Question 1: Palette 1/50

AP-EAPCET 2025 MCQ

If $\theta=\tan ^{-1}\left(\frac{1}{3}\right)+\tan ^{-1}\left(\frac{1}{7}\right)+\tan ^{-1}\left(\frac{1}{13}\right) +\tan ^{-1}\left(\frac{1}{21}\right)+\tan ^{-1}\left(\frac{1}{31}\right)$, then $\tan \theta=$

A

$\frac{3}{5}$

B

1

C

$\frac{5}{7}$

D

$\frac{7}{9}$

Question 2: Palette 2/50

AP-EAPCET 2025 MCQ

If $\tan ^{-1} x=\cot h^{-1} y=\log \sqrt{5}$, then $\tan ^{-1}(x y)=$

A

$\frac{\pi}{4}$

B

$\frac{\pi}{3}$

C

$\frac{\pi}{6}$

D

$\frac{3 \pi}{4}$

Question 3: Palette 3/50

AP-EAPCET 2025 MCQ

If $f(x)=2+\left|\sin ^{-1} x\right|$ and $A=\left\{x \in R / f^1(x)\right.$ exists $\}$, then $A=$

A

$\{0\}$

B

$[-1,1]$

C

$(-\infty,-1) \cup(1, \infty)$

D

$(-1,0) \cup(0,1)$

Question 4: Palette 4/50

AP-EAPCET 2025 MCQ

The equation $\cos ^{-1}(1-x)-2 \cos ^{-1} x=\frac{\pi}{2}$ has

A

no solution

B

only one solution

C

two solutions

D

more than two solutions

Question 5: Palette 5/50

AP-EAPCET 2025 MCQ

$ \tan \left(2 \tan ^{-1}\left(\frac{1}{3}\right)+\tan ^{-1}\left(\frac{1}{7}\right)\right)= $

A

$\frac{1}{\sqrt{3}}$

B

$\sqrt{3}$

C

1

D

$3 / 7$

Question 6: Palette 6/50

AP-EAPCET 2025 MCQ

$ \tanh ^{-1}\left(\frac{1}{3}\right)+\operatorname{coth}^{-1}(3)= $

A

$\operatorname{sech}^{-1}\left(\frac{1}{3}\right)$

B

$\operatorname{cosech}^{-1}\left(\frac{1}{3}\right)$

C

$\cosh ^{-1}\left(\frac{4}{3}\right)$

D

$\sinh ^{-1}\left(\frac{3}{4}\right)$

Question 7: Palette 7/50

AP-EAPCET 2025 MCQ

If $y=\sin ^{-1}\left(\frac{2 x}{1+x^2}\right)$ and $\left(\frac{d^2 y}{d x^2}\right)_{x=2}=k$, then $25 k=$

A

$(-3)^2$

B

$(-2)^3$

C

3

D

$(-2)^5$

Question 8: Palette 8/50

AP-EAPCET 2025 MCQ

If $f(x)=\sec ^{-1}\left(\frac{1}{2 x^2-1}\right)$ and $g(x)=\tan ^{-1}\left(\frac{\sqrt{1+x^2}-1}{x}\right)$, then the derivative of $f(x)$ with respect to $g(x)$ is

A

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

B

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

C

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

D

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

Question 9: Palette 9/50

AP-EAPCET 2025 MCQ

If $A=\left\{x \in R / \sin ^{-1}\left(\sqrt{x^2+x+1}\right) \in\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)\right\}$ and $B=\left\{y \in R / y=\sin ^{-1}\left(\sqrt{x^2+x+1}\right), x \in A\right\}$, then

A

$A \cap B \neq \phi$

B

$A \cap B^C=[0,1]$

C

$A^C \cap B=\left[\frac{\pi}{3}, \frac{\pi}{2}\right]$

D

$A \cup B=R-\left\{[-1,0] \cup\left[\frac{\pi}{3}, \frac{\pi}{2}\right]\right\}$

Question 10: Palette 10/50

AP-EAPCET 2025 MCQ

The domain of the function, $f(x)=\sqrt{\log _e\left(\frac{1}{x^2-4 x+4}\right)}+\sin ^{-1}\left(x^2-2\right)$ is

