Chapter 27

If \(\mathrm{A}=\cot ^{-1} \sqrt{\tan \theta}-\tan ^{-1} \sqrt{\tan \theta}\), then \(\tan \left(\frac{\pi}{4}-\frac{\mathrm{A}}{2}\right)\) is equal to (A) \(\sqrt{\cot \theta} \theta\) (B) \(\tan \theta\) (C) \(\sqrt{\tan \theta}\) (D) none of these

\(\cos \left\\{\tan ^{-1}\left[\sin \left(\cot ^{-1} \sqrt{3}\right)\right]\right\\}\) is equal to (A) \(\frac{4}{\sqrt{5}}\) (B) \(\frac{2}{\sqrt{5}}\) (C) \(-\frac{2}{\sqrt{c}}\) (D) none of these

The value of \(\sin \left(4 \tan ^{-1} \frac{1}{3}\right)-\cos \left(2 \tan ^{-1} \frac{1}{7}\right)\) is (A) \(\frac{4}{7}\) (B) 0 (C) \(\frac{7}{8}\) (D) none of these

If \(\cos ^{-1}\left(\frac{n}{2 \pi}\right)>\frac{2 \pi}{3}\) then the minimum and the maximum values of integer \(\mathrm{n}\) are respectively. \(\begin{array}{ll}\text { (A) }-6 \text { and }-4 & \text { (B) } 4 \text { and } 6\end{array}\) (C) \(-6\) and \(-3\) (D) none of these

The value of \(\tan ^{-1} \sqrt{\frac{a(a+b+c)}{b c}}+\tan ^{-1} \sqrt{\frac{b(a+b+c)}{c a}}+\) \(\tan ^{-1} \sqrt{\frac{c(a+b+c)}{a b}}\) is (A) \(\frac{\pi}{4}\) (B) \(\frac{\pi}{2}\) (C) \(\pi\) (D) 0

The number of real solutions of the equations \(\tan ^{-1} \sqrt{x^{2}-3 x+2}+\cos ^{-1} \sqrt{4 x-x^{2}-3}=\pi\) is (A) one (B) two (C) zero (D) infinite

\(\sum_{m=1}^{n} \tan ^{-1} \frac{2 m}{m^{4}+m^{2}+2}=\) (A) \(\tan ^{-1}\left(n^{2}+n+1\right)\) (B) \(\tan ^{-1}\left(n^{2}-n+1\right)\) (C) \(\tan ^{-1} \frac{n^{2}+n}{n^{2}+n+2}\) (D) none of these

The number of solutions of the equation \(2 \sin ^{-1} \sqrt{x^{2}-x+1}+\cos ^{-1}\left(\sqrt{x^{2}-x}\right)=\frac{3 \pi}{2}\) is (A) 0 (B) infinite (C) 2 (D) 4

\(\cot ^{-1}(\sqrt{\cos \alpha})-\tan ^{-1}(\sqrt{\cos \alpha})=x\), then \(\sin x=\) (A) \(\tan ^{2}\left(\frac{\alpha}{2}\right)\) (B) \(\cot ^{2}\left(\frac{\alpha}{2}\right)\) (C) \(\tan \alpha\) (D) \(\cot \left(\frac{\alpha}{2}\right)\)

If \(\cos ^{-1} \sqrt{p}+\cos ^{-1} \sqrt{1-p}+\cos ^{-1} \sqrt{1-q}=\frac{3 \pi}{2}\), then the value of \(q\) is (A) 1 (B) \(\frac{1}{\sqrt{2}}\) (C) \(\frac{1}{3}\) (D) \(\frac{1}{2}\)

The 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 \(\cos ^{-1} \frac{x}{2}+\cos ^{-1} \frac{y}{3}=\theta\), then \(9 x^{2}-12 x y \cos \theta+4 y^{2}=\) (A) 36 (B) \(-36 \sin ^{2} \theta\) (C) \(36 \sin ^{2} \theta\) (D) \(36 \cos ^{2} \theta\)

If \(\theta=\tan ^{-1} \alpha, \varphi=\tan ^{-1} b\) and \(a b=-1\) then \(\theta-\varphi=\) (A) 0 (B) \(\frac{\pi}{4}\) (C) \(\frac{\pi}{2}\) (D) none of these

