Question 1: Palette 1/56

AP-EAPCET 2025 MCQ

If $\sin x \sqrt{\cos y}-\cos y \sqrt{\sin x}=0$, then $\frac{d y}{d x}=$

A

$\tan x$

B

1

C

-1

D

$-\cot x$

Question 2: Palette 2/56

AP-EAPCET 2025 MCQ

If $y=\left(\log _x \sin x\right)^x$, then $\frac{d y}{d x}=$

A

$y\left[\frac{x \sin x}{\log \cos x}+\log (\log \sin x)+\frac{1}{\log x}-\log (\log x)\right]$

B

$y\left[\frac{x \cos x}{\log \sin x}-\log (\log \sin x)+\frac{1}{\log x}+\log (\log x)\right]$

C

$y\left[\frac{x \cot x}{\log \sin x}+\log (\log \sin x)-\frac{1}{\log x}-\log (\log x)\right]$

D

$y\left[\frac{x \cot x}{\log \sin x}-\log (\log \sin x)+\frac{1}{\log x}-(\log x)\right]$

Question 3: Palette 3/56

AP-EAPCET 2025 MCQ

If $x=\sqrt{2^{\operatorname{cosec}^{-1} t}}$ and $y=\sqrt{2^{\sec ^{-1} t}},|t| \geq 1$, then $\frac{d y}{d x}=$

A

$\frac{x}{y}$

B

$\frac{y}{x}$

C

$-\frac{y}{x}$

D

$-\frac{x}{y}$

Question 4: Palette 4/56

AP-EAPCET 2025 MCQ

If $(a+\sqrt{2} b \cos x)(a-\sqrt{2} b \cos y) =a^2-b^2$, where $a>b>0$, then at $\left(\frac{\pi}{4}, \frac{\pi}{4}\right), \frac{d y}{d x}=$

A

$\frac{a+b}{a-b}$

B

$\frac{a-b}{a+b}$

C

$\frac{a-2 b}{a+2 b}$

D

$\frac{2 a+b}{2 a-b}$

Question 5: Palette 5/56

AP-EAPCET 2025 MCQ

If $f(x)=x^{\sec ^{-1} x}$, then $f^{\prime}(2)=$

A

$\frac{2^{\pi / 3}}{6}(\pi-\sqrt{3} \log 2)$

B

$\frac{2^{\pi / 6}}{6}(\pi+\sqrt{3} \log 2)$

C

$\frac{2^{\pi / 3}}{6}(\pi+\sqrt{3} \log 2)$

D

$\frac{2^{\pi / 6}}{6}(\pi-\sqrt{3} \log 2)$

Question 6: Palette 6/56

AP-EAPCET 2025 MCQ

If $y=\tan ^{-1}\left(\frac{3 x-x^3}{1-3 x^2}\right)+\tan ^{-1}\left(\frac{7 x}{1-12 x^2}\right)$, then at $x=0, \frac{d y}{d x}=$

A

6

B

7

C

9

D

10

Question 7: Palette 7/56

AP-EAPCET 2025 MCQ

If $y=\sqrt{\frac{x^4 \sqrt{3 x-5}}{\left(x^2-3\right)(2 x-3)}}$, then $\left(\frac{d y}{d x}\right)_{x=2}=$

A

5

B

0

C

1

D

-5

Question 8: Palette 8/56

AP-EAPCET 2025 MCQ

If $x^2+y^2+\sin y=4$, then the value of $\frac{d^2 y}{d x^2}$ at $x=-2$ is

A

-30

B

-34

C

-32

D

-18

Question 9: Palette 9/56

AP-EAPCET 2025 MCQ

If $y=\sqrt{\cosh x+\sqrt{\cosh x}}$, then $\frac{d y}{d x}=$

A

$\frac{\sinh x\left(2 y^2+2 \cosh x+1\right)}{4 y\left(y^2+\cosh x\right)}$

B

$\frac{\sinh x\left(2 y^2-2 \cosh x-1\right)}{4 y\left(y^2-\cosh x\right)}$

C

$\frac{\sinh x(1-2 \sqrt{\cosh x})}{4 y \sqrt{\cosh x}}$

D

$\frac{\sinh x(1+2 \sqrt{\cosh x})}{4 y \sqrt{\cosh x})}$

Question 10: Palette 10/56

AP-EAPCET 2025 MCQ

If $y=\tan ^{-1} \sqrt{x^2-1}+\sinh ^{-1} \sqrt{x^2-1}, x>1$, then $\frac{d y}{d x}=$

