Chapter 27

\(\frac{\alpha^{3}}{2} \operatorname{cosec}^{2}\left(\frac{1}{2} \tan ^{-1} \frac{\alpha}{\beta}\right)+\frac{\beta^{3}}{2} \sec ^{2}\left(\frac{1}{2} \tan ^{-1} \frac{\beta}{\alpha}\right)\) is equal to (A) \((\alpha-\beta)\left(\alpha^{2}+\beta^{2}\right)\) (B) \((\alpha+\beta)\left(\alpha^{2}-\beta^{2}\right)\) (C) \((\alpha+\beta)\left(\alpha^{2}+\beta^{2}\right)\) (D) none of these

If \(\left(\sin ^{-1} x\right)^{2}+\left(\cos ^{-1} x\right)^{2}=\frac{5 \pi^{2}}{8}\), then \(x\) is equal to (A) 1 (B) \(-1\) (C) \(\frac{1}{\sqrt{2}}\) (D) \(-\frac{1}{\sqrt{2}}\)

Solution of the equation \(\sin \left[2 \cos ^{-1}\left\\{\cot \left(2 \tan ^{-1} x\right)\right\\}\right]=0\) is (A) \(x=\pm 1\) (B) \(1 \pm \sqrt{2}\) (C) \(-(1 \pm \sqrt{3})\) (D) \(1 \pm \sqrt{2}\)

If \(\tan ^{-1} y=4 \tan ^{-1} x\), then \(y\) is finite if (A) \(x^{2} \neq 3+2 \sqrt{2}\) (B) \(x^{2} \neq 3-2 \sqrt{2}\) (C) \(x^{4} \neq 6 x^{2}-1\) (D) \(x^{4} \neq 6 x^{2}+1\)

If \(x=\operatorname{cosec}\left(\tan ^{-1}\left(\cos \left(\cot ^{-1}\left(\sec \left(\sin ^{-1} a\right)\right)\right)\right)\right)\) and \(y=\sec \left(\cot ^{-1}\left(\sin \left(\tan ^{-1}\left(\operatorname{cosec}\left(\cos ^{-1} a\right)\right)\right)\right)\right.\), where \(a \in\) \([0,1]\). Then (A) \(x>y\) (B) \(x=y\) (C) \(y^{2}+a^{2}=3\) (D) \(x^{2}+a^{2}=3\)

If \(\cos ^{-1} x+\left(\sin ^{-1} y\right)^{2}=\frac{p \pi^{2}}{4}\) and \(\left(\cos ^{-1} x\right)\left(\sin ^{-1} y\right)^{2}=\frac{\pi^{2}}{16}\), (A) \(0 \leq p \leq \frac{4}{\pi}+1\) (B) \(p=2\) is the only integral value of \(p\) (C) \(p=0,1,2\) (integral values). (D) \(p=1\) is the only integral value of \(p\)

I. \(\cot ^{-1} 9+\operatorname{cosec}^{-1} \frac{\sqrt{41}}{4}=\) (A) \(\frac{3 \pi}{4}\) II. \(\sin ^{-1}\left(\cos \left(\sin ^{-1} x\right)\right)+\cos ^{-1}\left(\sin \left(\cos ^{-1} x\right)\right)=\) (B) \(\pi\) III. \(\cos ^{-1}\left[\cos \left\\{2 \cot ^{-1}(\sqrt{2}-1)\right\\}\right]=\) (C) \(\frac{\pi}{4}\) IV. \(\sin ^{-1} \frac{12}{13}+\cos ^{-1} \frac{4}{5}+\tan ^{-1} \frac{63}{16}=\) (D) \(\frac{\pi}{2}\)