A

$[1,3]$

B

$[1,3)$

C

$[1, \sqrt{3}]$

D

$[1, \sqrt{3})$

Question 11: Palette 11/50

AP-EAPCET 2025 MCQ

If $\cot \left(\cos ^{-1} x\right)=\sec \left\{\tan ^{-1}\left(\frac{a}{\sqrt{b^2-a^2}}\right)\right\}, b>a$ then $x=$

A

$\frac{b}{\sqrt{2 b^2-a^2}}$

B

$\frac{a}{\sqrt{2 b^2-a^2}}$

C

$\frac{\sqrt{b^2-a^2}}{a}$

D

$\frac{\sqrt{b^2-a^2}}{b}$

Question 12: Palette 12/50

AP-EAPCET 2025 MCQ

If $\sinh ^{-1} x=\log 3$ and $\cosh ^{-1} y=\log \frac{3}{2}$, then $\tanh ^{-1}(x-y)=$

A

$\log \sqrt{\frac{5}{3}}$

B

$\log \frac{5}{3}$

C

$\log \frac{4}{3}$

D

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

Question 13: Palette 13/50

AP-EAPCET 2025 MCQ

The number of solution of $\tan ^{-1} 1+\frac{1}{2} \cos ^{-1} x^2-\tan ^{-1} \left(\frac{\sqrt{1+x^2}+\sqrt{1-x^2}}{\sqrt{1+x^2}-\sqrt{1-x^2}}\right)=0$ is

A

3

B

0

C

1

D

infinitely many

Question 14: Palette 14/50

AP-EAPCET 2025 MCQ

$ \tanh ^{-1}(\sin \theta)= $

A

$\sinh ^{-1}(\operatorname{cosec} \theta)$

B

$\sinh ^{-1}(\sec \theta)$

C

$\cosh ^{-1}(\operatorname{cosec} \theta)$

D

$\cosh ^{-1}(\sec \theta)$

Question 15: Palette 15/50

AP-EAPCET 2025 MCQ

The interval in which the function $f(x)=\tan ^{-1}(\sin x+\cos x)$ is an increasing function is

A

$\left(0, \frac{\pi}{2}\right)$

B

$\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$

C

$\left(-\frac{3 \pi}{4}, \frac{\pi}{4}\right)$

D

$\left(\frac{\pi}{4}, \frac{\pi}{2}\right)$

Question 16: Palette 16/50

AP-EAPCET 2025 MCQ

The range of the real valued function $f(x)=\cos ^{-1}\left(\frac{3}{\sqrt{9 x^2-12 x+22}}\right)$ is

A

$\left(0, \frac{\pi}{4}\right]$

B

$\left[\frac{\pi}{4}, \frac{\pi}{2}\right)$

C

$[0, \pi]$

D

$\left[\frac{\pi}{6}, \frac{\pi}{2}\right)$

Question 17: Palette 17/50

AP-EAPCET 2025 MCQ

If the equation $2 \cot ^{-1}\left(x^2+2 x+k\right)=\pi-3 \tan ^{-1} \left(x^2+2 x+k\right)$ has two distinct real solutions, then all the values of $k$ lie in the interval

A

$(-1,2)$

B

$(1, \infty)$

C

$(-\infty, \infty)$

D

$(-\infty, 1)$

Question 18: Palette 18/50

AP-EAPCET 2025 MCQ

$ \sec h^{-1}(\sin \alpha)= $

A

$\log \left(\sin \alpha+\sqrt{\sin ^2 \alpha-1}\right)$

B

$\log (\tan \alpha+1)$

C

$\log \left(\cot \frac{\alpha}{2}\right)$

D

$\log \left(\frac{1+\tan \alpha}{2 \sin \alpha}\right)$

Question 19: Palette 19/50

AP-EAPCET 2025 MCQ

If $y=\log \left(\sec \left(\tan ^{-1} x\right)\right)(x>0)$, then $\frac{d y}{d x}$ at $x=1$ is

