91Ó°ÊÓ

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) \(36 \cos ^{2} \theta\)

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 \(x\)