I. The value of \(\cos \left(2 \cos ^{-1} x+\sin ^{-1} x\right)\) at \(x=\frac{1}{5}\) is (A) \(\frac{2}{3 \sqrt{5}}\) II. If \(\sin \left(\sin ^{-1} \frac{1}{5}+\cos ^{-1} x\right)=1\), then \(x=\) (B) \(-\frac{2 \sqrt{6}}{5}\) III. The value of \(\tan \left\\{\cos ^{-1}\left(-\frac{2}{7}\right)-\frac{\pi}{2}\right\\}\) is (C) \(\frac{3 \pi}{4}\) IV. If \(\sqrt{p}+\cos ^{-1} \sqrt{1-p}+\cos ^{-1} \sqrt{1-q}=\frac{3 \pi}{4}\), then \(q=\) (D) \(\frac{1}{5}\)

Assertion: If \(\cot ^{-1}(\sqrt{\cos \alpha})-\tan ^{-1}(\sqrt{\cos \alpha})=x\), then \(\sin x=\tan ^{2} \frac{\alpha}{2}\) Reason: \(\tan ^{-1} x-\tan ^{-1} y=\tan ^{-1}\left(\frac{x-y}{1+x y}\right)\)

Assertion: \(\sin ^{-1} \frac{4}{5}+\sin ^{-1} \frac{5}{13}+\sin ^{-1} \frac{16}{65}=\frac{\pi}{2}\) Reason: \(\sin ^{-1} x+\sin ^{-1} y\) \(=\sin ^{-1}\left(x \sqrt{1-y^{2}}+y \sqrt{1-x^{2}}\right)\)

10\. \(\cot ^{-1}(\sqrt{\cos \alpha})-\tan ^{-1}(\sqrt{\cos \alpha})=\mathrm{x}\), then \(\sin \mathrm{x}\) is equal to: (A)tan \(^{2}\left(\frac{\alpha}{2}\right)\) (B) \(\cot ^{2}\left(\frac{\alpha}{2}\right)\) (C) \(\tan \alpha\) (D) \(\cot \left(\frac{\alpha}{2}\right)\)

\(\tan ^{-1}\left(\frac{1}{4}\right)+\tan ^{-1}\left(\frac{2}{9}\right)\) is equal to: (A) \(\frac{1}{2} \cos ^{-1}\left(\frac{3}{5}\right)\) (B) \(\frac{1}{2} \sin ^{-1}\left(\frac{3}{5}\right)\) (C) \(\frac{1}{2} \tan ^{-1}\left(\frac{3}{5}\right)\) (D) \(\tan ^{-1}\left(\frac{1}{2}\right)\)

The trigonometric equation \(\sin ^{-1} \mathrm{x}=2 \sin ^{-1} a\), has a solution for (A) \(\frac{1}{2}<|a|<\frac{1}{\sqrt{2}}\) (B) all real values of \(a\) (C) \(|a|<\frac{1}{2}\) (D) \(|a| \leq \frac{1}{\sqrt{2}}\)

If \(\sin ^{-1}\left(\frac{x}{5}\right)+\cos e c^{-1}\left(\frac{5}{4}\right)=\frac{\pi}{2}\) then a value of \(x\) is (A) 1 (B) 3 (C) 4 (D) 5

The function \(f(x)=\tan ^{-1}(\sin x+\cos x)\) is an increas- ing function in (A) \(\left(\frac{\pi}{4}, \frac{\pi}{2}\right)\) (B) \(\left(-\frac{\pi}{2}, \frac{\pi}{4}\right)\) (C) \(\left(0, \frac{\pi}{2}\right)\) (D) \(\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)\)

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

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) \(2 x=3 y=6 z\) (B) \(6 x=3 y=2 z\) (C) \(6 x=4 y=3 z\) (D) \(x=y=z\)

Let \(\tan ^{-1} y=\tan ^{-1} x+\tan ^{-1}\left(\frac{2 x}{1-x^{2}}\right)\), where \(|x|<\frac{1}{\sqrt{3}}\), Then a value of \(y\) 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) \(\frac{3 x-x^{3}}{1-3 x^{2}}\)