A

1

B

3

C

$\frac{1}{2}$

D

$\frac{3}{2}$

Question 20: Palette 20/50

AP-EAPCET 2025 MCQ

If $y=\sin ^{-1} \frac{\sqrt{1+\sin x}+\sqrt{1-\sin x}}{\sqrt{1+\sin x}-\sqrt{1-\sin x}}$ and $\frac{-3 \pi}{2}

A

$-\frac{\left|\operatorname{cosec} \frac{x}{2}\right|}{2 \sqrt{\sin ^2 \frac{x}{2}-\cos ^2 \frac{x}{2}}}$

B

$\frac{\left|\sec \frac{x}{2}\right|}{2 \sqrt{\cos x}}$

C

$\frac{\cos \frac{x}{2}}{2 \sqrt{\cos x}}$

D

$\frac{\cos \frac{x}{2}}{\sqrt{\cos x}}$

Question 21: Palette 21/50

AP-EAPCET 2025 MCQ

If $\frac{1}{2} \sin ^{-1}\left(\frac{3 \sin 2 \theta}{5+4 \cos 2 \theta}\right)=\tan ^{-1} x$, then $x=$

A

$\tan \frac{\theta}{3}$

B

$\frac{1}{3} \tan \theta$

C

$\tan 3 \theta$

D

$\frac{1}{3} \tan 3 \theta$

Question 22: Palette 22/50

AP-EAPCET 2025 MCQ

If $\operatorname{sech}^{-1} x=\log 2$ and $\operatorname{cosech}^{-1} y=-\log 3$, then $(x+y)=$

A

$\frac{1}{6}$

B

$\frac{1}{20}$

C

6

D

20

Question 23: Palette 23/50

AP-EAPCET 2025 MCQ

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

A

$\frac{4}{16 x^2+1}-\frac{3}{9 x^2+1}$

B

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

C

$\frac{3}{9 x^2+1}-\frac{2}{4 x^2+1}$

D

$\frac{1}{9 x^2+1}-\frac{1}{x^2+1}$

Question 24: Palette 24/50

AP-EAPCET 2025 MCQ

The range of the real valued function $f(x)=\cos ^{-1}(-x)+\sin ^{-1}(-x)+\operatorname{cosec}^{-1}(x)$ is

A

$\left\{0, \frac{\pi}{2}\right\}$

B

$\left[0, \frac{\pi}{2}\right) \cup\left(\frac{\pi}{2}, \pi\right]$

C

$\left(0, \frac{\pi}{2}\right)$

D

$\{0, \pi\}$

Question 25: Palette 25/50

AP-EAPCET 2025 MCQ

The horizontal distance between a tower and a building is $10 \sqrt{3}$ units. If the angle of depression of the foot of the building from the top of the tower is $60^{\circ}$ and the angle of elevation of the top of the building from the foot of the tower is $30^{\circ}$, then the sum of the heights of the tower and the building is

A

60

B

50

C

40

D

30

Question 26: Palette 26/50

AP-EAPCET 2025 MCQ

If $x$ is a real number, then the number of solutions of $\tan ^{-1}(\sqrt{x(x+1)})+\sin ^{-1}\left(\sqrt{x^2+x+1}\right)=\frac{\pi}{2}$ is

A

1

B

2

C

3

D

4

Question 27: Palette 27/50

AP-EAPCET 2025 MCQ

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

A

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

B

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

C

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

D

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

Question 28: Palette 28/50

AP-EAPCET 2025 MCQ

$ \tan ^{-1} \frac{\sqrt{8-2 \sqrt{15}}}{\sqrt{15}+1}+\tan ^{-1} \frac{1}{\sqrt{5}}= $

A

$\frac{\pi}{6}$

B

$\frac{\pi}{4}$

C

$\frac{\pi}{3}$

D

$\frac{\pi}{2}$

Question 29: Palette 29/50

AP-EAPCET 2025 MCQ

The derivative of $\sec ^{-1}\left(\frac{1}{2 x^2-1}\right)$ with respect to $\sqrt{1-x^2}$ at $x=\frac{1}{2}$ is

