Question 1: Palette 1/82

JEE Mains 2026 MCQ

Considering the principal values of inverse trigonometric functions, the value of the expression

$\tan \left( 2 \sin^{-1}\left( \frac{2}{\sqrt{13}} \right) - 2 \cos^{-1}\left( \frac{3}{\sqrt{10}} \right) \right)$

is equal to :

A

$ \frac{33}{56} $

B

$ -\frac{33}{56} $

C

$ -\frac{16}{63} $

D

$ \frac{16}{63} $

Question 2: Palette 2/82

JEE Mains 2026 MCQ

If the domain of the function $f(x)=\sin ^{-1}\left(\frac{1}{x^2-2 x-2}\right)$, is $(-\infty, \alpha] \cup[\beta, \gamma] \cup[\delta, \infty)$, then $\alpha+\beta+\gamma+\delta$ is equal to

A

4

B

2

C

5

D

3

Question 3: Palette 3/82

JEE Mains 2026 MCQ

The number of solutions of $\tan ^{-1} 4 x+\tan ^{-1} 6 x=\frac{\pi}{6}$, where $-\frac{1}{2 \sqrt{6}} < x < \frac{1}{2 \sqrt{6}}$, is equal to :

A

2

B

0

C

3

D

1

Question 4: Palette 4/82

JEE Mains 2026 MCQ

If the domain of the function $f(x)=\cos ^{-1}\left(\frac{2 x-5}{11-3 x}\right)+\sin ^{-1}\left(2 x^2-3 x+1\right)$ is the interval $[\alpha, \beta]$, then $\alpha+2 \beta$ is equal to :

A

5

B

2

C

3

D

1

Question 5: Palette 5/82

JEE Mains 2026 MCQ

Let $\alpha=3 \sin ^{-1}\left(\frac{6}{11}\right)$ and $\beta=3 \cos ^{-1}\left(\frac{4}{9}\right)$, where inverse trigonometric functions take only the principal values.

Given below are two statements :

Statement I : $\quad \cos (\alpha+\beta)>0$.

Statement II : $\quad \cos (\alpha)<0$.

In the light of the above statements, choose the correct answer from the options given below :

A

Both Statement I and Statement II are true

B

Both Statement I and Statement II are false

C

Statement I is true but Statement II is false

D

$ \text { Statement I is false but Statement II is true } $

Question 6: Palette 6/82

JEE Mains 2026 MCQ

If $\sin \left(\tan ^{-1}(x \sqrt{2})\right)=\cot \left(\sin ^{-1} \sqrt{1-x^2}\right), x \in(0,1)$, then the value of $x$ is:

A

$\frac{1}{2}$

B

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

C

$\frac{2}{3}$

D

$\frac{5}{8}$

Question 7: Palette 7/82

JEE Mains 2026 MCQ

Let $0<\alpha<1, \beta=\frac{1}{3 \alpha}$ and $\tan ^{-1}(1-\alpha)+\tan ^{-1}(1-\beta)=\frac{\pi}{4}$. Then $6(\alpha+\beta)$ is equal to:

A

6

B

7

C

8

D

9

Question 8: Palette 8/82

JEE Mains 2025 MCQ

The value of $ \cot^{-1} \left( \frac{\sqrt{1 + \tan^2(2)} - 1}{\tan(2)} \right) - \cot^{-1} \left( \frac{\sqrt{1 + \tan^2\left(\frac{1}{2}\right)} + 1}{\tan\left(\frac{1}{2}\right)} \right) $ is equal to

A

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

B

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

C

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

D

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

Question 9: Palette 9/82

JEE Mains 2025 MCQ

The sum of the infinite series $\cot ^{-1}\left(\frac{7}{4}\right)+\cot ^{-1}\left(\frac{19}{4}\right)+\cot ^{-1}\left(\frac{39}{4}\right)+\cot ^{-1}\left(\frac{67}{4}\right)+\ldots$. is :

A

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

B

$\frac{\pi}{2}-\cot ^{-1}\left(\frac{1}{2}\right)$

C

$\frac{\pi}{2}-\tan ^{-1}\left(\frac{1}{2}\right)$

D

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

Question 10: Palette 10/82

JEE Mains 2025 MCQ

Considering the principal values of the inverse trigonometric functions, $\sin ^{-1}\left(\frac{\sqrt{3}}{2} x+\frac{1}{2} \sqrt{1-x^2}\right),-\frac{1}{2}< x<\frac{1}{\sqrt{2}}$, is equal to

A

$\frac{-5 \pi}{6}-\sin ^{-1} x$

B

$\frac{5 \pi}{6}-\sin ^{-1} x$

C

$\frac{\pi}{6}+\sin ^{-1} x$

D

$\frac{\pi}{4}+\sin ^{-1} x$

Question 11: Palette 11/82

JEE Mains 2025 MCQ

Let [x] denote the greatest integer less than or equal to x. Then the domain of $ f(x) = \sec^{-1}(2[x] + 1) $ is:

