 2018
30
If $\sin x=\frac{2 t}{1+t^2}, \tan y+\frac{2 t}{1-t^2}$, then $\frac{d y}{d x}$ is equal to
KCET 2017
31
If $y=\log (\log x)$, then $\frac{d^2 y}{d x^2}$ is equal to
KCET 2017
32
If $y=\tan ^{-1}\left(\frac{\sin x+\cos x}{\cos x-\sin x}\right)$, then $\frac{d y}{d x}$ is equal to
KCET 2017
33
The derivative of $\cos ^{-1}\left(2 x^2-1\right)$ w.r.t. $\cos ^{-1} x$ is
KCET 2017

MCQ (More than One Correct Answer)

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