Chapter 6

If \(\left(\tan ^{-1} x\right)^{2}+\left(\cot ^{-1} x\right)^{2}=\frac{5 \pi^{2}}{8}\), then the value of \(x\) is (a) 0 (b) \(-1\) (c) \(-2\) (d) \(-3\)

Find the number of real roots of \(\sqrt{\sin (x)}=\cos ^{-1}(\cos x)\) in \((0,2 \pi)\).

The number of real solutions of \(\cos ^{-1} x+\cos ^{-1} 2 x=-\pi\) is (a) 0 (b) 1 (c) 2 (d) infinitely many

If \(\tan ^{-1}\left(\frac{1}{3}\right)+\tan ^{-1}\left(\frac{1}{4}\right)+\tan ^{-1}\left(\frac{1}{5}\right)+\tan ^{-1}\left(\frac{1}{n}\right)=\frac{\pi}{4}\) where \(n \in N\), then find \(n\).

Let \(a, b, c\) be positive real numbers and \(\begin{aligned} \theta=\tan ^{-1}\left(\sqrt{\frac{a(a+b+c)}{b c}}\right)+&+\tan ^{-1}\left(\sqrt{\frac{b(a+b+c)}{a c}}\right) \\ &+\tan ^{-1}\left(\sqrt{\frac{c(a+b+c)}{b a}}\right), \end{aligned}\) then the value of \(\tan \theta\) is (a) 0 (b) 1 (c) \(-1\) (d) None

The set of values of \(x\) satisfying the inequation \(\tan ^{2}\left(\sin ^{-1} x\right)>1\) is (a) \([-1,1]\) (b) \(\left[-\frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}}\right]\) (c) \((-1,1)-\left[-\frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}}\right]\) (d) \([-1,1]-\left(-\frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}}\right)\)

The value of a for which \(a x^{2}+\sin ^{-1}\left(x^{2}-2 x+2\right)+\cos ^{-1}\left(x^{2}-2 x+2\right)=0\) has a real solution, is (a) \(\pi / 2\) (b) \(-\pi / 2\) (c) \(2 / \pi\) (d) \(-2 / \pi\)

Solve the following inequalities: $$ \sin ^{-1} x>\cos ^{-1} x $$

The value of \(\sin ^{-1}\left[\cot \left(\sin ^{-1} \sqrt{\left(\frac{2-\sqrt{3}}{4}\right)}+\cos ^{-1}\left(\frac{\sqrt{12}}{4}\right)+\sec ^{-1} \sqrt{2}\right)\right]\) is (a) 0 (b) \(\pi / 4\) (c) \(\pi / 6\) (d) \(\pi / 2\)

Solve the following inequalities: $$ \cos ^{-1} x>\sin ^{-1} x $$

The number of positive integral solutions of \(\tan ^{-1} x+\cot ^{-1}\left(\frac{1}{y}\right)=\sin ^{-1}\left(\frac{3}{\sqrt{10}}\right)\) is (a) 0 (b) 1 (c) \(\underline{2}\) (d) 3

Solve the following inequalities: $$ \left(\cot ^{-1} x\right)^{2}-5\left(\cot ^{-1} x\right)+6>0 $$

The value of \(\cos \left[\frac{1}{2} \cos \left\\{\cos \left(\sin ^{-1}\left(\frac{\sqrt{63}}{8}\right)\right)\right\\}\right]\) is (a) \(\frac{3}{16}\) (b) \(\frac{3}{8}\) (c) \(\frac{3}{4}\) (d) \(\frac{3}{2}\)

Solve the following inequalities: $$ \tan ^{2}\left(\sin ^{-1} x\right)>1 $$

If \(\tan ^{-1} x+\tan ^{-1} y+\tan ^{-1} z=\pi\), then the value of \(\frac{1}{y z}+\frac{1}{z x}+\frac{1}{x y}\) is (a) 0 (b) 1 (c) \(\frac{1}{x y z}\) (d) \(x y z\)

Solve the following inequalities: $$ 4\left(\tan ^{-1} x\right)^{2}-8\left(\tan ^{-1} x\right)+3<0 $$

If \(x<0\), then \(\tan ^{-1}\left(\frac{1}{x}\right)\) is (a) \(\cot ^{-1}(x)\) (b) \(-\cot ^{-1}(x)\) (c) \(-\pi+\cot ^{-1}(x)\) (d) None

