Question 1: Palette 1/74

TS-EAMCET 2025 MCQ

If $x=t-\sin t, y=1-\cos t$ and $\frac{d^2 y}{d x^2}=-1$ at $t=k, k>0$ then $\lim _{i \rightarrow K} \frac{y}{x}=$

A

$\frac{2}{\pi}$

B

$\frac{\pi-2}{2}$

C

$\frac{2}{\pi-2}$

D

$\frac{\pi}{2}$

Question 2: Palette 2/74

TS-EAMCET 2025 MCQ

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

A

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

B

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

C

$-\frac{4 x}{\left(1+x^2\right)^2}$

D

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

Question 3: Palette 3/74

TS-EAMCET 2025 MCQ

If $y=x^{\log x}+(\log x)^x, x>1$, then $\left(\frac{d y}{d x}\right)_{x=e}=$

A

0

B

1

C

2

D

3

Question 4: Palette 4/74

TS-EAMCET 2025 MCQ

If $y=\sqrt{\log \left(x^2+1\right)+\sqrt{\log \left(x^2+1\right)+\sqrt{\log \left(x^2+1\right)+\ldots+\infty}}, \text {, } 100.00}$, $|x|<1$, then $\frac{d y}{d x}=$

A

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

B

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

C

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

D

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

Question 5: Palette 5/74

TS-EAMCET 2025 MCQ

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

A

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

B

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

C

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

D

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

Question 6: Palette 6/74

TS-EAMCET 2025 MCQ

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

A

$\frac{2 \cos 5 \theta+\sin 3 \theta \sin 2 \theta}{2 \cos 5 \theta-\cos 3 \theta \cos 2 \theta}$

B

$\frac{2 \cos 5 \theta-\sin 3 \theta \sin 2 \theta}{2 \cos 5 \theta+\cos 3 \theta \cos 2 \theta}$

C

$\frac{2 \cos 5 \theta+\cos 3 \theta \cos 2 \theta}{2 \cos 5 \theta-\sin 3 \theta \sin 2 \theta}$

D

$\frac{2 \cos 5 \theta-\sin 3 \theta \sin 2 \theta}{2 \cos 5 \theta-\cos 3 \theta \cos 2 \theta}$

Question 7: Palette 7/74

TS-EAMCET 2025 MCQ

If $3^x y^x=x^{3 y}$, then the value of $\frac{d y}{d x}$ at $x=1$ is

A

-3

B

3

C

$-\frac{1}{3}$

D

$\frac{1}{3}$

Question 8: Palette 8/74

TS-EAMCET 2025 MCQ

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

A

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

B

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

C

$\frac{2(x y)}{1-x^2}$

D

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

Question 9: Palette 9/74

TS-EAMCET 2025 MCQ

If $f(x)=\log _{\left(x^2-2 x+1\right)}\left(x^2-3 x+2\right), x \in R-[1,2]$ and $x \neq 0$, then $f^{\prime}(3)=$

A

1

B

0

C

$\log _e 4$

D

$\log _4 \mathrm{e}$

Question 10: Palette 10/74

TS-EAMCET 2025 MCQ

If $\frac{d}{d x}\left\{\left(\frac{x-1}{x-\sqrt{x}}\right) e^{2 x+1}\right\}=\frac{x-1}{x-\sqrt{x}} e^{2 x+1} f(x)$, then $f(4)=$

A

0

B

1

C

$\frac{35}{24}$

D

$\frac{47}{24}$

Question 11: Palette 11/74

TS-EAMCET 2025 MCQ

If $y=f(\cosh x)$ and $f^{\prime}(x)=\log \left(x+\sqrt{x^2-1}\right)$, then $\frac{d^2 y}{d x^2}=$

A

$\sinh x+x \cosh x$

B

$x \sinh x$

C

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

D

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

Question 12: Palette 12/74

TS-EAMCET 2025 MCQ

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

A

2

B

-2

C

1

D

-1

Question 13: Palette 13/74

TS-EAMCET 2025 MCQ

If $x=\frac{t^2}{1+t^5}, y=\frac{2 t^3}{1+t^5}$ and $t \neq-1$ is a perimeter, then $\frac{d y}{d x}=$