The number of real solutions of \(\tan ^{-1} \sqrt{x(x+1)}+\sin ^{-1} \sqrt{x^{2}+x+1}=\frac{\pi}{2}\) is (A) 0 (B) 1 (C) 2 (D) infinite

The number of solutions of \(\sin ^{-1} x+\sin ^{-1}(1-x)=\cos ^{-1} x\) is (A) 1 (B) 0 (C) 2 (D) 4

The domain of \(\sin ^{-1}[x]\) is given by (A) \([-1,1]\) (B) \([-1,2)\) (C) \(\\{-1,0,1\\}\) (D) none of these

If \(\mathrm{A}=\tan ^{-1}\left(\frac{x \sqrt{3}}{2 k-x}\right)\) and \(\mathrm{B}=\tan ^{-1}\left(\frac{2 x-k}{k \sqrt{3}}\right)\) then the value of \(\mathrm{A}-\mathrm{B}\) is (A) \(0^{\circ}\) (B) \(45^{\circ}\) (C) \(60^{\circ}\) (D) \(30^{\circ}\)

If \(\tan ^{-1} y=4 \tan ^{-1} x\), then \(1 / y\) is zero for (A) \(x=1 \pm \sqrt{2}\) (B) \(x=\sqrt{2} \pm \sqrt{3}\) (C) \(x=3 \pm 2 \sqrt{2}\) (D) all values of \(x\)

\(\cos ^{-1} \sqrt{\frac{a-x}{a-b}}=\sin ^{-1} \sqrt{\frac{x-b}{a-b}}\) is possible if (A) \(a>x>b\) or \(ab\) and \(x\) takes any value (D) \(a

The greater of the two angles \(A=2 \tan ^{-1}(2 \sqrt{2}-1)\) and \(B=3 \sin ^{-1} \frac{1}{3}+\sin ^{-1} \frac{3}{5}\) is (A) \(\underline{B}\) (B) \(A\) (C) \(C\) (D) none of these

If \(x=\sin \left(2 \tan ^{-1} 2\right), y=\sin \left(\frac{1}{2} \tan ^{-1} \frac{4}{3}\right)\), then (A) \(x=1-y\) (B) \(x^{2}=1-y\) (C) \(x^{2}=1+y\) (D) \(y^{2}=1-x\)

Sum of infinite terms of the series \(\cot ^{-1}\left(1^{2}+\frac{3}{4}\right)+\cot ^{-1}\left(2^{2}+\frac{3}{4}\right)+\cot ^{-1}\left(3^{2}+\frac{3}{4}\right)+\ldots\) is (A) \(\frac{\pi}{4}\) (B) \(\tan ^{-1} 2\) (C) \(\tan ^{-1} 3\) (D) none of these

The value of \(\cot \left(\operatorname{cosec}^{-1} \frac{5}{3}+\tan ^{-1} \frac{2}{3}\right)\) is (A) \(6 / 17\) (B) \(3 / 17\) (C) \(4 / 17\) (D) \(5 / 17\)

Solution of the equation \(\tan \left(\cos ^{-1} x\right)=\sin \left(\cot ^{-1} \frac{1}{2}\right)\) is (A) \(x=\pm \frac{\sqrt{7}}{3}\) (B) \(x=\pm \frac{\sqrt{5}}{3}\) (C) \(x=\pm \frac{3 \sqrt{5}}{2}\) (D) none of these

\(\cos \left[\tan ^{-1}\left[\sin \left(\cot ^{-1} x\right)\right]\right]=\) (A) \(\sqrt{\frac{x^{2}+2}{x^{2}+3}}\) (B) \(\sqrt{\frac{x^{2}+2}{x^{2}+1}}\) (C) \(\sqrt{\frac{x^{2}+1}{x^{2}+2}}\) (D) none of these

If \(\sum_{i=1}^{2 n} \cos ^{-1} x_{i}=0\), then \(\sum_{i=1}^{2 n} x_{i}\) is (A) \(n\) (B) \(2 n\) (C) \(\frac{n(n+1)}{2}\) (D) none of these