A

$(-\infty, \infty)$

B

$(-\infty, \infty)- \{0\}$

C

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

D

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

Question 12: Palette 12/82

JEE Mains 2025 MCQ

$\cos \left(\sin ^{-1} \frac{3}{5}+\sin ^{-1} \frac{5}{13}+\sin ^{-1} \frac{33}{65}\right)$ is equal to:

A

$\frac{33}{65}$

B

1

C

$\frac{32}{65}$

D

0

Question 13: Palette 13/82

JEE Mains 2025 MCQ

If $\alpha>\beta>\gamma>0$, then the expression $\cot ^{-1}\left\{\beta+\frac{\left(1+\beta^2\right)}{(\alpha-\beta)}\right\}+\cot ^{-1}\left\{\gamma+\frac{\left(1+\gamma^2\right)}{(\beta-\gamma)}\right\}+\cot ^{-1}\left\{\alpha+\frac{\left(1+\alpha^2\right)}{(\gamma-\alpha)}\right\}$ is equal to :

A

$3 \pi$

B

$\frac{\pi}{2}-(\alpha+\beta+\gamma)$

C

$\pi$

D

0

Question 14: Palette 14/82

JEE Mains 2025 MCQ

If $\frac{\pi}{2} \leq x \leq \frac{3 \pi}{4}$, then $\cos ^{-1}\left(\frac{12}{13} \cos x+\frac{5}{13} \sin x\right)$ is equal to

A

$x+\tan ^{-1} \frac{5}{12}$

B

$x-\tan ^{-1} \frac{4}{3}$

C

$x+\tan ^{-1} \frac{4}{5}$

D

$x-\tan ^{-1} \frac{5}{12}$

Question 15: Palette 15/82

JEE Mains 2025 MCQ

Using the principal values of the inverse trigonometric functions, the sum of the maximum and the minimum values of $16\left(\left(\sec ^{-1} x\right)^2+\left(\operatorname{cosec}^{-1} x\right)^2\right)$ is :

A

$24 \pi^2$

B

$18 \pi^2$

C

$22 \pi^2$

D

$31 \pi^2$

Question 16: Palette 16/82

JEE Mains 2024 MCQ

Given that the inverse trigonometric function assumes principal values only. Let $x, y$ be any two real numbers in $[-1,1]$ such that $\cos ^{-1} x-\sin ^{-1} y=\alpha, \frac{-\pi}{2} \leq \alpha \leq \pi$. Then, the minimum value of $x^2+y^2+2 x y \sin \alpha$ is

A

0

B

$-$1

C

$\frac{1}{2}$

D

$\frac{-1}{2}$

Question 17: Palette 17/82

JEE Mains 2024 MCQ

If the domain of the function $\sin ^{-1}\left(\frac{3 x-22}{2 x-19}\right)+\log _{\mathrm{e}}\left(\frac{3 x^2-8 x+5}{x^2-3 x-10}\right)$ is $(\alpha, \beta]$, then $3 \alpha+10 \beta$ is equal to:

A

95

B

100

C

97

D

98

Question 18: Palette 18/82

JEE Mains 2024 MCQ

If $a=\sin ^{-1}(\sin (5))$ and $b=\cos ^{-1}(\cos (5))$, then $a^2+b^2$ is equal to

A

25

B

$4 \pi^2+25$

C

$8 \pi^2-40 \pi+50$

D

$4 \pi^2-20 \pi+50$

Question 19: Palette 19/82

JEE Mains 2024 MCQ

For $\alpha, \beta, \gamma \neq 0$, if $\sin ^{-1} \alpha+\sin ^{-1} \beta+\sin ^{-1} \gamma=\pi$ and $(\alpha+\beta+\gamma)(\alpha-\gamma+\beta)=3 \alpha \beta$, then $\gamma$ equals

A

$\sqrt{3}$

B

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

C

$\frac{1}{\sqrt{2}}$

D

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

Question 20: Palette 20/82

JEE Mains 2024 MCQ

Let $x=\frac{m}{n}$ ($m, n$ are co-prime natural numbers) be a solution of the equation $\cos \left(2 \sin ^{-1} x\right)=\frac{1}{9}$ and let $\alpha, \beta(\alpha >\beta)$ be the roots of the equation $m x^2-n x-m+ n=0$. Then the point $(\alpha, \beta)$ lies on the line

A

$3 x-2 y=-2$

B

$3 x+2 y=2$

C

$5 x+8 y=9$

D

$5 x-8 y=-9$

Question 21: Palette 21/82

JEE Mains 2024 MCQ

Considering only the principal values of inverse trigonometric functions, the number of positive real values of $x$ satisfying $\tan ^{-1}(x)+\tan ^{-1}(2 x)=\frac{\pi}{4}$ is :

A

more than 2

B

2

C

0

D

1

Question 22: Palette 22/82

JEE Mains 2023 MCQ

If the domain of the function

$f(x)=\log _{e}\left(4 x^{2}+11 x+6\right)+\sin ^{-1}(4 x+3)+\cos ^{-1}\left(\frac{10 x+6}{3}\right)$ is $(\alpha, \beta]$, then

$36|\alpha+\beta|$ is equal to :

A

72

B

54

C

45

D

63

Question 23: Palette 23/82

JEE Mains 2023 MCQ

Let $S = \left\{ {x \in R:0 < x < 1\,\mathrm{and}\,2{{\tan }^{ - 1}}\left( {{{1 - x} \over {1 + x}}} \right) = {{\cos }^{ - 1}}\left( {{{1 - {x^2}} \over {1 + {x^2}}}} \right)} \right\}$.

