Question 1: Palette 1/38

TS-EAMCET 2025 MCQ

The number of real solution of $\tan ^{-1} x+\tan ^{-1} 2 x=\frac{\pi}{4}$ is

A

2

B

1

C

0

D

infinitely many

Question 2: Palette 2/38

TS-EAMCET 2025 MCQ

Consider the following

Assertion

$ \begin{aligned} & \text { (A) } \begin{array}{r} \sqrt{x-3}\left(\sin ^{-1}(\log x)+\cos ^{-1}\right. \\ (\log x) d x=\frac{\pi}{3}(x-3)^{3 / 2}+c \end{array} \end{aligned} $

Reason $(\mathrm{R}) \sin ^{-1}(f(x))+\cos ^{-1}(f(x))=\frac{\pi}{2},|f(x)|<1$

The correct answer is

A

Both $(A)$ and $(B)$ are true and $(R)$ is the correct explanation of $(A)$.

B

Both (A) and (R) are true and (R) is not the correct explanation of (A).

C

(A) is true, but (R) is false.

D

(A) is false, but (R) is true.

Question 3: Palette 3/38

TS-EAMCET 2025 MCQ

$ \sin ^{-1}(-\cos 2)+\cos ^{-1}(\sin 3)+\tan ^{-1}(\cot 5)= $

A

7

B

5

C

$\frac{\pi}{2}$

D

$\pi$

Question 4: Palette 4/38

TS-EAMCET 2025 MCQ

The domain of the derivative of the function $f(x)=\cos ^{-1}(2 x-5)-\sin ^{-1}(x-2)$ is

A

$[2,3]$

B

$(-\infty, 2] \cup[3, \infty)$

C

$(-\infty, 2) \cup(3, \infty)$

D

$(2,3)$

Question 5: Palette 5/38

TS-EAMCET 2025 MCQ

The number of values of $x$ satisfying the equation, $\tan ^{-1}\left(x+\frac{\sqrt{2}}{x}\right)+\tan ^{-1}\left(x-\frac{\sqrt{2}}{x}\right)=\tan ^{-1}(x)$ is

A

0

B

1

C

2

D

3

Question 6: Palette 6/38

TS-EAMCET 2025 MCQ

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

A

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

B

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

C

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

D

$\frac{1+2 x^2}{x|x|\left(x^2-1\right)^{\frac{3}{2}}}$

Question 7: Palette 7/38

TS-EAMCET 2025 MCQ

If $0 \leq x<\frac{3}{4}$, then the number of values of $x$ satisfying the equation $\tan ^{-1}(2 x-1)+\tan ^{-1} 2 x= \tan ^{-1} 4 x-\tan ^{-1}(2 x+1)$ is

A

0

B

1

C

2

D

3

Question 8: Palette 8/38

TS-EAMCET 2025 MCQ

If $\sinh ^{-1} x=\cosh ^{-1} y=\log (1+\sqrt{2})$, then $\tan ^{-1}(x+y)$

A

$67 \frac{1}{2}^{\circ}$

B

$75^{\circ}$

C

$22 \frac{1}{2}^{\circ}$

D

$15^{\circ}$

Question 9: Palette 9/38

TS-EAMCET 2025 MCQ

Consider the following statements

Assertion (A) : When $x, y, z$ are positive numbers, then

$ \begin{aligned} & \tan ^{-1}\left(\sqrt{\frac{x(x+y+z)}{y z}}\right)+\tan ^{-1}\left(\sqrt{\frac{y(x+y+z)}{x z}}\right) +\tan ^{-1}\left(\sqrt{\frac{z(x+y+z)}{x y}}\right)=\pi \end{aligned} $

Reason (R) : $\tan ^{-1} a+\tan ^{-1} b=\tan ^{-1}\left(\frac{a+b}{1-a b}\right)$, if $a>0$ and $b>0$

The correct answer is

A

Both (A) and (R) are true, (R) is the correct explanation of (A).

B

Both $(A)$ and $(R)$ are true, $(R)$ is not the correct explanation of $(A)$.

