Chapter 17

If \(\alpha=\cos ^{-1}\left(\frac{3}{5}\right), \beta=\tan ^{-1}\left(\frac{1}{3}\right)\), where \(0<\alpha, \beta<\frac{\pi}{2}\), then \(\alpha-\beta\) is equal to : (a) \(\tan ^{-1}\left(\frac{9}{5 \sqrt{10}}\right)\) (b) \(\cos ^{-1}\left(\frac{9}{5 \sqrt{10}}\right)\) (c) \(\tan ^{-1}\left(\frac{9}{14}\right)\) (d) \(\sin ^{-1}\left(\frac{9}{5 \sqrt{10}}\right)\)

A value of \(x\) satisfying the equation \(\sin \left[\cot ^{-1}(1+x)\right]=\cos\) \(\left[\tan ^{-1} x\right]\), is : (a) \(-\frac{1}{2}\) (b) \(-1\) (c) 0 (d) \(\frac{1}{2}\)

The principal value of \(\tan ^{-1}\left(\cot \frac{43 \pi}{4}\right)\) is: (a) \(-\frac{3 \pi}{4}\) (b) \(\frac{3 \pi}{4}\) (c) \(-\frac{\pi}{4}\) (d) \(\frac{\pi}{4}\)

4 \(\Delta\) 4 The number of solutions of the equation, \(\sin ^{-1} x=2 \tan ^{-1} x\) (in principal values) is : (a) 1 (b) 4 (c) 2 (d) 3

A value of \(\tan ^{-1}\left(\sin \left(\cos ^{-1}\left(\sqrt{\frac{2}{3}}\right)\right)\right)\) is (a) \(\frac{\pi}{4}\) (b) \(\frac{\pi}{2}\) (c) \(\frac{\pi}{3}\) (d) \(\frac{\pi}{6}\)

The largest interval lying in \(\left(\frac{-\pi}{2}, \frac{\pi}{2}\right)\) for which the function, \(f(x)=4^{-x^{2}}+\cos ^{-1}\left(\frac{x}{2}-1\right)+\log (\cos x)\), is defined, is (a) \(\left[-\frac{\pi}{4}, \frac{\pi}{2}\right)\) (b) \(\left[0, \frac{\pi}{2}\right)\) (c) \([0, \pi]\) (d) \(\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)\)

The trigonometric equation \(\sin ^{-1} x=2 \sin ^{-1} a\) has a solution for \(\quad\) [2003] (a) \(|a| \leq \frac{1}{\sqrt{2}}\) (b) \(\frac{1}{2}<|a|<\frac{1}{\sqrt{2}}\) (c) all real values of a (d) \(|a|<\frac{1}{2}\)

The domain of \(\sin ^{-1}\left[\log _{3}(x / 3)\right]\) is (a) \([1,9]\) (b) \([-1,9]\) (c) \([-9,1]\) (d) \([-9,-1]\)

\(2 \pi-\left(\sin ^{-1} \frac{4}{5}+\sin ^{-1} \frac{5}{13}+\sin ^{-1} \frac{16}{65}\right)\) is equal to : (a) \(\frac{\pi}{2}\) (b) \(\frac{5 \pi}{4}\) (c) \(\frac{3 \pi}{2}\) (d) \(\frac{7 \pi}{4}\)

If \(\mathrm{S}\) is the sum of the first 10 terms of the series \(\tan ^{-1}\left(\frac{1}{3}\right)+\tan ^{-1}\left(\frac{1}{7}\right)+\tan ^{-1}\left(\frac{1}{13}\right)+\tan ^{-1}\left(\frac{1}{21}\right)+\ldots\) then \(\tan (\mathrm{S})\) is equal to: (a) \(\frac{5}{6}\) (b) \(\frac{5}{11}\) (c) \(-\frac{6}{5}\) (d) \(\frac{10}{11}\)

The value of \(\sin ^{-1}\left(\frac{12}{13}\right)-\sin ^{-1}\left(\frac{3}{5}\right)\) is equal to : (a) \(\pi-\sin ^{-1}\left(\frac{63}{65}\right)\) (b) \(\frac{\pi}{2}-\sin ^{-1}\left(\frac{56}{65}\right)\) (c) \(\frac{\pi}{2}-\cos ^{-1}\left(\frac{9}{65}\right)\) (d) \(\pi-\cos ^{-1}\left(\frac{33}{65}\right)\)

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 for all \(x, y, 4 x^{2}-4 x y \cos \alpha+y^{2}\) is equal to: (a) \(4 \sin ^{2} \alpha\) (b) \(2 \sin ^{2} \alpha\) (c) \(4 \sin ^{2} \alpha-2 x^{2} y^{2}\) (d) \(4 \cos ^{2} \alpha+2 x^{2} y^{2}\)

Considering only the principal values of inverse functions, the set \(A=\left\\{x \geq 0: \tan ^{-1}(2 x)+\tan ^{-1}(3 x)=\frac{\pi}{4}\right\\}\) (a) contains two elements (b) contains more than two elements (c) is a singleton (d) is an empty set

All \(x\) satisfying the inequality \(\left(\cot ^{-1} x\right)^{2}-7\left(\cot ^{-1} x\right)+10>\) 0 , lie in the interval: (a) \((-\infty, \cot 5) \cup(\cot 4, \cot 2)\) (b) \((\cot 2, \infty)\) (c) \((-\infty, \cot 5) \cup(\cot 2, \infty)\) (d) \((\cot 5, \cot 4)\)