If $\mathrm{n(S)}$ denotes the number of elements in $\mathrm{S}$ then :

A

$\mathrm{n}(\mathrm{S})=0$

B

$\mathrm{n}(\mathrm{S})=1$ and only one element in $\mathrm{S}$ is less than $\frac{1}{2}$.

C

$\mathrm{n}(\mathrm{S})=1$ and the elements in $\mathrm{S}$ is more than $\frac{1}{2}$.

D

$\mathrm{n}(\mathrm{S})=1$ and the element in $\mathrm{S}$ is less than $\frac{1}{2}$.

Question 24: Palette 24/82

JEE Mains 2023 MCQ

Let $S$ be the set of all solutions of the equation $\cos ^{-1}(2 x)-2 \cos ^{-1}\left(\sqrt{1-x^{2}}\right)=\pi, x \in\left[-\frac{1}{2}, \frac{1}{2}\right]$. Then $\sum_\limits{x \in S} 2 \sin ^{-1}\left(x^{2}-1\right)$ is equal to :

A

$\pi-2 \sin ^{-1}\left(\frac{\sqrt{3}}{4}\right)$

B

$\pi-\sin ^{-1}\left(\frac{\sqrt{3}}{4}\right)$

C

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

D

None

Question 25: Palette 25/82

JEE Mains 2023 MCQ

Let (a, b) $\subset(0,2 \pi)$ be the largest interval for which $\sin ^{-1}(\sin \theta)-\cos ^{-1}(\sin \theta)>0, \theta \in(0,2 \pi)$, holds.

If $\alpha x^{2}+\beta x+\sin ^{-1}\left(x^{2}-6 x+10\right)+\cos ^{-1}\left(x^{2}-6 x+10\right)=0$ and $\alpha-\beta=b-a$, then $\alpha$ is equal to :

A

$\frac{\pi}{16}$

B

$\frac{\pi}{48}$

C

$\frac{\pi}{8}$

D

$\frac{\pi}{12}$

Question 26: Palette 26/82

JEE Mains 2023 MCQ

If ${\sin ^{ - 1}}{\alpha \over {17}} + {\cos ^{ - 1}}{4 \over 5} - {\tan ^{ - 1}}{{77} \over {36}} = 0,0 < \alpha < 13$, then ${\sin ^{ - 1}}(\sin \alpha ) + {\cos ^{ - 1}}(\cos \alpha )$ is equal to :

A

16

B

$\pi$

C

16 $-$ 5$\pi$

D

0

Question 27: Palette 27/82

JEE Mains 2023 MCQ

${\tan ^{ - 1}}\left( {{{1 + \sqrt 3 } \over {3 + \sqrt 3 }}} \right) + {\sec ^{ - 1}}\left( {\sqrt {{{8 + 4\sqrt 3 } \over {6 + 3\sqrt 3 }}} } \right)$ is equal to :

A

${\pi \over 2}$

B

${\pi \over 3}$

C

${\pi \over 6}$

D

${\pi \over 4}$

Question 28: Palette 28/82

JEE Mains 2023 MSQ

Let $a_{1}=1, a_{2}, a_{3}, a_{4}, \ldots .$. be consecutive natural numbers.

Then $\tan ^{-1}\left(\frac{1}{1+a_{1} a_{2}}\right)+\tan ^{-1}\left(\frac{1}{1+a_{2} a_{3}}\right)+\ldots . .+\tan ^{-1}\left(\frac{1}{1+a_{2021} a_{2022}}\right)$ is equal to :

A

$\frac{\pi}{4}-\cot ^{-1}(2022)$

B

$\frac{\pi}{4}-\tan ^{-1}(2022)$

C

$\cot ^{-1}(2022)-\frac{\pi}{4}$

D

$\tan ^{-1}(2022)-\frac{\pi}{4}$

Question 29: Palette 29/82

JEE Mains 2022 MCQ

The domain of the function $f(x)=\sin ^{-1}\left(\frac{x^{2}-3 x+2}{x^{2}+2 x+7}\right)$ is :

A

$[1, \infty)$

B

$[-1,2]$

C

$[-1, \infty)$

D

$(-\infty, 2]$

Question 30: Palette 30/82

JEE Mains 2022 MCQ

The sum of the absolute maximum and absolute minimum values of the function $f(x)=\tan ^{-1}(\sin x-\cos x)$ in the interval $[0, \pi]$ is :

A

0

B

$\tan ^{-1}\left(\frac{1}{\sqrt{2}}\right)-\frac{\pi}{4}$

C

$\cos ^{-1}\left(\frac{1}{\sqrt{3}}\right)-\frac{\pi}{4}$

D

$\frac{-\pi}{12}$

Question 31: Palette 31/82

JEE Mains 2022 MCQ

Considering only the principal values of the inverse trigonometric functions, the domain of the function $f(x)=\cos ^{-1}\left(\frac{x^{2}-4 x+2}{x^{2}+3}\right)$ is :