C

(A) is true, but (R) is false.

D

(A) is false, but (R) is true.

Question 10: Palette 10/38

TS-EAMCET 2025 MCQ

If $e^{\left(\sinh ^{-1} 2+\cosh ^{-1} \sqrt{6}\right)}=(a+(b+\sqrt{c}) \sqrt{a}+b \sqrt{c})$, then $a+b+c=$

A

13

B

15

C

17

D

11

Question 11: Palette 11/38

TS-EAMCET 2025 MCQ

Consider the following statements

Assertion (A) For $x \in R-\{1\}$;

$ \frac{d}{d x}\left(\tan ^{-1}\left(\frac{1+x}{1-x}\right)\right)=\frac{d}{d x}\left(\tan ^{-1} x\right) $

Reason (R) For $x<1, \tan ^{-1}\left(\frac{1+x}{1-x}\right)=\frac{\pi}{4}+\tan ^{-1} x$, for

$ x>1, \tan ^{-1}\left(\frac{1+x}{1-x}\right)=-\frac{3 \pi}{4}+\tan ^{-1} x $

The correct answer is

A

Both $(A)$ and $(R)$ are true, $(R)$ is the correct explanation of $(A)$.

B

Both (A) and (R) are true, (R) is not the correct explanation of (A).

C

(A) is true, but (R) is false.

D

(A) is false, but (R) is true.

Question 12: Palette 12/38

TS-EAMCET 2025 MCQ

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

A

$\frac{1}{2}$

B

2

C

$-\frac{1}{2}$

D

4

Question 13: Palette 13/38

TS-EAMCET 2025 MCQ

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

A

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

B

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

C

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

D

$\left[\frac{\pi}{3}, \frac{\pi}{2}\right]$

Question 14: Palette 14/38

TS-EAMCET 2025 MCQ

$ \tan ^{-1} \frac{3}{5}+\tan ^{-1} \frac{6}{41}+\tan ^{-1} \frac{9}{191}= $

A

$\tan ^{-1} \frac{9}{10}$

B

$\tan ^{-1} \frac{18}{19}$

C

$\tan ^{-1} \frac{3}{191}$

D

$\tan ^{-1} \frac{6}{205}$

Question 15: Palette 15/38

TS-EAMCET 2025 MCQ

If $2 \tanh ^{-1} x=\sinh ^{-1}\left(\frac{4}{3}\right)$, then $\cosh ^{-1}\left(\frac{1}{x}\right)=$

A

$\log (\sqrt{2}+1)$

B

$\log (\sqrt{2}-1)$

C

$\log (2+\sqrt{3})$

D

$\log (2-\sqrt{3})$

Question 16: Palette 16/38

TS-EAMCET 2025 MCQ

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

A

$\sqrt{\frac{2}{\pi}}$

B

$\sqrt{\frac{\pi}{2}}$

C

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

D

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

Question 17: Palette 17/38

TS-EAMCET 2024 MCQ

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

A

$\frac{\pi}{6}$

B

$\frac{\pi}{4}$

C

$\frac{\pi}{3}$

D

$\frac{\pi}{2}$

Question 18: Palette 18/38

TS-EAMCET 2024 MCQ

$\operatorname{sech}^{-1}\left(\frac{3}{5}\right)-\tanh ^{-1}\left(\frac{3}{5}\right)=$

A

$\log _{e} 6$

B

$\log _{e} 5$

C

$\log _{e}\left(\frac{3}{2}\right)$

D

$\log _{e}\left(\frac{2}{3}\right)$

Question 19: Palette 19/38

TS-EAMCET 2024 MCQ

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

A

$[-2,0) \cup(1,2]$

B

$[-2,-1] \cup[1,2]$

C

$[-1,0] \cup[1,2]$

D

$[1, \infty) \cup(-2,0)$

Question 20: Palette 20/38

TS-EAMCET 2024 MCQ

The trigonometric equation $\sin ^{-1} x=2 \sin ^{-1} a$, has a solution

A

only when $\frac{1}{\sqrt{2}} < a < \frac{1}{2}$

B

for all real values of (a)