The value of \(\cot \left(\sum_{n=1}^{19} \cot ^{-1}\left(1+\sum_{p=1}^{n} 2 p\right)\right)\) is: (a) \(\frac{21}{19}\) (b) \(\frac{19}{21}\) (c) \(\frac{22}{23}\) (d) \(\frac{23}{22}\)

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: (a) \(\frac{\sqrt{145}}{12}\) (b) \(\frac{\sqrt{145}}{10}\) (c) \(\frac{\sqrt{146}}{12}\) (d) \(\frac{\sqrt{145}}{11}\)

The value of \(\tan ^{-1}\left[\frac{\sqrt{1+x^{2}}+\sqrt{1-x^{2}}}{\sqrt{1+x^{2}}-\sqrt{1-x^{2}}}\right],|x|<\frac{1}{2}, x \neq 0\), is equal to [Online April 8, 2017] (a) \(\frac{\pi}{4}+\frac{1}{2} \cos ^{-1} \mathrm{x}^{2}\) (b) \(\frac{\pi}{4}+\cos ^{-1} x^{2}\) (c) \(\frac{\pi}{4}-\frac{1}{2} \cos ^{-1} \mathrm{x}^{2}\) (d) \(\frac{\pi}{4}-\cos ^{-1} x^{2}\)

Let \(\tan ^{-1} \mathrm{y}=\tan ^{-1} \mathrm{x}+\tan ^{-1}\left(\frac{2 \mathrm{x}}{1-\mathrm{x}^{2}}\right)\) where or \(|\mathrm{x}|<\frac{1}{\sqrt{3}}\). Then a value of \(\mathrm{y}\) is : (a) \(\frac{3 x-x^{3}}{1+3 x^{2}}\) (b) \(\frac{3 x+x^{3}}{1+3 x^{2}}\) (c) \(\frac{3 x-x^{3}}{1-3 x^{2}}\) (d) \(\frac{3 x+x^{3}}{1-3 x^{2}}\)

If \(f(x)=2 \tan ^{-1} x+\sin ^{-1}\left(\frac{2 x}{1+x^{2}}\right), x>1\) then \(f(5)\) is equal to: (a) \(\tan ^{-1}\left(\frac{65}{156}\right)\) (b) \(\frac{\pi}{2}\) (c) \(\pi\) (d) \(4 \tan ^{-1}(5)\)

Statement \(\mathbf{I}:\) The equation \(\left(\sin ^{-1} x\right)^{3}+\left(\cos ^{-1} x\right)^{3}-a \pi^{3}=0\) has a solution for all \(\mathrm{a} \geq \frac{1}{32}\). Statement II: For any \(x \in R, \sin ^{-1} x+\cos ^{-1} x=\frac{\pi}{2}\) and \(0 \leq\left(\sin ^{-1} x-\frac{\pi}{4}\right)^{2} \leq \frac{9 \pi^{2}}{16} \quad\) [Online April 12, 2014] (a) Both statements I and II are true. (b) Both statements I and II are false. (c) Statement I is true and statement II is false. (d) Statement I is false and statement II is true.

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) \(2 x=3 y=6 z\) (c) \(6 x=3 y=2 z\) (d) \(6 x=4 y=3 z\)

Let \(x \in(0,1) .\) The set of all \(x\) such that \(\sin ^{-1} x>\cos ^{-1} x\), is the interval: \([\) Online April \(25,2013 \mid\) (a) \(\left(\frac{1}{2}, \frac{1}{\sqrt{2}}\right)\) (b) \(\left(\frac{1}{\sqrt{2}}, 1\right)\) (c) \((0,1)\) (d) \(\left(0, \frac{\sqrt{3}}{2}\right)\)

\(S=\tan ^{-1}\left(\frac{1}{n^{2}+n+1}\right)+\tan ^{-1}\left(\frac{1}{n^{2}+3 n+3}\right)+\ldots\) \(+\tan ^{-1}\left(\frac{1}{1+(n+19)(n+20)}\right)\), then \(\tan S\) is equal to: (a) \(\frac{20}{401+20 n}\) (b) \(\frac{n}{n^{2}+20 n+1}\) (c) \(\frac{20}{n^{2}+20 n+1}\) (d) \(\frac{n}{401+20 n}\)

A value of \(x\) for which \(\sin \left(\cot ^{-1}(1+x)\right)=\cos \left(\tan ^{-1} x\right)\), is (a) \(-\frac{1}{2}\) (b) 1 (c) 0 (d) \(\frac{1}{2}\)

If \(\sin ^{-1}\left(\frac{x}{5}\right)+\operatorname{cosec}^{-1}\left(\frac{5}{4}\right)=\frac{\pi}{2}\), then the values of \(\mathrm{x}\) is (a) 4 (b) \(5 \) (c) 1 (d) 3

If \(\cos ^{-1} x-\cos ^{-1} \frac{y}{2}=\alpha\), then \(4 x^{2}-4 x y \cos \alpha+y^{2}\) is equal to (a) \(2 \sin 2 \alpha\) (b) 4 (c) \(4 \sin ^{2} \alpha\) (d) \(-4 \sin ^{2} \alpha\)