A

$\frac{2\left(3+2 t^5\right)}{\left(2-3 t^5\right)}$

B

$\frac{2 t\left(3-2 t^5\right)}{\left(2-3 t^5\right)}$

C

$\frac{2 t\left(3-2 t^5\right)}{\left(2+3 t^5\right)}$

D

$\frac{2\left(3+2 t^5\right)}{\left(2+3 t^5\right)}$

Question 14: Palette 14/74

TS-EAMCET 2024 MCQ

A function $f: R \rightarrow R$ is such that $y f(x+y)+\cos m x y=1+y f(x)$. If $m=2$, then $f^{\prime}(x)=$

A

$-2 \sin 2 x y$

B

$4 x$

C

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

D

$2 x^{2}$

Question 15: Palette 15/74

TS-EAMCET 2024 MCQ

If $y=\sqrt{\sin (\log 2 x)+\sqrt{\sin (\log 2 x)+\sqrt{\sin (\log 2 x)+\ldots \infty,}}}$ then $\frac{d y}{d x}=$

A

$\frac{\cos (\log 2 x)}{2 x(2 y-1)}$

B

$\frac{\cos (\log 2 x)}{(2 y-1)}$

C

$\frac{\cos (\log 2 x)}{x(2 y-1)}$

D

$\frac{\sin (\log 2 x)}{x(2 y-1)}$

Question 16: Palette 16/74

TS-EAMCET 2024 MCQ

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

A

$\frac{6 x \cos (2 x)-3 \sin (2 x)}{x^{2}-\sin ^{2}(2 x)}$

B

$\frac{6 x \sin (2 x)-3 \cos (2 x)}{x^{2}+\sin ^{2}(2 x)}$

C

$\frac{2 x \cos (2 x)-\sin (2 x)}{x^{2}+\sin ^{2}(2 x)}$

D

$\frac{6 x \cos (2 x)-3 \sin (2 x)}{x^{2}+\sin ^{2}(2 x)}$

Question 17: Palette 17/74

TS-EAMCET 2024 MCQ

Derivative of $(\sin x)^{x}$ with respect to $x^{(\sin x)}$ is

A

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

B

$\frac{(\sin x)^{x}[(\sin x)(\log (\sin x)+x \cos x)]}{x^{(\sin x)}[x \cos x(\log x)+\sin x]}$

C

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

D

$\frac{x^{\sin x}[x \cos x(\log x)+\sin x]}{(\sin x)^{x}[(\sin x) \log (\sin x)+x \cos x]}$

Question 18: Palette 18/74

TS-EAMCET 2024 MCQ

If $y=\log \left(x-\sqrt{x^{2}-1}\right)$, then $\left(x^{2}-1\right) y^{\prime \prime}+x y^{\prime}+e^{y}+\sqrt{x^{2}-1}=$

A

0

B

1

C

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

D

$x$

Question 19: Palette 19/74

TS-EAMCET 2024 MCQ

If $y=\log \left[\tan \sqrt{\frac{2^x-1}{2^x+1}}\right], x>0$, then $\left(\frac{d y}{d x}\right)_{x=1}=$

A

$\frac{4 \sqrt{2} \log 2}{9 \sin \left(\frac{2}{\sqrt{3}}\right)}$

B

$\frac{4 \sqrt{3} \log 2}{9 \sin \left(\frac{\sqrt{3}}{2}\right)}$

C

$\frac{4 \sqrt{3} \log 2}{9 \sin \left(\frac{2}{\sqrt{3}}\right)}$

D

$\frac{4 \sqrt{2} \log 2}{9 \sin \left(\frac{\sqrt{3}}{2}\right)}$

Question 20: Palette 20/74

TS-EAMCET 2024 MCQ

If $\log y=y^{\log x}$, then $\frac{d y}{d x}=$

A

$\frac{y(\log y)^2}{x(1-\log x \log y)}$

B

$\frac{x(\log x)^2}{y(1-\log x \log y)}$

C

$\frac{x(1-\log x \log y)}{y(\log y)^2}$

D

$\frac{y(1-\log x \log y)}{x(\log x)^2}$

Question 21: Palette 21/74

TS-EAMCET 2024 MCQ

If $y=a \cos 3 x+b e^{-x}$, then $y^{\prime \prime}(3 \sin 3 x-\cos 3 x)=$

A

$10 y^{\prime} \sin 3 x+3 y(\sin 3 x+3 \cos 3 x)$

B

$10 y^{\prime} \cos 3 x+3 y(\sin 3 x+3 \cos 3 x)$

C

$10 y \cos 3 x+3 y(\cos 3 x+3 \sin 3 x)$

D

$10 y^{\prime} \cos 3 x+3 y(\sin 3 x-3 \cos 3 x)$

Question 22: Palette 22/74

TS-EAMCET 2024 MCQ

If $y=\frac{\tan x \cos ^{-1} x}{\sqrt{1-x^2}}$, then the value of $\frac{d y}{d x}$, when $x=0$ is