C

only when $|a| \leq \frac{1}{\sqrt{2}}$

D

only when $|a| \geq \frac{1}{\sqrt{2}}$

Question 21: Palette 21/38

TS-EAMCET 2024 MCQ

If the real valued function $f(x)=\sin ^{-1}\left(x^2-1\right)-3 \log _3\left(3^x-2\right)$ is not defined for all $x \in(-\infty, a) \cup(b, \infty)$, then $3^a+b^2=$

A

5

B

6

C

3

D

4

Question 22: Palette 22/38

TS-EAMCET 2024 MCQ

If $\sin ^{-1}(4 x)-\cos ^{-1}(3 x)=\frac{\pi}{6}$, then $x=$

A

$\frac{\sqrt{3}}{2 \sqrt{7}}$

B

$\frac{\sqrt{3}}{4 \sqrt{7}}$

C

$\frac{\sqrt{3}}{2 \sqrt{13}}$

D

$\frac{\sqrt{3}}{4 \sqrt{13}}$

Question 23: Palette 23/38

TS-EAMCET 2024 MCQ

If $\sin h^{-1}(-\sqrt{3})+\cos ^{-1}(2)=K$, then $\cosh K=$

A

$\log (2-\sqrt{3})$

B

$\log (2+\sqrt{3})$

C

0

D

1

Question 24: Palette 24/38

TS-EAMCET 2024 MCQ

If $y=\cos ^{-1}\left(\frac{6 x-2 x^2-4}{2 x^2-6 x+5}\right)$, then $\frac{d y}{d x}=$

A

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

B

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

C

$\frac{2}{\sqrt{2 x^2-6 x+5}}$

D

$\frac{2}{2 x^2-6 x+5}$

Question 25: Palette 25/38

TS-EAMCET 2024 MCQ

If $2 \tan ^{-1} x=3 \sin ^{-1} x$ and $x \neq 0$, then $8 x^2+1=$

A

13

B

5

C

$\sqrt{7}$

D

$\sqrt{17}$

Question 26: Palette 26/38

TS-EAMCET 2024 MCQ

Match the functions given in List I with their relevant characteristics from List II.

List I	List II
(A) sinh x	(I) Domains is (-1,1), even function
(B) sec hx	(II) Domain is [1,∞), neither even nor odd function
(C) tan hx	(III) Even function
(D) cosec h⁻¹x	(IV) Range is R, odd function
	(V) Range is (-1,1), odd function

The correct answer is

A

A-II, B-III, C-IV, D-V

B

A-V, B-I, C-II, D-III

C

A-IV, B-II, C-I, D-V

D

A-IV, B-III, C-V, D-II

Question 27: Palette 27/38

TS-EAMCET 2024 MCQ

$\cos ^{-1} \frac{3}{5}+\sin ^{-1} \frac{5}{13}+\tan ^{-1} \frac{16}{63}=$

A

$\frac{\pi}{2}$

B

$\frac{\pi}{3}$

C

$\frac{\pi}{4}$

D

$\frac{\pi}{6}$

Question 28: Palette 28/38

TS-EAMCET 2024 MCQ

If $\cosh ^{-1}\left(\frac{5}{3}\right)+\sinh ^{-1}\left(\frac{3}{4}\right)=k$, then $e^k=$

A

$\frac{2}{3}$

B

$\frac{3}{2}$

C

6

D

5

Question 29: Palette 29/38

TS-EAMCET 2020 MCQ

For the least possible value of $n \in \mathbf{Z}$ the solution $(x, y)$ of the equations $\cos ^{-1} x+\left(\sin ^{-1} y\right)^2=\frac{n \pi^2}{4}$ and $\cos ^{-1} x\left(\sin ^{-1} y\right)^2=\frac{\pi^4}{16}$, is

A

$\left(\frac{\pi^2}{4}, \pm 1\right)$

B

$\left(\frac{\pi^2}{4}, \sin \frac{\pi^2}{16}\right)$

C

$\left(\cos \left(\frac{\pi^2}{4}\right), \pm 1\right)$

D

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

Question 30: Palette 30/38

TS-EAMCET 2020 MCQ

If $x=\left(\tan ^{-1} \frac{1}{5}+\tan ^{-1} \frac{1}{8}\right)$, then $\frac{\sin x+\cos x}{\tan x}=$