A

$\frac{1}{x \sqrt{x^2-1}}$

B

$\frac{x+1}{x \sqrt{x^2-1}}$

C

$\frac{x+1}{x^2 \sqrt{x^2-1}}$

D

$\frac{x}{\sqrt{x^2-1}}$

Question 11: Palette 11/56

AP-EAPCET 2025 MCQ

If $y=(\log x)^{1 / x}+x^{\log x}$, at $x=e, \frac{d y}{d x}=$

A

$2+\frac{1}{e}$

B

$e^2+\frac{1}{2}$

C

$\frac{1}{e^2}+2$

D

$e+\frac{1}{e}$

Question 12: Palette 12/56

AP-EAPCET 2025 MCQ

If $x=\sqrt{2} e^t(\sin t-\cos t)$ and $y=\sqrt{2} e^t(\sin t+\cos t)$, then $\left(\frac{d^2 y}{d x^2}\right)_{t=\frac{\pi}{4}}=$

A

$-e^{\frac{-\pi}{4}}$

B

$\sqrt{2} e^{\frac{\pi}{4}}$

C

$\sqrt{2} e^{\frac{-\pi}{4}}$

D

$e^{\frac{-\pi}{4}}$

Question 13: Palette 13/56

AP-EAPCET 2025 MCQ

If $g$ is the inverse of the function $f(x)$ and $g(x)=x+\tan x$, then $f^{\prime}(x)=$

A

$1+\sec ^2 x$

B

$\frac{1}{1+\sec ^2 f(x)}$

C

$\frac{1}{1+\sec ^2 g(x)}$

D

$1+\sec ^2 f(x)$

Question 14: Palette 14/56

AP-EAPCET 2025 MCQ

If $\sqrt{x-x y}+\sqrt{y-x y}=1$, then $\frac{d y}{d x}=$

A

$-\sqrt{\frac{y-y^2}{x-x^2}}$

B

$-\sqrt{\frac{1-y^2}{1-x^2}}$

C

$-\sqrt{\frac{1-y}{1-x}}$

D

$-\sqrt{\frac{x-y}{x+y}}$

Question 15: Palette 15/56

AP-EAPCET 2025 MCQ

If $x=2 \cos ^3 \theta$ and $y=3 \sin ^2 \theta$, then $\frac{d y}{d x}=$

A

$-\sec \theta$

B

$\cos \theta$

C

$-\operatorname{cosec} \theta$

D

$\sin \theta$

Question 16: Palette 16/56

AP-EAPCET 2025 MCQ

Assertion (A) If $y=f(x)=(|x|-|x-1|)^2$, then $\left(\frac{d y}{d x}\right)_{x=1}=1$

Reason (R) $\mathop {\lim }\limits_{x \to a} \frac{f(x)-f(a)}{x-a}$ exist, then it is called derivative of $f(x)$ at $x=a$.

A

(A) is true, (R) is true, (R) is correct explanation to (A)

B

(A) is true, (R) is true, (R) is not the correct explanation to (A)

C

(A) is true, (R) is false

D

(A) if false, (R) is true

Question 17: Palette 17/56

AP-EAPCET 2025 MCQ

If $x^2+y^2=t-\frac{1}{t}$ and $x^4+y^4=t^2+\frac{1}{t^2}$, then $\frac{d y}{d x}=$