A

-2

B

1

C

2

D

4

Question 30: Palette 30/50

AP-EAPCET 2024 MCQ

If $0 < x < \frac{1}{2}$ and $\alpha=\sin ^{-1} x+\cos ^{-1}\left(\frac{x}{2}+\frac{\sqrt{3-3 x^2}}{2}\right)$, then $\tan \alpha+\cot \alpha$ is equal to

A

$\frac{4}{\sqrt{3}}$

B

$4 \sqrt{3}$

C

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

D

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

Question 31: Palette 31/50

AP-EAPCET 2024 MCQ

$\cot \left(\sum\limits_{n=1}^{50} \tan ^{-1}\left(\frac{1}{1+n+n^2}\right)\right)$ is equal to

A

$\frac{26}{25}$

B

$\frac{25}{26}$

C

$\frac{50}{51}$

D

$\frac{52}{51}$

Question 32: Palette 32/50

AP-EAPCET 2024 MCQ

The value of $x$ such that $\sin \left(2 \tan ^{-1} \frac{3}{4}\right)=\cos \left(2 \tan ^{-1} x)\right.$

A

7

B

$\frac{3}{7}$

C

$\frac{1}{7}$

D

$\frac{4}{7}$

Question 33: Palette 33/50

AP-EAPCET 2024 MCQ

The range of the real valued function $f(x)=\sin ^{-1}\left(\frac{1+x^2}{2 x}\right)+\cos ^{-1}\left(\frac{2 x}{1+x^2}\right)$ is

A

$\left\{\frac{\pi}{2}\right\}$

B

$R$

C

$Q$

D

$\left\{-\frac{\pi}{2}, \frac{\pi}{2}\right\}$

Question 34: Palette 34/50

AP-EAPCET 2024 MCQ

The real values of $x$ that satisfy the equation $\tan ^{-1} x+\tan ^{-1} 2 x=\frac{\pi}{4}$ is

A

$\frac{-3 \pm \sqrt{17}}{4}$

B

$-1 \pm \sqrt{3}$

C

$\sqrt{3}-1$

D

$\frac{\sqrt{17}-3}{4}$

Question 35: Palette 35/50

AP-EAPCET 2024 MCQ

$2 \operatorname{coth}^{-1}(4)+\sec h^{-1}\left(\frac{3}{5}\right)=$

A

$\log 5$

B

$2 \log 3$

C

$3 \log 2$

D

$\log \frac{5}{3}$

Question 36: Palette 36/50

AP-EAPCET 2024 MCQ

If $y=\sin ^{-1} x$, then $\left(1-x^2\right) y_2-x y_1=$

A

0

B

1

C

2

D

$2 y$

Question 37: Palette 37/50

AP-EAPCET 2024 MCQ

If $\cos ^{-1} 2 x+\cos ^{-1} 3 x=\frac{\pi}{3}$ and $4 x^2=\frac{a}{b}$, then $a+b$ is equal to

A

12

B

11

C

31

D

10

Question 38: Palette 38/50

AP-EAPCET 2024 MCQ

If $\theta=\sec ^{-1}(\cosh u)$, then $u=$

A

$\log _e\left(\cot \left(\frac{\theta}{2}-\frac{\pi}{4}\right)\right)$

B

$\log _e\left(\tan \left(\frac{\theta}{2}-\frac{\pi}{4}\right)\right)$

C

$\log _\theta\left(\tan \left(\frac{\pi}{4}-\frac{\theta}{2}\right)\right)$

D

$\log _e\left(\tan \left(\frac{\pi}{4}+\frac{\theta}{2}\right)\right)$

Question 39: Palette 39/50

AP-EAPCET 2024 MCQ

If $\cos ^4 \frac{\pi}{8}+\cos ^4 \frac{3 \pi}{8}+\cos ^4 \frac{5 \pi}{8}+\cos ^4 \frac{7 \pi}{8}=k$, then $\sin ^{-1}\left(\sqrt{\frac{k}{2}}\right)+\cos ^{-1}\left(\frac{k}{3}\right)=$