A

$\left(-\infty, \frac{1}{4}\right]$

B

$\left[-\frac{1}{4}, \infty\right)$

C

$(-1 / 3, \infty)$

D

$\left(-\infty, \frac{1}{3}\right]$

Question 32: Palette 32/82

JEE Mains 2022 MCQ

Considering the principal values of the inverse trigonometric functions, the sum of all the solutions of the equation $\cos ^{-1}(x)-2 \sin ^{-1}(x)=\cos ^{-1}(2 x)$ is equal to :

A

0

B

1

C

$\frac{1}{2}$

D

$-\frac{1}{2}$

Question 33: Palette 33/82

JEE Mains 2022 MCQ

The domain of the function $f(x)=\sin ^{-1}\left[2 x^{2}-3\right]+\log _{2}\left(\log _{\frac{1}{2}}\left(x^{2}-5 x+5\right)\right)$, where [t] is the greatest integer function, is :

A

$ \left(-\sqrt{\frac{5}{2}}, \frac{5-\sqrt{5}}{2}\right) $

B

$ \left(\frac{5-\sqrt{5}}{2}, \frac{5+\sqrt{5}}{2}\right) $

C

$ \left(1, \frac{5-\sqrt{5}}{2}\right) $

D

$ \left[1, \frac{5+\sqrt{5}}{2}\right) $

Question 34: Palette 34/82

JEE Mains 2022 MCQ

If $0 < x < {1 \over {\sqrt 2 }}$ and ${{{{\sin }^{ - 1}}x} \over \alpha } = {{{{\cos }^{ - 1}}x} \over \beta }$, then the value of $\sin \left( {{{2\pi \alpha } \over {\alpha + \beta }}} \right)$ is :

A

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

B

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

C

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

D

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

Question 35: Palette 35/82

JEE Mains 2022 MCQ

$\tan \left(2 \tan ^{-1} \frac{1}{5}+\sec ^{-1} \frac{\sqrt{5}}{2}+2 \tan ^{-1} \frac{1}{8}\right)$ is equal to :

A

1

B

2

C

$\frac{1}{4}$

D

$\frac{5}{4}$

Question 36: Palette 36/82

JEE Mains 2022 MCQ

Let m and M respectively be the minimum and the maximum values of $f(x) = {\sin ^{ - 1}}2x + \sin 2x + {\cos ^{ - 1}}2x + \cos 2x,\,x \in \left[ {0,{\pi \over 8}} \right]$. Then m + M is equal to :

A

$1 + \sqrt 2 + \pi $

B

$\left( {1 + \sqrt 2 } \right)\pi $

C

$\pi + \sqrt 2 $

D

$1 + \pi $

Question 37: Palette 37/82

JEE Mains 2022 MCQ

Let $\alpha = \tan \left( {{{5\pi } \over {16}}\sin \left( {2{{\cos }^{ - 1}}\left( {{1 \over {\sqrt 5 }}} \right)} \right)} \right)$ and $\beta = \cos \left( {{{\sin }^{ - 1}}\left( {{4 \over 5}} \right) + {{\sec }^{ - 1}}\left( {{5 \over 3}} \right)} \right)$ where the inverse trigonometric functions take principal values. Then, the equation whose roots are $\alpha$ and $\beta$ is :

A

$15{x^2} - 8x - 7 = 0$

B

$5{x^2} - 12x + 7 = 0$

C

$25{x^2} - 18x - 7 = 0$

D

$25{x^2} - 32x + 7 = 0$

Question 38: Palette 38/82

JEE Mains 2022 MCQ

The domain of the function ${\cos ^{ - 1}}\left( {{{2{{\sin }^{ - 1}}\left( {{1 \over {4{x^2} - 1}}} \right)} \over \pi }} \right)$ is :

A

$R - \left\{ { - {1 \over 2},{1 \over 2}} \right\}$

B

$( - \infty , - 1] \cup [1,\infty ) \cup \{ 0\} $

C

$\left( { - \infty ,{{ - 1} \over 2}} \right) \cup \left( {{1 \over 2},\infty } \right) \cup \{ 0\} $

D

$\left( { - \infty ,{{ - 1} \over {\sqrt 2 }}} \right] \cup \left[ {{1 \over {\sqrt 2 }},\infty } \right) \cup \{ 0\} $

Question 39: Palette 39/82

JEE Mains 2022 MCQ

The value of $\cot \left( {\sum\limits_{n = 1}^{50} {{{\tan }^{ - 1}}\left( {{1 \over {1 + n + {n^2}}}} \right)} } \right)$ is :

A

${{26} \over {25}}$

B

${{25} \over {26}}$

C

${{50} \over {51}}$

D

${{52} \over {51}}$

Question 40: Palette 40/82

JEE Mains 2022 MCQ

${\sin ^1}\left( {\sin {{2\pi } \over 3}} \right) + {\cos ^{ - 1}}\left( {\cos {{7\pi } \over 6}} \right) + {\tan ^{ - 1}}\left( {\tan {{3\pi } \over 4}} \right)$ is equal to :