A

$\frac{y}{x}$

B

$\frac{y^2}{x^2}$

C

$\sqrt{\frac{y}{x}}$

D

$-\frac{y}{x}$

Question 18: Palette 18/56

AP-EAPCET 2025 MCQ

If $y=(a x+b) \cos x$, then

$ y_2+y_1 \sin 2 x+y\left(1+\sin ^2 x\right)= $

A

$y_2 \cos ^2 x$

B

$y_2 \sin ^2 x$

C

$y_1 \sin ^2 x$

D

$y \sin ^2 x$

Question 19: Palette 19/56

AP-EAPCET 2025 MCQ

If $5 f(x)+3 f\left(\frac{1}{x}\right)=x+2$ and $y=x f(x)$, then $\frac{d y}{d x}$ at $x=1$ is equal to

A

14

B

$\frac{7}{8}$

C

1

D

7

Question 20: Palette 20/56

AP-EAPCET 2024 MCQ

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

A

$\frac{-\sqrt{2}}{|1+x| \sqrt{1+x^2}}$

B

$\frac{-1}{(1+x) \sqrt{x}}$

C

$\frac{1}{\left(1+x^2\right) \sqrt{1+x}}$

D

$\frac{-\sqrt{2}}{(1+x) \sqrt{1-x}}$

Question 21: Palette 21/56

AP-EAPCET 2024 MCQ

If $y=(x-1)(x+2)\left(x^2+5\right)\left(x^4+8\right)$, then $\lim _{x \rightarrow-1}\left(\frac{d y}{d x}\right)$ is equal to

A

-30

B

30

C

52

D

-52

Question 22: Palette 22/56

AP-EAPCET 2024 MCQ

If $y=\left(\tan ^{-1} 2 x\right)^2+\left(\cot ^{-1} 2 x\right)^2$, then $\left(1+4 x^2\right)^2 y^{\prime \prime}-16$ is equal to

A

$8 x y^{\prime}$

B

$-8 x\left(1+4 x^2\right) y^{\prime}$

C

$8 x\left(1+4 x^2\right) y^{\prime}$

D

$-8 x y^{\prime}$

Question 23: Palette 23/56

AP-EAPCET 2024 MCQ

If $y=\tan ^{-1} \frac{x}{1+2 x^2}+\tan ^{-1} \frac{x}{1+6 x^2}+\tan ^{-1} \frac{x}{1+12 x^2}$, then $\left(\frac{d y}{d x}\right)_{x=\frac{1}{2}}$ is equal to

A

1

B

-1

C

0

D

$1 / 2$

Question 24: Palette 24/56

AP-EAPCET 2024 MCQ

If $f(x)=5 \cos ^3 x-3 \sin ^2 x$ and $g(x)=4 \sin ^3 x+\cos ^2 x$, then the derivative of $f(x)$ with respect to $g(x)$ is

A

$\frac{5 \cos +2}{6 \cos x-1}$

B

$-\left(\frac{5 \cos x+2}{6 \cos x-1}\right)$

C

$\frac{15 \cos x-6}{12 \sin x+2}$

D

$-\left(\frac{15 \cos x+6}{12 \sin x-2}\right)$

Question 25: Palette 25/56

AP-EAPCET 2024 MCQ

If $y=1+x+x^2+x^3+\ldots \ldots \infty$ and $|x|<1$, then $y^{\prime \prime}$ is equal to

A

$2 y^{\prime}$

B

$\frac{2 y}{y^{\prime}}$

C

$\frac{y^{\prime}}{2 y}$

D

$2 y^2 y^{\prime}$

Question 26: Palette 26/56

AP-EAPCET 2024 MCQ

If $y=\sqrt{\sin x+\sqrt{\sin x+\sqrt{\sin x+\ldots \infty}}}$, then the value of $\frac{d^2 y}{d x^2}$ at the point $(\pi, 1)$ is

A

2

B

-2

C

$-\frac{1}{2}$

D

$\frac{1}{2}$

Question 27: Palette 27/56

AP-EAPCET 2024 MCQ

64. If $f(0)=0, f^{\prime}(0)=3$, then the derivative of $y=f(f(f(f(f(x)))))$ at $x=0$ is

A

16

B

32

C

81

D

243

Question 28: Palette 28/56

AP-EAPCET 2024 MCQ

If $\frac{d}{d x}\left(\frac{1+x^2+x^4}{1+x+x^2}\right)=a x+b$, then $(a, b)=$

A

$(-1,2)$

B

$(-2,1)$

C

$(2,-1)$

D

$(1,2)$

Question 29: Palette 29/56

AP-EAPCET 2024 MCQ

The rate of change of $x^{\sin x}$ with respect to $(\sin x)^x$ is

A

$\frac{x^{\sin x}\left(\frac{\sin x}{x}+\cos x \cdot \log x\right)}{(\sin x)^x(x \cdot \cot x+\log \sin x)}$