A

0

B

$\frac{\pi}{2}$

C

1

D

$\frac{\pi}{6}$

Question 23: Palette 23/74

TS-EAMCET 2024 MCQ

If $y(\cos x)^{\sin x}=(\sin x)^{\sin x}$, then the value of $\frac{d y}{d x}$ at $x=\frac{\pi}{4}$ is

A

0

B

1

C

$\sqrt{2}$

D

$\frac{\sqrt{3}}{2}$

Question 24: Palette 24/74

TS-EAMCET 2024 MCQ

If $y=44 x^{45}+45 x^{-44}$, then $y^n=$

A

$\frac{1980 y}{x^2}$.

B

$\frac{2020 x^2}{y}$

C

$\frac{2024 y}{x^2}$

D

$\frac{1990 x^2}{y}$

Question 25: Palette 25/74

TS-EAMCET 2024 MCQ

If $2 x^2-3 x y+4 y^2+2 x-3 y+4=0$, then $\left(\frac{d y}{d x}\right)_{(3,2)}=$

A

-5

B

$\frac{5}{7}$

C

-2

D

$\frac{2}{7}$

Question 26: Palette 26/74

TS-EAMCET 2024 MCQ

If $x=\frac{9 t^2}{1+t^4}$ and $y=\frac{16 t^2}{1-t^4}$ then $\frac{d y}{d x}=$

A

$\frac{16}{9}\left(\frac{1-t^4}{1+t^4}\right)^3$

B

$\frac{16}{9} \frac{\left(1-t^4\right)}{\left(1+t^4\right)}$

C

$\frac{9}{16} \frac{\left(1-t^4\right)}{\left(1+t^4\right)}$

D

$\frac{16}{9}\left(\frac{1+t^4}{1-t^4}\right)^3$

Question 27: Palette 27/74

TS-EAMCET 2024 MCQ

If $y=\sin a x+\cos b x$, then $y^{\prime \prime}+b^2 y=$

A

$\left(b^2-a^2\right) \sin a x$

B

$\left(b^2-a^2\right) \cos b x$

C

$\left(a^2-b^2\right) \tan a x$

D

$\left(b^2-a^2\right) \cot b x$

Question 28: Palette 28/74

TS-EAMCET 2024 MCQ

A particle moving from a fixed point on a straight line travels a distance $S$ metres in $t \mathrm{sec}$. If $S=t^3-t^2-t+3$, then the distance (in mts) travelled by the particle when it comes to rest, is

A

5

B

4

C

2

D

3

Question 29: Palette 29/74

TS-EAMCET 2023 MCQ

If $f(x)=\sqrt{x}(x \geq 0)$ and $g(x)=1+x^2$, then $(f \circ g)^{\prime}(1)=$

A

1

B

$1 / 2$

C

$\sqrt{2}$

D

$1 / \sqrt{2}$

Question 30: Palette 30/74

TS-EAMCET 2023 MCQ

Match the values of $\frac{d y}{d x}$ at $x=\frac{\pi}{3}$ for the following system of curves in parametric form given in List-I with those of the items in List-II