A

${{11\pi } \over {12}}$

B

${{17\pi } \over {12}}$

C

${{31\pi } \over {12}}$

D

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

Question 41: Palette 41/82

JEE Mains 2022 MCQ

If the inverse trigonometric functions take principal values then

${\cos ^{ - 1}}\left( {{3 \over {10}}\cos \left( {{{\tan }^{ - 1}}\left( {{4 \over 3}} \right)} \right) + {2 \over 5}\sin \left( {{{\tan }^{ - 1}}\left( {{4 \over 3}} \right)} \right)} \right)$ is equal to :

A

0

B

${\pi \over 4}$

C

${\pi \over 3}$

D

${\pi \over 6}$

Question 42: Palette 42/82

JEE Mains 2022 MCQ

The value of ${\tan ^{ - 1}}\left( {{{\cos \left( {{{15\pi } \over 4}} \right) - 1} \over {\sin \left( {{\pi \over 4}} \right)}}} \right)$ is equal to :

A

$ - {\pi \over 4}$

B

$ - {\pi \over 8}$

C

$ - {{5\pi } \over {12}}$

D

$ - {{4\pi } \over 9}$

Question 43: Palette 43/82

JEE Mains 2022 MCQ

Let $x * y = {x^2} + {y^3}$ and $(x * 1) * 1 = x * (1 * 1)$.

Then a value of $2{\sin ^{ - 1}}\left( {{{{x^4} + {x^2} - 2} \over {{x^4} + {x^2} + 2}}} \right)$ is :

A

${\pi \over 4}$

B

${\pi \over 3}$

C

${\pi \over 2}$

D

${\pi \over 6}$

Question 44: Palette 44/82

JEE Mains 2022 MCQ

The set of all values of k for which

${({\tan ^{ - 1}}x)^3} + {({\cot ^{ - 1}}x)^3} = k{\pi ^3},\,x \in R$, is the interval :

A

$\left[ {{1 \over {32}},{7 \over 8}} \right)$

B

$\left( {{1 \over {24}},{{13} \over {16}}} \right)$

C

$\left[ {{1 \over {48}},{{13} \over {16}}} \right]$

D

$\left[ {{1 \over {32}},{9 \over 8}} \right)$

Question 45: Palette 45/82

JEE Mains 2022 MCQ

The domain of the function

$f(x) = {{{{\cos }^{ - 1}}\left( {{{{x^2} - 5x + 6} \over {{x^2} - 9}}} \right)} \over {{{\log }_e}({x^2} - 3x + 2)}}$ is :

A

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

B

$(2,\infty )$

C

$\left[ { - {1 \over 2},1} \right) \cup (2,\infty )$

D

$\left[ { - {1 \over 2},1} \right) \cup (2,\infty ) - \left\{ 3,{{{3 + \sqrt 5 } \over 2},{{3 - \sqrt 5 } \over 2}} \right\}$

Question 46: Palette 46/82

JEE Mains 2021 MCQ

${\cos ^{ - 1}}(\cos ( - 5)) + {\sin ^{ - 1}}(\sin (6)) - {\tan ^{ - 1}}(\tan (12))$ is equal to :

(The inverse trigonometric functions take the principal values)

A

3$\pi$ $-$ 11

B

4$\pi$ $-$ 9

C

4$\pi$ $-$ 11

D

3$\pi$ + 1

Question 47: Palette 47/82

JEE Mains 2021 MCQ

The domain of the function

$f(x) = {\sin ^{ - 1}}\left( {{{3{x^2} + x - 1} \over {{{(x - 1)}^2}}}} \right) + {\cos ^{ - 1}}\left( {{{x - 1} \over {x + 1}}} \right)$ is :

A

$\left[ {0,{1 \over 4}} \right]$

B

$[ - 2,0] \cup \left[ {{1 \over 4},{1 \over 2}} \right]$

C

$\left[ {{1 \over 4},{1 \over 2}} \right] \cup \{ 0\} $

D

$\left[ {0,{1 \over 2}} \right]$

Question 48: Palette 48/82

JEE Mains 2021 MCQ

Let M and m respectively be the maximum and minimum values of the function
f(x) = tan^$-$1 (sin x + cos x) in $\left[ {0,{\pi \over 2}} \right]$, then the value of tan(M $-$ m) is equal to :

A

$2 + \sqrt 3 $

B

$2 - \sqrt 3 $

C

$3 + 2\sqrt 2 $

D

$3 - 2\sqrt 2 $

Question 49: Palette 49/82

JEE Mains 2021 MCQ

If ${({\sin ^{ - 1}}x)^2} - {({\cos ^{ - 1}}x)^2} = a$; 0 < x < 1, a $\ne$ 0, then the value of 2x² $-$ 1 is :

A

$\cos \left( {{{4a} \over \pi }} \right)$

B

$\sin \left( {{{2a} \over \pi }} \right)$

C

$\cos \left( {{{2a} \over \pi }} \right)$

D

$\sin \left( {{{4a} \over \pi }} \right)$

Question 50: Palette 50/82

JEE Mains 2021 MCQ

The domain of the function ${{\mathop{\rm cosec}\nolimits} ^{ - 1}}\left( {{{1 + x} \over x}} \right)$ is :