B

$\frac{(\sin x)^x(x \cot x+\log \sin x)}{x^{\sin x}\left(\frac{\sin x}{x}+\cos x \cdot \log x\right)}$

C

$y\left(\frac{\sin x}{x}+\cos x \log x\right)$

D

$(\sin x)^x(x \cot x+\log \sin x)$

Question 30: Palette 30/56

AP-EAPCET 2024 MCQ

If $y=\frac{\alpha x+\beta}{\gamma \alpha+\delta}$, then $2 y_1 y_3=$

A

$2 y_2{ }^3$

B

$3 y_2{ }^2$

C

$y_2{ }^2$

D

$3 y_3{ }^2$

Question 31: Palette 31/56

AP-EAPCET 2024 MCQ

Which one of the following is false ?

A

$\frac{d}{d x}\left[\sec ^{-1}(\cosh x)\right]=\operatorname{sech} x$

B

$\frac{d}{d x}\left[\cos ^{-1}(\operatorname{sech} x)\right]=\operatorname{sech} x$

C

$\frac{d}{d x}\left[\tan ^{-1}(\sinh x)\right]=\operatorname{sech} x$

D

$\frac{d}{d x}\left[\tan ^{-1}\left(\tan \frac{x}{2}\right)\right]=\sec x$

Question 32: Palette 32/56

AP-EAPCET 2024 MCQ

If $y=t^2+t^3$ and $x=t-t^4$, then $\frac{d^2 y}{d x^2}$ at $t=1$ is

A

$\frac{-2}{3}$

B

$\frac{-4}{3}$

C

$\frac{8}{3}$

D

4

Question 33: Palette 33/56

AP-EAPCET 2024 MCQ

If $y=\tan (\log x)$, then $\frac{d^2 y}{d x^2}=$

A

$\frac{-\sec ^2(\log x)[1+2 \tan x]}{x^2}$

B

$\frac{\sec ^2(\log x)[1+\tan (\log x)]}{x^2}$

C

$\frac{\sec (\log x)[2 \tan (\log x)-1]}{x^2}$

D

$\frac{\sec ^2(\log x)[2 \tan (\log x)-1]}{x^2}$

Question 34: Palette 34/56

AP-EAPCET 2024 MCQ

For $x<0, \frac{d}{d x}\left[|x|^x\right]=$

A

$(-x)^x[-1+\log (-x)]$

B

$(-x)^x[1+\log (-x)]$

C

$(-x)^x[1-\log (-x)]$

D

$(-x)^x[-1-\log (-x)]$

Question 35: Palette 35/56

AP-EAPCET 2024 MCQ

If $y=x-x^2$, then the rate of change of $y^2$ with respect to $x^2$ at $x=2$ is

A

0

B

-1

C

3

D

9

Question 36: Palette 36/56

AP-EAPCET 2024 MCQ

If $y=f(x)$ is a thrice differentiable function and a bijection, then $\frac{d^2 x}{d y^2}\left(\frac{d y}{d x}\right)^3+\frac{d^2 y}{d x^2}=$

A

$y$

B

$-y$

C

$x$

D

0

Question 37: Palette 37/56

AP-EAPCET 2024 MCQ

If $y=\tan ^{-1}\left(\frac{2-3 \sin x}{3-2 \sin x}\right)$, then $\frac{d y}{d x}=$