A

$\frac{12}{\sqrt{10}}$

B

$\frac{15}{\sqrt{10}}$

C

$\frac{1}{\sqrt{10}}$

D

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

Question 31: Palette 31/38

TS-EAMCET 2020 MCQ

If for $|x|>1, \tanh ^{-1}\left(\frac{1}{x}\right)+\operatorname{coth}^{-1}(x)=\log _e(f(x))$, then $f(-5)=$

A

$\frac{3}{2}$

B

$\frac{-2}{3}$

C

$\frac{2}{3}$

D

$\frac{1}{3}$

Question 32: Palette 32/38

TS-EAMCET 2020 MCQ

Domain of $\cos ^{-1}\left[\log _5\left(x^2+7 x+15\right)\right]$ is

A

The set of all real numbers

B

$(-\infty,-5] \cup[-2, \infty)$

C

$R-\{-5,-2\}$, where $R$ is the set of real numbers

D

$[-5,-2]$

Question 33: Palette 33/38

TS-EAMCET 2020 MCQ

If $\sum_{n=1}^k \tan ^{-1}\left(\frac{1}{n^2+3 n+3}\right)=\tan ^{-1} \alpha$, then $\alpha=$

A

$\frac{k}{k+2}$

B

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

C

$\frac{k}{2 k+5}$

D

$\frac{3 k}{4 k+5}$

Question 34: Palette 34/38

TS-EAMCET 2020 MCQ

The set of values of $x$ such that $\tan ^{-1}\left(\frac{x}{x-2}\right)-\tan ^{-1}\left(\frac{x}{2 x-1}\right)=\tan ^{-1}\left(\frac{2}{3}\right)$ is

A

$\phi$

B

$\left\{\frac{1}{2}\right\}$

C

$\left\{\frac{1}{3}, 2\right\}$

D

$\left\{\frac{1}{3}, 4\right\}$

Question 35: Palette 35/38

TS-EAMCET 2020 MCQ

If $\sin ^{-1}\left(\frac{12}{x}\right)+\sin ^{-1}\left(\frac{5}{x}\right)=\frac{\pi}{2}$, then $x=$

A

5

B

7

C

13

D

17

Question 36: Palette 36/38

TS-EAMCET 2020 MCQ

$ \operatorname{cosec}^{-1}\left[\left(\frac{\tan ^2\left(\frac{\alpha-\pi}{4}\right)-1}{\tan ^2\left(\frac{\alpha-\pi}{4}\right)+1}+\cos \frac{\alpha}{2} \cdot \cot 5 \alpha\right) \sec \frac{11 \alpha}{2}\right] $

A

$2 \alpha$

B

$5 \alpha$

C

$\frac{\pi}{2}-4 \alpha$

D

$\frac{5}{2} \alpha$

Question 37: Palette 37/38

TS-EAMCET 2020 MCQ

If $\tan ^{-1} \frac{1}{5}+\frac{1}{2} \sec ^{-1} x+\tan ^{-1} \frac{1}{8}=\frac{\pi}{8}$, then $x^2=$

A

$\frac{12}{7}$

B

$\frac{50}{49}$

C

$\frac{13}{12}$

D

$\frac{1}{2}$

Question 38: Palette 38/38

TS-EAMCET 2020 MCQ

Assertion $(\mathrm{A}) \operatorname{cosech}^{-1}(3)=\log \left(\frac{1+\sqrt{10}}{3}\right)$

Reason (R) $e^{\operatorname{cosech}^{-1} x}$ is a root of the quadratic equation $x p^2-2 p-x=0$

The correct option among the following is

A

(A) is true, (R) is true and (R) is the correct explanation for (A)

B

(A) is true, (R) is true but (R) is not the correct explanation for (A)

C

(A) is true but (R) is false

D

(A) is false but (R) is true