A

$\left( { - 1, - {1 \over 2}} \right] \cup (0,\infty )$

B

$\left[ { - {1 \over 2},0} \right) \cup [1,\infty )$

C

$\left( { - {1 \over 2},\infty } \right) - \{ 0\} $

D

$\left[ { - {1 \over 2},\infty } \right) - \{ 0\} $

Question 51: Palette 51/82

JEE Mains 2021 MCQ

If $\sum\limits_{r = 1}^{50} {{{\tan }^{ - 1}}{1 \over {2{r^2}}} = p} $, then the value of tan p is :

A

${{101} \over {102}}$

B

${{50} \over {51}}$

C

100

D

${{51} \over {50}}$

Question 52: Palette 52/82

JEE Mains 2021 MCQ

If the domain of the function $f(x) = {{{{\cos }^{ - 1}}\sqrt {{x^2} - x + 1} } \over {\sqrt {{{\sin }^{ - 1}}\left( {{{2x - 1} \over 2}} \right)} }}$ is the interval ($\alpha$, $\beta$], then $\alpha$ + $\beta$ is equal to :

A

${3 \over 2}$

B

2

C

${1 \over 2}$

D

1

Question 53: Palette 53/82

JEE Mains 2021 MCQ

The value of $\tan \left( {2{{\tan }^{ - 1}}\left( {{3 \over 5}} \right) + {{\sin }^{ - 1}}\left( {{5 \over {13}}} \right)} \right)$ is equal to :

A

${{ - 181} \over {69}}$

B

${{220} \over {21}}$

C

${{ - 291} \over {76}}$

D

${{151} \over {63}}$

Question 54: Palette 54/82

JEE Mains 2021 MCQ

The number of real roots of the equation ${\tan ^{ - 1}}\sqrt {x(x + 1)} + {\sin ^{ - 1}}\sqrt {{x^2} + x + 1} = {\pi \over 4}$ is :

A

1

B

2

C

4

D

0

Question 55: Palette 55/82

JEE Mains 2021 MCQ

The number of solutions of the equation

${\sin ^{ - 1}}\left[ {{x^2} + {1 \over 3}} \right] + {\cos ^{ - 1}}\left[ {{x^2} - {2 \over 3}} \right] = {x^2}$, for x$\in$[$-$1, 1], and [x] denotes the greatest integer less than or equal to x, is :

A

0

B

Infinite

C

2

D

4

Question 56: Palette 56/82

JEE Mains 2021 MCQ

The sum of possible values of x for

tan^$-$1(x + 1) + cot^$-$1$\left( {{1 \over {x - 1}}} \right)$ = tan^$-$1$\left( {{8 \over {31}}} \right)$ is :

A

$-$${{{32} \over 4}}$

B

$-$${{{33} \over 4}}$

C

$-$${{{31} \over 4}}$

D

$-$${{{30} \over 4}}$

Question 57: Palette 57/82

JEE Mains 2021 MCQ

If cot^$-$1($\alpha$) = cot^$-$1 2 + cot^$-$1 8 + cot^$-$1 18 + cot^$-$1 32 + ...... upto 100 terms, then $\alpha$ is :

A

1.02

B

1.03

C

1.01

D

1.00

Question 58: Palette 58/82

JEE Mains 2021 MCQ

Given that the inverse trigonometric functions take principal values only. Then, the number of real values of x which satisfy

${\sin ^{ - 1}}\left( {{{3x} \over 5}} \right) + {\sin ^{ - 1}}\left( {{{4x} \over 5}} \right) = {\sin ^{ - 1}}x$ is equal to :

A

2

B

0

C

3

D

1

Question 59: Palette 59/82

JEE Mains 2021 MCQ

If 0 < a, b < 1, and tan^$-$1a + tan^$-$1b = ${\pi \over 4}$, then the value of

$(a + b) - \left( {{{{a^2} + {b^2}} \over 2}} \right) + \left( {{{{a^3} + {b^3}} \over 3}} \right) - \left( {{{{a^4} + {b^4}} \over 4}} \right) + .....$ is :

A

${\log _e}$2

B

e

C

${\log _e}\left( {{e \over 2}} \right)$

D

e² = 1

Question 60: Palette 60/82

JEE Mains 2021 MCQ

If ${{{{\sin }^1}x} \over a} = {{{{\cos }^{ - 1}}x} \over b} = {{{{\tan }^{ - 1}}y} \over c}$; $0 < x < 1$,
then the value of $\cos \left( {{{\pi c} \over {a + b}}} \right)$ is :

A

${{1 - {y^2}} \over {2y}}$

B

${{1 - {y^2}} \over {y\sqrt y }}$

C

$1 - {y^2}$

D

${{1 - {y^2}} \over {1 + {y^2}}}$

Question 61: Palette 61/82

JEE Mains 2021 MCQ

cosec$\left[ {2{{\cot }^{ - 1}}(5) + {{\cos }^{ - 1}}\left( {{4 \over 5}} \right)} \right]$ is equal to :