A

$\frac{(3-2 \sin x)^2}{13 \sin ^2 x-24 \sin x+13}$

B

$\frac{-5 \cos x}{13 \sin ^2 x-24 \sin x+19}$

C

$\frac{5 \sin x}{13 \sin ^2 x-24 \sin x+13}$

D

$\frac{-5 \sin x}{13 \sin ^2 x-24 \sin x+13}$

Question 38: Palette 38/56

AP-EAPCET 2024 MCQ

If $x=3\left[\sin t-\log \left(\cot \frac{t}{2}\right)\right]$ and $y=6\left[\cos t+\log \left(\operatorname{tin} \frac{t}{2}\right)\right]$ then $\frac{d y}{d x}=$

A

$\frac{2 \sin ^2 t}{1+\sin t \cos t}$

B

$\frac{2 \cos ^2 t}{1+\sin 2 t}$

C

$\frac{2 \cos ^2 t}{1+\sin t \cos t}$

D

$\frac{1+\cos g}{1+\sin a}$

Question 39: Palette 39/56

AP-EAPCET 2024 MCQ

The length of the tangent drawn at the point $P\left(\frac{\pi}{4}\right)$ on the curve $x^{2 / 3}+y^{2 / 3}=2^{2 / 3}$ is

A

$\frac{2}{3}$

B

1

C

$\frac{4}{3}$

D

2

Question 40: Palette 40/56

AP-EAPCET 2022 MCQ

Assertion (A) $\frac{d}{d x}\left(\frac{x^2 \sin x}{\log x}\right)=\frac{x^2 \sin x}{\log x}\left(\cot x+\frac{2}{x}-\frac{1}{x \log x}\right)$

Reason (R) $\frac{d}{d x}\left(\frac{u v}{w}\right)=\frac{u v}{w}\left[\frac{u^{\prime}}{u}+\frac{v^{\prime}}{v}+\frac{w^{\prime}}{w}\right]$

A

A is true, R is true and R is correct explanation of A

B

A is true, R is true and R is not correct explanation of A

C

A is true, R is not correct

D

A is not correct, R is correct

Question 41: Palette 41/56

AP-EAPCET 2022 MCQ

If $x=f(\theta)$ and $y=g(\theta)$, then $\frac{d^2 y}{d x^2}=$

A

$\frac{g^{\prime \prime}(\theta)}{f^{\prime}(\theta)}$

B

$\frac{f^{\prime \prime}(\theta)}{x(\theta)}$

C

$\frac{f^{\prime}(\theta) g^{\prime \prime}(\theta)-g^{\prime}(\theta) f^{\prime \prime}(\theta)}{\left(f^{\prime}(\theta)\right)^3}$

D

$\frac{g^{\prime}(\theta) f^{\prime \prime}(\theta)-g^{\prime \prime}(\theta) f^{\prime \prime}(\theta)}{\left(g^{\prime \prime}(\theta)\right)^3}$

Question 42: Palette 42/56

AP-EAPCET 2022 MCQ

$y=x^3-a x^2+48 x+7$ is an increasing function for all real values of $x$, then $a$ lies in the interval

A

$(-14,14)$

B

$(-12,12)$

C

$(-16,16)$

D

$(-21,-21)$

Question 43: Palette 43/56

AP-EAPCET 2022 MCQ

If $x \neq 0$ and $f(x)$ satisfies $8 f(x)+6 f(1 / x) =x+5$, then $\frac{d}{d x}\left(x^2 f(x)\right)$ at $x=1$ is

A

$-1 / 14$

B

$25 / 14$

C

$9 / 14$

D

$19 / 14$

Question 44: Palette 44/56

AP-EAPCET 2022 MCQ

If $f(x)=\cot ^{-1}\left(\frac{x^x+x^{-x}}{2}\right)$, then $f^{\prime}(1)=$

A

1

B

$-$1

C

2

D

$-$2

Question 45: Palette 45/56

AP-EAPCET 2021 MCQ

If $x=\sec \theta-\cos \theta$ and $y=\sec ^n \theta-\cos ^n \theta$, then $\left(x^2+4\right)\left(\frac{d y}{d x}\right)^2$ is equal to

A

$n(y+4)$

B

$n^2\left(y^2+4\right)$

C

$n(y+2)$

D

$n^2\left(y^2+2\right)$

Question 46: Palette 46/56

AP-EAPCET 2021 MCQ

If $y=\log _{\cot x} \tan x-\log _{\tan x} \cot x +\tan ^{-1}\left(\frac{4 x}{4-x^2}\right)$, then $\frac{d y}{d x}$ is equal to