If \(\alpha=\sin ^{-1} \frac{\sqrt{3}}{2}+\sin ^{-1} \frac{1}{3}\) and \(\beta=\cos ^{-1} \frac{\sqrt{3}}{2}+\cos ^{-1} \frac{1}{3}\), then (A) \(\alpha>\beta\) (B) \(\alpha=\beta\) (C) \(\alpha<\beta\) (D) \(\alpha+\beta=2 \pi\)

If \(-1

If \(A=\cot ^{-1} \sqrt{\tan \theta}-\tan ^{-1} \sqrt{\tan \theta}\), then \(\tan \left(\frac{\pi}{4}-\frac{A}{2}\right)\) is equal to (A) \(\sqrt{\cot \theta}\) (B) \(\tan \theta\) (C) \(\sqrt{\tan \theta}\) (D) none of these

If \(\sum_{i=1}^{2 n} \sin ^{-1} x_{i}=n \pi\), then \(\sum_{i=1}^{2 n} x_{i}\) is equal to (A) \(n\) (B) \(2 n\) (C) \(\frac{n(n+1)}{2}\) (D) none of these

If \(\cos ^{-1}\left(\frac{n}{2 \pi}\right)>\frac{2 \pi}{3}\) then the minimum and the maximum values of integer \(n\) are respectively (A) \(-6\) and - 4 (B) 4 and 6 (C) \(-6\) and \(-3\) (D) none of these

The number of real solutions of the equations \(\tan ^{-1} \sqrt{x^{2}-3 x+2}+\cos ^{-1} \sqrt{4 x-x^{2}-3}=\pi\) is (A) one (B) two (C) zero (D) infinite

If \(\sin ^{-1} x+\sin ^{-1} y+\sin ^{-1} z=\frac{3 \pi}{2}\) and \(f(1)=1, f(p+q)=\) \(f(p) \cdot f(q) \mathrm{v}-p, q \in R\) then, \(x^{(1)}+y^{(2)}+z^{(3)}-\frac{x+y+z}{x^{f(1)}+y^{f(2)}+z^{f(3)}}=\) (A) 0 (B) 1 (C) 2 (D) 3

The 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 \(\tan ^{-1} y=4 \tan ^{-1} x\), then \(1 / y\) is zero for (A) \(x=1 \pm \sqrt{2}\) (B) \(x=\sqrt{2} \pm \sqrt{3}\) (C) \(x=3 \pm 2 \sqrt{2}\) (D) all values of \(x\)

The greater of the two angles \(A=2 \tan ^{-1}(2 \sqrt{2}-1)\) and \(B=3 \sin ^{-1} \frac{1}{3}+\sin ^{-1} \frac{3}{5}\) is (A) \(B\) (B) \(A\) (C) \(\bar{C}\) (D) none of these

If \(x=\sin \left(2 \tan ^{-1} 2\right), y=\sin \left(\frac{1}{2} \tan ^{-1} \frac{4}{3}\right)\), then (A) \(x=1-y\) (B) \(x^{2}=1-y\) (C) \(x^{2}=1+y\) (D) \(y^{2}=1-x\)

If \(r=x+y+z\), then \(\tan ^{-1} \sqrt{\frac{x r}{y z}}+\tan ^{-1} \sqrt{\frac{y r}{z x}}+\tan ^{-1} \sqrt{\frac{z r}{x y}}=\) (A) \(\pi\) (B) \(2 \pi\) (C) \(\frac{\pi}{2}\) (D) none of these

If \(u=\cot ^{-1} \sqrt{\cos 2 \theta}-\tan ^{-1} \sqrt{\cos 2 \theta}\), then \(\sin u=\) (A) \(\sin ^{2} \theta\) (B) \(\cos ^{2} \theta\) (C) \(\tan ^{2} \theta\) (D) \(\tan ^{2} 2 \theta\)

\(2 \tan ^{-1}\left(\tan \frac{\theta}{2} \tan \frac{\phi}{2}\right)=\) (A) \(\cos ^{-1}\left(\frac{\cos \theta+\cos \phi}{1+\cos \theta \cos \phi}\right)\) (B) \(\cos ^{-1}\left(\frac{\cos \theta-\cos \phi}{1+\cos \theta \cos \phi}\right)\) (C) \(\cos ^{-1}\left(\frac{\cos \theta+\cos \phi}{1-\cos \theta \cos \phi}\right)\) (D) none of these