A

${{75} \over {56}}$

B

${{65} \over {56}}$

C

${{56} \over {33}}$

D

${{65} \over {33}}$

Question 62: Palette 62/82

JEE Mains 2021 MCQ

A possible value of $\tan \left( {{1 \over 4}{{\sin }^{ - 1}}{{\sqrt {63} } \over 8}} \right)$ is :

A

$\sqrt 7 - 1$

B

${1 \over {\sqrt 7 }}$

C

$2\sqrt 2 - 1$

D

${1 \over {2\sqrt 2 }}$

Question 63: Palette 63/82

JEE Mains 2020 MCQ

If S is the sum of the first 10 terms of the series

${\tan ^{ - 1}}\left( {{1 \over 3}} \right) + {\tan ^{ - 1}}\left( {{1 \over 7}} \right) + {\tan ^{ - 1}}\left( {{1 \over {13}}} \right) + {\tan ^{ - 1}}\left( {{1 \over {21}}} \right) + ....$

then tan(S) is equal to :

A

${10 \over {11}}$

B

${5 \over {11}}$

C

-${6 \over {5}}$

D

${5 \over {6}}$

Question 64: Palette 64/82

JEE Mains 2020 MCQ

2$\pi $ - $\left( {{{\sin }^{ - 1}}{4 \over 5} + {{\sin }^{ - 1}}{5 \over {13}} + {{\sin }^{ - 1}}{{16} \over {65}}} \right)$ is equal to :

A

${{7\pi } \over 4}$

B

${{5\pi } \over 4}$

C

${{3\pi } \over 2}$

D

${\pi \over 2}$

Question 65: Palette 65/82

JEE Mains 2020 MCQ

The domain of the function
f(x) = ${\sin ^{ - 1}}\left( {{{\left| x \right| + 5} \over {{x^2} + 1}}} \right)$ is (– $\infty $, -a]$ \cup $[a, $\infty $). Then a is equal to :

A

${{\sqrt {17} - 1} \over 2}$

B

${{1 + \sqrt {17} } \over 2}$

C

${{\sqrt {17} } \over 2} + 1$

D

${{\sqrt {17} } \over 2}$

Question 66: Palette 66/82

JEE Mains 2019 MCQ

The value of ${\sin ^{ - 1}}\left( {{{12} \over {13}}} \right) - {\sin ^{ - 1}}\left( {{3 \over 5}} \right)$ is equal to :

A

$\pi - {\sin ^{ - 1}}\left( {{{63} \over {65}}} \right)$

B

${\pi \over 2} - {\sin ^{ - 1}}\left( {{{56} \over {65}}} \right)$

C

${\pi \over 2} - {\cos ^{ - 1}}\left( {{9 \over {65}}} \right)$

D

$\pi - {\cos ^{ - 1}}\left( {{{33} \over {65}}} \right)$

Question 67: Palette 67/82

JEE Mains 2019 MCQ

If ${\cos ^{ - 1}}x - {\cos ^{ - 1}}{y \over 2} = \alpha $,where –1 $ \le $ x $ \le $ 1, – 2 $ \le $ y $ \le $ 2, x $ \le $ ${y \over 2}$ , then for all x, y, 4x² – 4xy cos $\alpha $ + y² is equal to :

A

4 sin² $\alpha $

B

2 sin² $\alpha $

C

4 sin² $\alpha $ - 2x²y²

D

4 cos² $\alpha $ + 2x²y²

Question 68: Palette 68/82

JEE Mains 2019 MCQ

If $\alpha = {\cos ^{ - 1}}\left( {{3 \over 5}} \right)$, $\beta = {\tan ^{ - 1}}\left( {{1 \over 3}} \right)$ where $0 < \alpha ,\beta < {\pi \over 2}$ , then $\alpha $ - $\beta $ is equal to :

A

${\tan ^{ - 1}}\left( {{9 \over {14 }}} \right)$

B

${\sin ^{ - 1}}\left( {{9 \over {5\sqrt {10} }}} \right)$

C

${\cos ^{ - 1}}\left( {{9 \over {5\sqrt {10} }}} \right)$

D

${\tan ^{ - 1}}\left( {{9 \over {5\sqrt {10} }}} \right)$

Question 69: Palette 69/82

JEE Mains 2019 MCQ

Considering only the principal values of inverse functions, the set
A = { x $ \ge $ 0: tan^$-$1(2x) + tan^$-$1(3x) = ${\pi \over 4}$}

A

contains two elements

B

contains more than two elements

C

is an empty set

D

is a singleton

Question 70: Palette 70/82

JEE Mains 2019 MCQ

All x satisfying the inequality (cot^–1 x)²– 7(cot^–1 x) + 10 > 0, lie in the interval :

A

(cot 2, $\infty $)

B

(–$\infty $, cot 5) $ \cup $ (cot 2, $\infty $)

C

(cot 5, cot 4)

D

(– $\infty $, cot 5) $ \cup $ (cot 4, cot 2)

