Question

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

JEE Main 2021 (17 Mar Shift 1)

Options

• A: $1.01$
• B: $1.00$
• C: $1.02$
• D: $1.03$

Question

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

JEE Main 2021 (20 Jul Shift 1)

Options

• A: $1$
• B: $2$
• C: $4$
• D: $0$

Question

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

JEE Main 2022 (26 Jul Shift 2)

Options

• A: $4\sqrt{\left(1-x^2\right)}\left(1-2x^2\right)$
• B: $4x\sqrt{\left(1-x^2\right)}\left(1-2x^2\right)$
• C: $2x\sqrt{\left(1-x^2\right)}\left(1-4x^2\right)$
• D: $4\sqrt{\left(1-x^2\right)}\left(1-4x^2\right)$

Question

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

JEE Main 2022 (27 Jun Shift 2)

Options

• A: $\frac{25}{26}$
• B: $\frac{50}{51}$
• C: $\frac{26}{25}$
• D: $\frac{52}{51}$

Question

Let $x=\sin \left(2{\tan }^{-1}\alpha \right)$ and $y=\sin \left(\frac{1}{2}{\tan }^{-1}\frac{4}{3}\right)$. If $S=\left\{\alpha \in \mathrm{ℝ}:y^2=1-x\right\}$, then $\sum _{\alpha \in S}16{\alpha }^3$ is equal to _______.

JEE Main 2022 (25 Jul Shift 2)

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Question

Let $\left(a,b\right)\subset \left(0,2\pi \right)$ be the largest interval for which ${\sin }^{-1}\left(\sin \theta \right)-{\cos }^{-1}\left(\sin \theta \right)>0,\theta \in \left(0,2\pi \right)$, holds . If $\alpha x^2+\beta x+{\sin }^{-1}\left(x^2-6x+10\right)+{\cos }^{-1}\left(x^2-6x+10\right)=0$ and $\alpha -\beta =b-a$, then $\alpha$ is equal to;

JEE Main 2023 (31 Jan Shift 2)

Options

• A: $\frac{\pi }{8}$
• B: $\frac{\pi }{48}$
• C: $\frac{\pi }{16}$
• D: $\frac{\pi }{12}$

Question

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

JEE Main 2024 (04 Apr Shift 2)

Options

• A: 0
• B: -1
• C: $\frac{1}{2}$
• D: $-\frac{1}{2}$

Question

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

JEE Main 2024 (06 Apr Shift 1)

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Question

For $\alpha ,\beta ,\gamma \ne 0$. If ${\sin }^{-1}\alpha +{\sin }^{-1}\beta +{\sin }^{-1}\gamma =\pi$ and $\left(\alpha +\beta +\gamma \right)\left(\alpha -\gamma +\beta \right)=3\alpha \beta$, then $\gamma$ equal to

JEE Main 2024 (31 Jan Shift 1)

Options

• A: $\frac{\sqrt{3}}{2}$
• B: $\frac{1}{\sqrt{2}}$
• C: $\frac{\sqrt{3}-1}{2\sqrt{2}}$
• D: $\sqrt{3}$