List-I			List-II
(i)	$x=a(\theta -\sin \theta ),y=a(1-\cos \theta )$$x=a(\theta -\sin \theta ),y=a(1-\cos \theta )$x=a(theta-sin theta),y=a(1-cos theta)	(a)	$4\sqrt{3}$$4\sqrt{3}$4sqrt3
(ii)	$x=3\cos \theta -2{\cos }^3\theta ,y=3\sin \theta -2{\sin }^3\theta$$x=3\cos \theta -2{\cos }^3\theta ,y=3\sin \theta -2{\sin }^3\theta$x=3cos theta-2cos^(3)theta,y=3sin theta-2sin^(3)theta	(b)	$\frac{-1}{3\sqrt{3}}$$\frac{-1}{3\sqrt{3}}$(-1)/(3sqrt3)
(iii)	$x=3\cos \theta -{\cos }^3\theta ,y=3\sin \theta -{\sin }^3\theta$$x=3\cos \theta -{\cos }^3\theta ,y=3\sin \theta -{\sin }^3\theta$x=3cos theta-cos^(3)theta,y=3sin theta-sin^(3)theta	(c)	$\sqrt{3}$$\sqrt{3}$sqrt3
(iv)	$x=a\log \sin \theta ,y=a\tan \theta$$x=a\log \sin \theta ,y=a\tan \theta$x=a log sin theta,y=a tan theta	(d)	$\frac{1}{\sqrt{3}}$$\frac{1}{\sqrt{3}}$(1)/(sqrt3)
		(e)	$\frac{1}{3\sqrt{3}}$$\frac{1}{3\sqrt{3}}$(1)/(3sqrt3)

A

(i) → c, (ii) → d, (iii) → b, (iv) → a

B

(i) → c, (ii) → e, (iii) → d, (iv) → a

C

(i) → d, (ii) → c, (iii) → b, (iv) → a

D

(i) →d, (ii) → c, (iii) → e, (iv) → b

Question 31: Palette 31/74

TS-EAMCET 2023 MCQ

If $y=x \sin x$ and $\frac{\frac{d y}{d x}-\frac{y}{x}}{x \frac{d y}{d x}-y}$ at $x=\alpha$ is 1 , then $\alpha=$

A

$\sqrt{2}$

B

2

C

1

D

$1 / \sqrt{2}$

Question 32: Palette 32/74

TS-EAMCET 2023 MCQ

On differentiation if we get $f(x, y) d y-g(x, y) d x=0$ from $2 x^2-3 x y+y^2+x+2 y-8=0$, then $\frac{g(2,2)}{f(1,1)}=$

A

$11 / 7$

B

-3

C

$-1 / 3$

D

7

Question 33: Palette 33/74

TS-EAMCET 2023 MCQ

If $f(x)=e^x, h(x)=(f \circ f)(x)$, then $\frac{h^{\prime}(x)}{h(x)}=$

A

$h(x)$

B

$\frac{1}{h(x)}$

C

$\log h(x)$

D

$-\log h(x)$

Question 34: Palette 34/74

TS-EAMCET 2023 MCQ

If $\sin y=\sin 3 t$ and $x=\sin t$, then $\frac{d y}{d x}=$

A

$\frac{3}{\sqrt{4-x^2}}$

B

$\frac{3}{\sqrt{1-x^2}}$

C

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

D

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

Question 35: Palette 35/74

TS-EAMCET 2023 MCQ

If $f(x)=\sqrt{\log \left(x^2+x+1\right)+\sqrt{\cosh (2 x-3)}}$, then $f^{\prime}(0)=$

A

$\frac{1}{2 \sqrt{\sqrt{\cosh (3)}}}\left(1+\frac{\sinh (3)}{\sqrt{\cosh (3)}}\right)$

B

$\frac{1}{2 \sqrt{\sqrt{\cosh (3)}}}\left(\log 3-\frac{\sinh (3)}{\sqrt{\cosh (3)}}\right)$

C

$\frac{\log 3 \sqrt{\cosh (3)}-\sinh (3)}{2(\cosh (3))^{\frac{3}{4}}}$

D

$\frac{\sqrt{\cosh (3)}-\sinh (3)}{2(\cosh (3))^{\frac{3}{4}}}$

Question 36: Palette 36/74

TS-EAMCET 2023 MCQ

65. If $x=\cos ^3 \theta-\sin ^3 \theta$ and $y=\sqrt[3]{\cos \theta}-\sqrt[3]{\sin \theta}$, then the value of $\frac{d y}{d x}$ at $\theta=\frac{\pi}{4}$ is

A

$\frac{2}{9} \sqrt[3]{2}$

B

$\frac{\sqrt[3]{2}}{3}$

C

$\frac{4}{9} \sqrt[3]{2}$

D

$\frac{\sqrt[3]{2}}{9}$

Question 37: Palette 37/74

TS-EAMCET 2023 MCQ

If $2 x^2+3 x y-y^2+4 x-5 y+6=0$, then the value of $\frac{d y}{d x}$ at $(x, y)=(1,-2)$ is