A

$\frac{1}{4+x^2}$

B

$\frac{4}{4+x^2}$

C

$\frac{1}{4-x^2}$

D

$\frac{4}{4-x^2}$

Question 47: Palette 47/56

AP-EAPCET 2021 MCQ

If $f(x)=2x^2+3x-5$, then the value of $f'(0)+3f'(-1)$ is equal to

A

1

B

0

C

3

D

2

Question 48: Palette 48/56

AP-EAPCET 2021 MCQ

If $y=\left(1+\frac{1}{x}\right)\left(1+\frac{2}{x}\right)\left(1+\frac{3}{x}\right) \ldots\left(1+\frac{n}{x}\right)$ and $x \neq 0$. When $x=-1, \frac{d y}{d x}$ is equal to

A

$n !$

B

$(n-1) !$

C

$(-1)^n(n-1)!$

D

$(-1)^n n!$

Question 49: Palette 49/56

AP-EAPCET 2021 MCQ

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

A

0

B

1

C

2

D

$-$1

Question 50: Palette 50/56

AP-EAPCET 2021 MCQ

If $y=e^{x^2+e^{x^2+e^{x^2+\cdots \infty}}}$, then $\frac{d y}{d x}$ is equal to

A

$\frac{2 x}{1-y}$

B

$\frac{2 x y}{y-1}$

C

$\frac{2 x y}{1-y}$

D

$\frac{2 y}{y-1}$

Question 51: Palette 51/56

AP-EAPCET 2021 MCQ

$\frac{d}{d x}\left[\tan ^{-1}\left(\frac{\cos x}{1+\sin x}\right)\right]$ is equal to

A

$\frac{1}{2}$

B

$\frac{-1}{2}$

C

1

D

$-$1

Question 52: Palette 52/56

AP-EAPCET 2021 MCQ

If $x^2+y^2=1$, then

A

$y\left(y^{\prime \prime}\right)-4\left(y^{\prime}\right)^2+1=0$

B

$y\left(y^{\prime \prime}\right)+\left(y^{\prime}\right)^2+1=0$

C

$y\left(y^{\prime \prime}\right)-\left(y^{\prime}\right)^2-1=0$

D

$y\left(y^{\prime \prime}\right)+2\left(y^{\prime}\right)^2+1=0$

Question 53: Palette 53/56

AP-EAPCET 2021 MCQ

If $y=x+\frac{1}{x}$, then which among the following holds?

A

$x^2 y^{\prime}+x y=0$

B

$x^2 y^{\prime}+x y+2=0$

C

$x^2 y^{\prime}-x y+2=0$

D

$x^2 y^{\prime}+x y-2=0$

Question 54: Palette 54/56

AP-EAPCET 2021 MCQ

If $3 \sin x y+4 \cos x y=5$, then $\frac{d y}{d x}$ is equal to

A

$\frac{3 \sin x y+4 \cos x y}{3 \cos x y-4 \sin x y}$

B

$\frac{3 \cos x y+4 \sin x y}{4 \cos x y-3 \sin x y}$

C

$\frac{-y}{x}$

D

$\frac{x}{y}$

Question 55: Palette 55/56

AP-EAPCET 2021 MCQ

$f(x)=\sqrt{x^2+1}: g(x)=\frac{x+1}{x^2+1}: h(x)=2 x-3$, then the value of $f^{\prime}\left[h^{\prime}\left(g^{\prime}(x)\right)\right]$ is equal to

A

$\sqrt{5}$

B

$\frac{2}{\sqrt{5}}$

C

$\frac{\sqrt{5}}{2}$

D

$\frac{1}{\sqrt{5}}$

Question 56: Palette 56/56

AP-EAPCET 2021 MCQ

For which value(s) of $a$ $f(x)=-x^3+4 a x^2+2 x-5$ is decreasing for every $x$ ?

A

(1, 2)

B

(3, 4)

C

R

D

no value of a