A

$\frac{2 \pi}{3}$

B

$\frac{3 \pi}{4}$

C

$\frac{\pi}{4}$

D

$\frac{\pi}{2}$

Question 40: Palette 40/50

AP-EAPCET 2024 MCQ

$4 \tan ^{-1} \frac{1}{5}-\tan ^{-1} \frac{1}{70}+\tan ^{-1} \frac{1}{99}=$

A

$\frac{\pi}{12}$

B

$\frac{\pi}{6}$

C

$\frac{\pi}{4}$

D

$\frac{\pi}{3}$

Question 41: Palette 41/50

AP-EAPCET 2024 MCQ

$ \cosh \left(\sinh ^{-1}(\sqrt{8})+\cosh ^{-1} 5\right)= $

A

$\sqrt{6}+4 \sqrt{2}$

B

$15+8 \sqrt{3}$

C

$6 \sqrt{6}+10 \sqrt{2}$

D

$8-15 \sqrt{3}$

Question 42: Palette 42/50

AP-EAPCET 2024 MCQ

$\tan^{-1} 2 + \tan^{-1} 3 = $

A

$\frac{\pi}{4}$

B

$\frac{\pi}{2}$

C

$\frac{3\pi}{4}$

D

$\frac{5\pi}{4}$

Question 43: Palette 43/50

AP-EAPCET 2021 MCQ

$\tan ^{-1}(-2)-\tan ^{-1}(3)$ is equal to

A

$\frac{3 \pi}{4}$

B

$\frac{-\pi}{6}$

C

$\frac{\pi}{6}$

D

$\frac{-3 \pi}{4}$

Question 44: Palette 44/50

AP-EAPCET 2021 MCQ

If $x=\sin \left(2 \tan ^{-1} 2\right), y=\cos \left(2 \tan ^{-1} 3\right)$ and $z=\sec \left(3 \tan ^{-1} 4\right)$, then

A

$x < y < z$

B

$y < z < x$

C

$z < x < y$

D

$z < y < x$

Question 45: Palette 45/50

AP-EAPCET 2021 MCQ

$\frac{d}{d x}\left\{\sin ^2\left(\cot ^{-1} \sqrt{\frac{1+x}{1-x}}\right)\right\}$ is equal to

A

0

B

$\frac{1}{2}$

C

$\frac{-1}{2}$

D

$-1$

Question 46: Palette 46/50

AP-EAPCET 2021 MCQ

If $y=\tan ^{-1}\left\{\frac{a x-b}{b x+a}\right\}$, then $y^{\prime}$ is equal to

A

$\frac{1}{1+x^2}+\frac{a^2}{a^2+b^2}$

B

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

C

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

D

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

Question 47: Palette 47/50

AP-EAPCET 2021 MCQ

For how many distinct values of $x$, the following $\sin \left[2 \cos ^{-1} \cot \left(2 \tan ^{-1} x\right)\right]=0$ holds?

A

8

B

2

C

6

D

4

Question 48: Palette 48/50

AP-EAPCET 2021 MCQ

If $\tan ^{-1}\left[\frac{1}{1+1 \cdot 2}\right]+\tan ^{-1}\left[\frac{1}{1+2 \cdot 3}\right]+\ldots+\tan ^{-1} \left[\frac{1}{1+n(1+1)}\right]=\tan ^{-1}[x]$, then $x$ is equal to

A

$\frac{1}{n+1}$

B

$\frac{n}{n+1}$

C

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

D

$\frac{n}{n+2}$

Question 49: Palette 49/50

AP-EAPCET 2021 MCQ

If $y=\tan ^{-1}\left(\frac{\sqrt{1+x^2}+\sqrt{1-x^2}}{\sqrt{1+x^2}-\sqrt{1-x^2}}\right)$, where $x^2 \leq 1$. Then, find $\frac{d y}{d x}$ is equal to

A

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

B

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

C

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

D

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

Question 50: Palette 50/50

AP-EAPCET 2021 MCQ

If $\int \frac{d x}{x\left(\sqrt{\left.x^4-1\right)}\right.}=\frac{1}{k} \sec ^{-1}\left(x^k\right)$, then the value of $k$ is equal to

A

1

B

2

C

3

D

4