A

1

B

-1

C

$\frac{7}{2}$

D

0

Question 38: Palette 38/74

TS-EAMCET 2023 MCQ

If $f(x)=|x-1|+|x-2|$, then

$ f^{\prime}(-2023)+f^{\prime}\left(\frac{2024}{2023}\right)+f^{\prime}(2023)= $

A

1

B

-1

C

0

D

3

Question 39: Palette 39/74

TS-EAMCET 2023 MCQ

If $f(x)=\frac{e^{2 x}-e^{-2 x}}{e^{3 x}+e^{-3 x}}$, then $f^{\prime}(0)=$

A

-1

B

0

C

1

D

2

Question 40: Palette 40/74

TS-EAMCET 2023 MCQ

If $f(x)=x^{\tan x}+(\tan x)^x$, then $f^{\prime}\left(\frac{\pi}{4}\right)=$

A

$1+\frac{\pi}{2} \log \left(\frac{e \pi}{4}\right)$

B

$\frac{\pi}{2}\left(\log \frac{\pi}{4}+1\right)$

C

1

D

0

Question 41: Palette 41/74

TS-EAMCET 2023 MCQ

If $\sec \left(\log _2 y^2\right)=\operatorname{cosec}\left(\log _2 x^2\right)$, then $\frac{d y}{d x}=$

A

$\frac{x}{y}$

B

$\frac{y}{x}$

C

$-\frac{y}{x}$

D

$-\frac{x}{y}$

Question 42: Palette 42/74

TS-EAMCET 2023 MCQ

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

A

$\sinh x$

B

$\cosh x$

C

$\tanh x$

D

$-\sinh x$

Question 43: Palette 43/74

TS-EAMCET 2023 MCQ

If $x=\log p$ and $y=\frac{1}{p}$, then $\frac{d y}{d x}=$

A

$-e^{-x}$

B

$e^x$

C

$x$

D

$y$

Question 44: Palette 44/74

TS-EAMCET 2023 MCQ

If $\tan y=\cot \left(\frac{\pi}{4}-x\right)$, then $\frac{d y}{d x}=$

A

$\frac{\operatorname{cosec}^2\left(\frac{\pi}{4}-x\right)}{1+\cot ^2\left(\frac{\pi}{4}+x\right)}$

B

$\frac{-\operatorname{cosec}^2\left(\frac{\pi}{4}-x\right)}{\sec ^2 y}$

C

$\frac{\operatorname{cosec}^2\left(\frac{\pi}{4}-x\right)}{1+\tan ^2\left(\frac{\pi}{4}+x\right)}$

D

$\frac{\sec ^2\left(\frac{\pi}{4}+x\right)}{1+\tan ^2\left(\frac{\pi}{4}+x\right)}$

Question 45: Palette 45/74

TS-EAMCET 2023 MCQ

If $x=3 \sqrt{2} \cos ^3 \theta$ and $y=4 \tan ^2 \theta$, then $\left(\frac{d y}{d x}\right)_{\theta=\pi / 4}=$

A

$\frac{32 \sqrt{2}}{9}$

B

$\frac{16}{9}$

C

$-\frac{16}{9}$

D

$-\frac{32}{9}$

Question 46: Palette 46/74

TS-EAMCET 2023 MCQ

The derivative of $\frac{1-x^2}{1+x^2}$ with respect to $\frac{2 x}{1+x^2}$ at $x=2$ is

A

0

B

$\frac{4}{3}$

C

1

D

$-\frac{4}{3}$

Question 47: Palette 47/74

TS-EAMCET 2023 MCQ

If the slope of the tangent drawn to the curve $y=e^{a+b x^2}$ at the point $P(1,1)$ is -2 , then the value of $2 a-3 b$ is

A

5

B

6

C

7

D

8

Question 48: Palette 48/74

TS-EAMCET 2022 MCQ

If $f(x)=\sum_{p=1}^7 p^2 \sin ^{-1}\left(\frac{4}{5} \sin (p x)-\frac{3}{5} \cos (p x)\right)$, then the value of $\frac{d f}{d x}$ at $x=1$ is [given that $\sin ^{-1}(\sin x)=x$ ])