Solution of the equation \(\sin ^{-1} x+\sin ^{-1} 2 x=\frac{\pi}{3}\) is (A) \(x=\frac{\sqrt{3}}{2 \sqrt{7}}\) (B) \(x=-\frac{\sqrt{3}}{2 \sqrt{7}}\) (C) \(x=\pm \frac{1}{\sqrt{2}}\) (D) none of these

If \(f(x)=2 \tan ^{-1} x+\sin ^{-1} \frac{2 x}{1+x^{2}}\), then for \(x \geq 1, f(x)\) is equal to (A) \(\pi\) (B) \(2 \pi\) (C) \(\frac{\pi}{2}\) (D) none of these

If \(\theta\) and \(\varphi\) are the roots of the equation \(8 x^{2}+22 x+5=\) 0 , then (A) both \(\sin ^{-1} \theta\) and \(\sin ^{-1} \varphi\) are real (B) both \(\sec ^{-1} \theta\) and \(\sec ^{-1} \varphi\) are real (C) both \(\tan ^{-1} \theta\) and \(\tan ^{-1} \varphi\) are real (D) none of these

The positive integral solution of the equation \(\tan ^{-1} x+\cos ^{-1}\left(\frac{y}{\sqrt{1+y^{2}}}\right)=\sin ^{-1}\left(\frac{3}{\sqrt{10}}\right)\) is (A) \(x=1, y=2\) (B) \(x=2, y=1\) (C) \(x=3, y=2\) (D) \(x=-2, y=-1\)

\(\tan ^{-1} \frac{1}{3}+\tan ^{-1} \frac{1}{7}+\tan ^{-1} \frac{1}{18}+\ldots+\tan ^{-1}\left(\frac{1}{n^{2}+n+1}\right)+\) (A) \(\frac{\pi}{2}\) (B) \(\frac{\pi}{4}\) (C) \(\frac{2 \pi}{3}\) (D) 0 .

The set of values of \(x\) for which the identity \(\cos ^{-1} x+\) \(\cos ^{-1}\left(\frac{\pi}{2}+\frac{1}{2} \sqrt{3-3 x^{2}}\right)=\frac{\pi}{3}\) holds good, is (A) \(\left[\frac{1}{2}, 1\right]\) (B) \(\left(\frac{1}{2}, 1\right)\) (C) \((0,1)\) (D) \([0,1]\)

If \(a x+b\left(\sec \left(\tan ^{-1} x\right)\right)=c\) and \(a y+b\left(\sec \left(\tan ^{-1} y\right)\right)=\) \(c\), then \(\frac{x+y}{1-x y}=\) (A) \(\frac{a c}{a^{2}-c^{2}}\) (B) \(\frac{2 a c}{a^{2}-c^{2}}\) (C) \(\frac{2 a c}{a^{2}+c^{2}}\) (D) \(\frac{a c}{a^{2}+c^{2}}\)

\(\cot ^{-1}\left(2^{2}+\frac{1}{2}\right)+\cot ^{-1}\left(2^{3}+\frac{1}{2^{2}}\right)+\cot ^{-1}\left(2^{4}+\frac{1}{2^{3}}\right)+\ldots\) \(\infty=\) (A) \(\tan ^{-1} 2\) (B) \(\cot ^{-1} 2\) (C) \(2 \tan ^{-1} 2\) (D) \(2 \tan ^{-1} 2\)

The set of values of \(x\) satisfying \(\left[\tan ^{-1} x\right]+\left[\cot ^{-1} x\right]=\) 2, where \([x]\) denotes the greatest integer less than or equal to \(x\), is (A) \((\cot 3,-\tan 1)\) (B) \((\cot 3, \cot 2)\) (C) \((\cot 3,0)\) (D) none of these

If \(a<\frac{1}{32}\), then the number of solutions of \(\left(\sin ^{-1} x\right)^{3}+\) \(\left(\cos ^{-1} x\right)^{3}=a \pi^{3}\) is (A) 0 (B) 1 (C) 2 (D) infinite