Q JEE Main 2024

Let the inverse trigonometric functions take principal values. The number of real solutions of the equation $...

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Q JEE Main 2024

If $\int \operatorname{cosec}^5 x d x=\alpha \cot x \operatorname{cosec} x\left(\operatorname{cosec}^2 x+\frac{3}{2}\right)+\beta \log _e\left|\tan \frac{x}{2}\right|+C$ where $\alpha, \beta \in \mathbb{R}$ and...

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Q JEE Main 2024

Given the inverse trigonometric function assumes principal values only. Let $x, y$ be any two real numbers in $[-1,1]$ such...

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Q JEE MAIN 2026

If $k=\tan \left(\frac{\pi}{4}+\frac{1}{2} \cos ^{-1}\left(\frac{2}{3}\right)\right)+\tan \left(\frac{1}{2} \sin ^{-1}\left(\frac{2}{3}\right)\right)$, then the number of solutions of the equation $...

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Q JEE MAIN 2026

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...

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Q JEE MAIN_2026_

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

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Q JEE MAIN 2026

Let the maximum value of $\left(\sin ^{-1} x\right)^2+\left(\cos ^{-1} x\right)^2$ for $x \in\left[-\frac{\sqrt{3}}{2}, \frac{1}{\sqrt{2}}\right]$ be $\frac{m}{n} \pi^2$, where $\operatorname{gcd}(m, n)=1$....

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Q JEE MAIN 2026

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 $...

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Q JEE MAIN 2020

If $y=\sum_{k=1}^6 k \cos ^{-1}\left\{\frac{3}{5} \cos k x-\frac{4}{5} \sin k x\right\}$, then $\frac{d y}{d x}$ at $x=0$ is

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Q JEE-MAIN 2019

Considering only the principal values of inverse functions, the set $A=\left\{x \geq 0 ; \tan ^{-1}(2 x)+\tan ^{-1}(3 x)=\frac{\pi}{4}\right\}$

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Q JEE MAIN 2019

If $x=\sin ^{-1}(\sin 10)$ and $y=\cos ^{-1}(\cos 10)$, then $y-x$ is equal to

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Q JEE MAIN 2020

If S is the sum of the first 10 terms of the series $\tan ^{-1}\left(\frac{1}{3}\right)+\tan ^{-1}\left(\frac{1}{7}\right)+\tan ^{-1}\left(\frac{1}{13}\right)+\tan ^{-1}\left(\frac{1}{21}\right)+\ldots \ldots$ then...

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Q JEE MAIN 2019

If $\cos ^{-1} x-\cos ^{-1} \frac{y}{2}=\alpha$, where $1 \leq x \leq 1,-2 \leq y \leq 2$, $x \leq \frac{y}{2}$, then...

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Q JEE MAIN 2019

All $x$ satisfying the inequality $\left(\cot ^{-1} x\right)^2-7\left(\cot ^{-1} x\right)+10>0$, lie in the interval

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Q JEE MAIN 2019

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

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Q JEE Main 2021

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

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Q JEE MAIN 2021

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Q JEE Main 2019

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

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Q JEE MAIN 2021

If $\sum_{r=1}^{50} \tan ^{-1} \frac{1}{2 r^2}=P$, then the value of $\tan p$ is :

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Q JEE MAIN 2019

If $2 y=\left(\cot ^{-1}\left(\frac{\sqrt{3} \cos x+\sin x}{\cos x-\sqrt{3} \sin x}\right)\right)^2, x \in\left(0, \frac{\pi}{2}\right)$ then $\frac{\mathrm{dy}}{\mathrm{dx}}$ is equal to:

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Q JEE MAIN 2021 S2

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

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Q JEE MAIN 2021

If $0

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Q JEE MAIN 2021

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

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Q JEE MAIN 2021

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

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Q JEE-MAIN 2021

The number of solutions of the equation $$ \sin ^{-1}\left[x^2+\frac{1}{3}\right]+\cos ^{-1}\left[x^2-\frac{2}{3}\right]=x^2 $$ for $x \in[-1,1]$ and $[x]$ denotes the greatest...