Question 71: Palette 71/82

JEE Mains 2019 MCQ

The value of $\cot \left( {\sum\limits_{n = 1}^{19} {{{\cot }^{ - 1}}} \left( {1 + \sum\limits_{p = 1}^n {2p} } \right)} \right)$ is :

A

${{22} \over {23}}$

B

${{23} \over {22}}$

C

${{21} \over {19}}$

D

${{19} \over {21}}$

Question 72: Palette 72/82

JEE Mains 2019 MCQ

If  x = sin^$-$1(sin10) and y = cos^$-$1(cos10), then y $-$ x is equal to :

A

0

B

10

C

7$\pi $

D

$\pi $

Question 73: Palette 73/82

JEE Mains 2019 MCQ

If ${\cos ^{ - 1}}\left( {{2 \over {3x}}} \right) + {\cos ^{ - 1}}\left( {{3 \over {4x}}} \right) = {\pi \over 2}$ (x > $3 \over 4$), then x is equal to :

A

${{\sqrt {145} } \over {10}}$

B

${{\sqrt {145} } \over {11}}$

C

${{\sqrt {145} } \over {12}}$

D

${{\sqrt {146} } \over {12}}$

Question 74: Palette 74/82

JEE Mains 2017 MCQ

A value of x satisfying the equation sin[cot⁻¹ (1+ x)] = cos [tan⁻¹ x], is :

A

$ - {1 \over 2}$

B

$-$ 1

C

0

D

$ {1 \over 2}$

Question 75: Palette 75/82

JEE Mains 2017 MCQ

The value of tan^-1 $\left[ {{{\sqrt {1 + {x^2}} + \sqrt {1 - {x^2}} } \over {\sqrt {1 + {x^2}} - \sqrt {1 - {x^2}} }}} \right],$ $\left| x \right| < {1 \over 2},x \ne 0,$ is equal to :

A

${\pi \over 4} + {1 \over 2}{\cos ^{ - 1}}\,{x^2}$

B

${\pi \over 4} + {\cos ^{ - 1}}\,{x^2}$

C

${\pi \over 4} - {1 \over 2}{\cos ^{ - 1}}\,{x^2}$

D

${\pi \over 4} - {\cos ^{ - 1}}\,{x^2}$

Question 76: Palette 76/82

JEE Mains 2015 MCQ

Let ${\tan ^{ - 1}}y = {\tan ^{ - 1}}x + {\tan ^{ - 1}}\left( {{{2x} \over {1 - {x^2}}}} \right),$
where $\left| x \right| < {1 \over {\sqrt 3 }}.$ Then a value of $y$ is :

A

${{3x - {x^3}} \over {1 + 3{x^2}}}$

B

${{3x + {x^3}} \over {1 + 3{x^2}}}$

C

${{3x - {x^3}} \over {1 - 3{x^2}}}$

D

${{3x + {x^3}} \over {1 - 3{x^2}}}$

Question 77: Palette 77/82

JEE Mains 2013 MCQ

If $x, y, z$ are in A.P. and ${\tan ^{ - 1}}x,{\tan ^{ - 1}}y$ and ${\tan ^{ - 1}}z$ are also in A.P., then :

A

$x=y=z$

B

$2x=3y=6z$

C

$6x=3y=2z$

D

$6x=4y=3z$

Question 78: Palette 78/82

JEE Mains 2008 MCQ

The value of $cot\left( {\cos e{c^{ - 1}}{5 \over 3} + {{\tan }^{ - 1}}{2 \over 3}} \right)$ is :

A

${{6 \over 17}}$

B

${{3 \over 17}}$

C

${{4 \over 17}}$

D

${{5 \over 17}}$

Question 79: Palette 79/82

JEE Mains 2007 MCQ

If sin^-1$\left( {{x \over 5}} \right)$ + cosec^-1$\left( {{5 \over 4}} \right)$ = ${\pi \over 2}$, then the value of x is :

A

4

B

5

C

1

D

3

Question 80: Palette 80/82

JEE Mains 2005 MCQ

If ${\cos ^{ - 1}}x - {\cos ^{ - 1}}{y \over 2} = \alpha ,$ then $4{x^2} - 4xy\cos \alpha + {y^2}$ is equal to :

A

$2\sin 2\alpha $

B

$4$

C

$4{\sin ^2}\alpha $

D

$-4{\sin ^2}\alpha $

Question 81: Palette 81/82

JEE Mains 2003 MCQ

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

A

$\left| a \right| \ge {1 \over {\sqrt 2 }}$

B

${1 \over 2} < \left| a \right| < {1 \over {\sqrt 2 }}$

C

all real values of $a$

D

$\left| a \right| \le {1 \over {\sqrt 2 }}$

Question 82: Palette 82/82

JEE Mains 2002 MCQ

${\cot ^{ - 1}}\left( {\sqrt {\cos \alpha } } \right) - {\tan ^{ - 1}}\left( {\sqrt {\cos \alpha } } \right) = x,$ then sin x is equal to :

A

${\tan ^2}\left( {{\alpha \over 2}} \right)$

B

${\cot ^2}\left( {{\alpha \over 2}} \right)$

C

$\tan \alpha $

D

$cot\left( {{\alpha \over 2}} \right)$