A

0

B

628

C

1140

D

784

Question 49: Palette 49/74

TS-EAMCET 2022 MCQ

If $y=\frac{a x+b}{c x+d}$, then $\frac{d x}{d y}=$

A

$\frac{a d-b c}{(a x+b)^2}$

B

$\frac{a d-b c}{(a-c y)^2}$

C

$\frac{a d+b c}{(c x+d)^2}$

D

$\frac{a d+b c}{(a+c y)^2}$

Question 50: Palette 50/74

TS-EAMCET 2022 MCQ

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

A

$\frac{x}{y}$

B

$\frac{-x}{y}$

C

$\frac{y}{x}$

D

$\frac{-y}{x}$

Question 51: Palette 51/74

TS-EAMCET 2022 MCQ

If $f(x)=\frac{e^{-x} \sin x}{\log _e x}$ and $f^{\prime}(x)=f(x) \cdot g(x)$, then $g^{\prime}(e)=$

A

$e^{-2}-\operatorname{cosec}^2(e)$

B

$2 e^2-\operatorname{cosec}^2(e)$

C

$2 e^{-2}-\operatorname{cosec}^2(e)$

D

$2 e^{-2}+\operatorname{cosec}^2(e)$

Question 52: Palette 52/74

TS-EAMCET 2022 MCQ

If $y=\frac{e^{\sin x}+\sinh ^3 x}{\cosh x-\tan x}$, then $y^{\prime}(0)=$

A

0

B

1

C

-1

D

2

Question 53: Palette 53/74

TS-EAMCET 2022 MCQ

If $\frac{d}{d x}\left(\frac{2 x+1}{(x+1)^2(x-2)}\right)=\frac{A}{(x-2)^2}+\frac{B}{(x+1)^3}+\frac{C}{(x+1)^2}$, then $A+B+C=$

A

$\frac{-2}{3}$

B

$\frac{2}{3}$

C

$\frac{1}{3}$

D

$\frac{-1}{3}$

Question 54: Palette 54/74

TS-EAMCET 2022 MCQ

$ \frac{d}{d x}\left[\left(x^{\frac{5}{2}}-x^{\frac{3}{2}}+1\right)\left(x^2-3 x+5\right)\right]= $

A

$\frac{9}{2} x^{7 / 2}-14 x^{5 / 2}+20 x^{3 / 2}-\frac{15}{2} x^{1 / 2}+2 x-3$

B

$\frac{9}{2} x^{7 / 2}-7 x^{5 / 2}+5 x^{3 / 2}-\frac{3}{2} x^{1 / 2}+2 x-3$

C

$9 x^{7 / 2}-14 x^{5 / 2}+20 x^{3 / 2}-15 x^{1 / 2}+2 x-3$

D

$\frac{9}{2} x^{7 / 2}-\frac{7}{2} x^{5 / 2}+\frac{5}{2} x^{3 / 2}-\frac{15}{2} x^{1 / 2}+2 x-3$

Question 55: Palette 55/74

TS-EAMCET 2022 MCQ

The value of $\frac{d}{d x}\left[\log \left(\sin \sqrt{\frac{x^2+1}{x^2+2}}\right)\right]$ when $x=\sqrt{2}$, is

A

$\frac{\sqrt{2} \cot \left(\frac{\sqrt{3}}{2}\right)}{6 \sqrt{3}}$

B

$\frac{\sqrt{2} \tan \left(\frac{\sqrt{3}}{2}\right)}{6 \sqrt{3}}$

C

$\frac{\sqrt{2} \cot \left(\frac{\sqrt{3}}{2}\right)}{8 \sqrt{3}}$

D

$\frac{\sqrt{2} \tan \left(\frac{\sqrt{3}}{2}\right)}{8 \sqrt{3}}$

Question 56: Palette 56/74

TS-EAMCET 2022 MCQ

If $f(x)=\frac{1+\sec x}{2(\sec x-1)}$ for $0

A

$\operatorname{cosec} x$

B

$-\operatorname{cosec} x$

C

$2 \operatorname{cosec} x$

D

$-2 \operatorname{cosec} x$

Question 57: Palette 57/74

TS-EAMCET 2022 MCQ

If $\frac{3 x+5}{(x+1)\left(2 x^2+3\right)}=\frac{A}{x+1}+\frac{B x+C}{2 x^2+3}$ and $f(x)=A x^3+B x^2+7 x+C$, then $5 C-f^{\prime}(-2)=$