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Q JEE MAIN 2021

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

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Q JEE MAIN 2021

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

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Q JEE MAIN 2021

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

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Q JEE MAIN 2021

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Q JEE MAIN 2022

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

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Q JEE MAIN 2022

The value of $\lim _{n \rightarrow \infty} 6 \tan \left\{\sum_{r=1}^n \tan ^{-1}\left(\frac{1}{r^2+3 r+3}\right)\right\}$

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Q JEE MAIN 2022

If the inverse trigonometric functions take principal values, then $\cos ^{-1}\left(\frac{3}{10} \cos \left(\tan ^{-1}\left(\frac{4}{3}\right)\right)+\frac{2}{5} \sin \left(\tan ^{-1}\left(\frac{4}{3}\right)\right)\right)$ is equal to...

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Q JEE MAIN 2022

If $y=\tan ^{-1}\left(\sec x^3-\tan x^3\right)$. $\frac{\pi}{2}$< $x^3$< $\frac{3 \pi}{2}$, then

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Q JEE MAIN 2022

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

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Q JEE MAIN 2022

Let $x * y=x^2+y^3$ and $(x * 1) * 1=x *(1 * 1)$. Then a value of $2 \sin ^{-1}\left(\frac{x^4+x^2-2}{x^4+x^2+2}\right)$

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Q JEE MAIN 2022

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

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Q JEE MAIN 2022

The set of all values of $k$ for which $\left(\tan ^{-1} x\right)^3+\left(\cot ^{-1} x\right)^3=k \pi^3, x \in R$, is the...

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Q JEE MAIN 2023

For $\mathrm{x} \in(-1,1]$, the number of solutions of the equation $\sin ^{-1} \mathrm{x}=2 \tan ^{-1} \mathrm{x}$ is equal to

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Q JEE MAIN 2025

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

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Q JEE MAIN 2025

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

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Q JEE MAIN 2025

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 :

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Q JEE MAIN 2023

If $S=\left\{x \in \mathbb{R}: \sin ^{-1}\left(\frac{x+1}{\sqrt{x^2+2 x+2}}\right)-\sin ^{-1}\left(\frac{x}{\sqrt{x^2+1}}\right)=\frac{\pi}{4}\right\}$, then $\sum_{x \in \mathbb{R}}\left(\sin \left(x^2+x+5\right) \frac{\pi}{2}\right)-\cos \left(\left(x^2+x+5\right) \pi\right)$ is equal to $\_\_\_\_$...

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Q JEE-Main 2024

For $n \in N$, if $\cot ^{-1} 3+\cot ^{-1} 4+\cot ^{-1} 5+\cot ^1 n=\frac{\pi}{4}$, then $n$ is equal to $\_\_\_\_$...

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Q JEE MAIN 2024

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 $\gamma$...

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Q JEE MAIN 2024

If the domain of the function f(x)=cos^(-1)⁡((2-|x|)/4)+(log_e⁡(3-x))^(-1) is [-α,β)-{y}, then α+β+γ is equal to :

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Q JEE MAIN 2025

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

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Q JEE MAIN 2025

Considering the principal values of the inverse trigonometric functions, $...

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Q JEE MAIN 2025

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 :

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Q JEE MAIN 2025

Let $\mathrm{S}=\left\{x: \cos ^{-1} x=\pi+\sin ^{-1} x+\sin ^{-1}(2 x+1)\right\}$. Then $\sum_{x \in \mathrm{~S}}(2 x-1)^2$ is equal to ______.

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Q JEE MAIN 2025

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

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Q JEE MAIN 2025

If for some $\alpha ,\beta ;\alpha \le \beta ,\alpha + \beta = 8$ and $...

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Q JEE MAIN 2025

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

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Q JEE MAIN 2025

Using the principal values of the inverse trigonometric functions, the sum of the maximum and the minimum values of $...

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