Embibe Experts Solutions for Chapter: Inverse Trigonometric Functions, Exercise 1: JEE Main - 26 August 2021 Shift 2

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

${\tan }^{-1}\left(\frac{1+\sqrt{3}}{3+\sqrt{3}}\right)+{\sec }^{-1}\sqrt{\frac{8+4\sqrt{3}}{6+3\sqrt{3}}}=$

If the sum of all the solutions of
${\tan }^{-1}\left(\frac{2x}{1-x^2}\right)+{\cot }^{-1}\left(\frac{1-x^2}{2x}\right)=\frac{\pi }{3}$, $-1<x<1,x\ne 0$, is $\alpha -\frac{4}{\sqrt{3}}$, then $\alpha$ is equal to _____ .

Let $a_1=1, a_2, a_3, a_4,\dots .$. be consecutive natural numbers. Then ${\tan }^{-1}\left(\frac{1}{1+a_1{a}_2}\right)+{\tan }^{-1}\left(\frac{1}{1+a_2{a}_3}\right)$ $+\dots ..+{\tan }^{-1}\left(\frac{1}{1+{\mathrm{a}}_{2021}{\mathrm{a}}_{2022}}\right)$ is equal to

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

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;

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

Let $S=\left\{x\in R:0<x<1 \text{and} 2{\tan }^{-1}\left(\frac{1-x}{1+x}\right)={\cos }^{-1}\left(\frac{1-x^2}{1+x^2}\right)\right\}$. If $n\left(S\right)$ denotes the number of elements in $S$ then :