A

19

B

15

C

4

D

34

Question 58: Palette 58/74

TS-EAMCET 2022 MCQ

Let $f(x)=\sin x, g(x)=\cos x, h(x)=x^2$, then $\lim _{x \rightarrow 1} \frac{f(g(h(x)))-f(g(h(1)))}{x-1}=$

A

0

B

$-2 \sin 1 \cos (\cos 1)$

C

$\infty$

D

$-2 \sin 1 \cos 1$

Question 59: Palette 59/74

TS-EAMCET 2022 MCQ

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

A

$\sin k$

B

$\cos k$

C

1

D

0

Question 60: Palette 60/74

TS-EAMCET 2022 MCQ

If $x=a(\cos \theta+\theta \sin \theta), y=f(\theta), f(2 \pi)=0$, $\frac{d y}{d x}=\frac{\tan \theta}{\theta}, \theta \neq 0$ and $\theta \neq(2 n+1) \frac{\pi}{2}$, then $f\left(\frac{\pi}{3}\right)=$

A

$2 a \pi$

B

$\frac{\pi}{2} a$

C

$\frac{a}{2}$

D

$-2 a$

Question 61: Palette 61/74

TS-EAMCET 2022 MCQ

If $a f(x)+b f\left(\frac{1}{x}\right)=x+1$, and $\frac{d}{d x}\left(x^2 f(x)\right)=2 x^2+2 x+\frac{1}{3}$, then $a-b$

A

2

B

3

C

0

D

1

Question 62: Palette 62/74

TS-EAMCET 2022 MCQ

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

A

$\frac{2}{9} \sinh \left(\frac{2}{3}\right) \cos \left(\cosh \left(\frac{2}{3}\right)\right)$

B

$\sinh \left(\frac{2}{3}\right) \cos \left(\cosh \left(\frac{2}{3}\right)\right)$

C

$\frac{2}{9} \cos \left(\cosh \left(\frac{2}{3}\right)\right)$

D

$\frac{2}{9} \cosh \left(\frac{2}{3}\right) \cos \left(\sinh \left(\frac{2}{3}\right)\right)$

Question 63: Palette 63/74

TS-EAMCET 2022 MCQ

If $f(x)=\log _e\left(e^{2 x}\left(\frac{3 x+5}{5-3 x}\right)^{2 / 3}\right), x \neq \frac{-5}{3}, \frac{5}{3}$, then the value of $\frac{d f}{d x}$ at $x=1$, is

A

$\frac{5}{4}$

B

$\frac{7}{4}$

C

$\frac{11}{4}$

D

$\frac{13}{4}$

Question 64: Palette 64/74

TS-EAMCET 2022 MCQ

If $x=\operatorname{cosec} \theta-\sin \theta, y=\operatorname{cosec}^{2022} \theta-\sin ^{2022} \theta$ and $\left(\frac{d y}{d x}\right)^2=\frac{k\left(y^2+4\right)}{g(x)}$ where $k \in R$, then $10+k-g(2022)=$

A

0

B

6

C

10

D

14

Question 65: Palette 65/74

TS-EAMCET 2020 MCQ

Match the functions of List-I with derivates given in List-II

$ \text { List-I } $		$ \text { List-II } $
A.	$ \sec ^{-1} x $	I.	$ \frac{1}{1-x^2}, x \in(-1,1) $
B.	$ \tanh ^{-1} x $	II.	$ \frac{-1}{|x| \sqrt{x^2+1}}, x \neq 0 $
C.	$ \operatorname{coth}^{-1} x $	III.	$ \frac{1}{|x| \sqrt{x^2-1}},|x|>1 $
D.	$ \operatorname{cosech}^{-1} x $	IV.	$ \frac{1}{1-x^2}, x \in \mathbf{R}-[-1,1] $
		V.	$ \frac{-1}{|x| \sqrt{1-x^2}},|x|<1, x \neq 0 $

A

A	B	C	D
V	II	I	III

B

A	B	C	D
I	III	V	II

C

A	B	C	D
III	I	II	V

D

A	B	C	D
III	I	IV	II

Question 66: Palette 66/74

TS-EAMCET 2020 MCQ

If $f(x)=\frac{x-1}{e^x}$, then $f^{\prime}(0)+f^{\prime \prime}(0)=$

A

0

B

1

C

-1

D

2

Question 67: Palette 67/74

TS-EAMCET 2020 MCQ

$ \begin{aligned} & \text { If }\left(\frac{d y}{d x}\right)=\frac{1}{\left(\frac{d x}{d y}\right)} \text { and } \frac{d^2 x}{d y^2}\left(\frac{d y}{d x}\right)^3+\frac{d^2 y}{d x^2}=k \text {, then } \\ & e^{k f(x)}-k f(x)= \end{aligned} $

A

1

B

0

C

$1 / 2$

D

2

Question 68: Palette 68/74

TS-EAMCET 2020 MCQ

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

A

$2|\sec 2 x|$

B

$\cos 2 x$

C

$-2|\operatorname{cosec} 2 x|$

D

$\sin 2 x$

Question 69: Palette 69/74

TS-EAMCET 2020 MCQ

Let $f: R \rightarrow R$ be defined by $f\left(\frac{x+y}{2}\right)=\frac{f(x)+f(y)}{2}$ for all $x$ and $y$. If $f^{\prime}(0)$ exists and equals -1 and $f(0)=1$, then $f(2)=$

A

-1

B

0

C

$1 / 2$

D

1

Question 70: Palette 70/74

TS-EAMCET 2020 MCQ

$ \begin{aligned} & \text { If } f(x)=\tan ^{-1}\left(\frac{1}{\sin ^2 x+\sin x+1}\right) \\ & \quad+\tan ^{-1}\left(\frac{1}{\sin ^2 x+3 \sin x+3}\right)+\tan ^{-1} \end{aligned} $

$\left(\frac{1}{\sin ^2 x+5 \sin x+7}\right)+\ldots+$ upto 10 terms, then $f^{\prime}(0)=$

A

$\frac{-1}{101}$

B

$\frac{100}{101}$

C

$\frac{-100}{101}$

D

0

Question 71: Palette 71/74

TS-EAMCET 2020 MCQ

If $\alpha$ is such a minimum value for which the inverse of $f(x)=x^2+3 x-3$ exists in $[\alpha, \infty)$ and $g$ is the inverse of the $f$, then at $x=\alpha+\frac{5}{2}, \frac{d g}{d x}$

A

$\frac{1}{2}$

B

$\frac{1}{3}$

C

$\frac{1}{4}$

D

$\frac{1}{5}$

Question 72: Palette 72/74

TS-EAMCET 2020 MCQ

let $g(x) \neq 0, g^{\prime}(x) \neq 0, f(x) \neq 0, f^{\prime}(x) \neq 0$. If

$F(x)=f(x) g(x), G(x)=f^{\prime}(x) g^{\prime}(x)$ and

$F^{\prime}(x)=G(x) H(x)=F(x) K(x)$, then $H(x)+K(x)=$

A

$\frac{f^{\prime}}{f}+\frac{f}{f^{\prime}}+\frac{g}{g^{\prime}}$

B

$\frac{f^{\prime}}{f}+\frac{g}{g^{\prime}}+\frac{g^{\prime}}{g}$

C

$\frac{f^{\prime} g^{\prime}+f g}{f f^{\prime} g g^{\prime}}$

D

$\frac{f^{\prime}}{f}+\frac{g}{g^{\prime}}+\frac{f}{f^{\prime}}+\frac{g^{\prime}}{g}$

Question 73: Palette 73/74

TS-EAMCET 2020 MCQ

If $y=\frac{x \sin ^{-1} x}{\sqrt{1-x^2}}+\log \sqrt{1-x^2}$, then $\frac{d y}{d x}=$

A

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

B

$\frac{\sin ^{-1} x}{\left(1-x^2\right)^{3 / 2}}$

C

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

D

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

Question 74: Palette 74/74

TS-EAMCET 2020 MCQ

Let $f(x)$ and $g(x)$ be twice differentiable functions such that $f(x)=x^2+g^{\prime}(1) x+g^{\prime \prime}(2)$ and $g(x)=f(1) x^2+x f^{\prime}(x)+f^{\prime \prime}(x)$. Then $f(x)-g(x)=$

A

$2 x+5$

B

$3 x^2+6 x+1$

C

$x^2-6 x+2$

D

$x^2-2$