Solve the following inequalities: $$ 4 \cot ^{-1} x-\left(\cot ^{-1} x\right)^{2}-3 \geq 0 $$

The number of triplets satisfying \(\sin ^{-1} x+\cos ^{-1} y+\sin ^{-1} z=2 \pi\), is (a) 0 (b) 2 (c) 1 (d) infinite

Solve the following inequalities: $$ \sin ^{-1}\left(\sin \left(\frac{2 x^{2}+4}{1+x^{2}}\right)\right)<\pi-2 $$

If \(x^{2}+y^{2}+z^{2}=r^{2}\), then \(\tan ^{-1}\left(\frac{x y}{z r}\right)+\tan ^{-1}\left(\frac{z y}{x r}\right)+\tan ^{-1}\left(\frac{z x}{y r}\right)\) is equal to.... (a) \(\bar{\pi}\) (b) \(\pi / 2\) (c) 0 (d) None

Find the maximum value of \(f(x)=\left\\{\sin ^{-1}(\sin x)\right\\}^{2}-\sin ^{-1}(\sin x)\)

If \(\tan ^{-1} x+\tan ^{-1} 2 x+\tan ^{-1} 3 x=\pi\), then the value of \(x\) is (a) 0 (b) \(-1\) (c) 1 (d) \(\phi\)

Find the minimum value of \(f(x)=8^{\sin ^{-1} x}+8^{\cos ^{-1} x}\)

The number of solutions of the equation \(1+x^{2}+2 x \sin \left(\cos ^{-1} y\right)=0\) is (a) 1 (b) 2 (c) 3 (d) 4 .

Find the set of values of \(k\) for which \(x^{2}-k x+\sin ^{-1}(\sin 4)>0\), for all real \(x\).

If \(\alpha\) is the only real root of the equation \(x^{3}+b x^{2}+c x+1=0(b

If \(A=2 \tan ^{-1}(2 \sqrt{2}-1)\) and \(B=3 \sin ^{-1}\left(\frac{1}{3}\right)+\sin ^{-1}\left(\frac{3}{5}\right)\), then prove that \(A>B\).

If \(\alpha, \beta, \gamma\) are the roots of \(x^{3}+p x^{2}+2 x+p=0\), the general value of \(\tan ^{-1} \alpha+\tan ^{-1} \beta+\tan ^{-1} \gamma\) is (a) \(n \pi\) (b) \(n \pi / 2\) (c) \((2 n+1) \pi / 2\) (d) depend on \(p\).

If \(\sin ^{-1}\left(\frac{\sqrt{x}}{2}\right)+\sin ^{-1}\left(\sqrt{1-\frac{y}{4}}\right)+\tan ^{-1} y=\frac{2 \pi}{3}\) then find the maximum value of \(\left(x^{2}+y^{2}+1\right)\).

Find the number of integral ordered pairs \((x, y)\) satisfying the equation \(\tan ^{-1}\left(\frac{1}{x}\right)+\tan ^{-1}\left(\frac{1}{y}\right)=\tan ^{-1}\left(\frac{1}{10}\right)\).

Let \(\left[\cot \left(\sum_{k=1}^{10} \cot ^{-1}\left(k^{2}+k+1\right)\right)\right]=\frac{a}{b}\) where \(a\) and \(b\) are co-prime, then find the value of \((a+b+10)\).

If \(p>q>0, p r<-1

Consider the equation \(\left(\sin ^{-1} x\right)^{3}+\left(\cos ^{-1} x\right)^{3}=a \pi^{3}\) find the values of 'a' so that the given equation has a solution.

If the range of the function \(f(x)=\cot ^{-1}\left(\frac{x^{2}}{x^{2}+1}\right)\) is \((a, b)\), find the value of \(\left(\frac{b}{a}+2\right)\).

If \(\tan ^{-1} y=4 \tan ^{-1} x,\left(|x|<\tan \left(\frac{\pi}{8}\right)\right)\), find \(y\) as an algebraix function of \(x\) and hence prove that \(\tan \left(\frac{\pi}{8}\right)\) is a root of the equation \(x^{4}-6 x^{2}+1=0\).

Prove that \(\tan ^{-1}\left(\sqrt{\frac{a(a+b+c)}{b c}}\right)+\tan ^{-1}\left(\sqrt{\frac{b(a+b+c)}{a c}}\right)\) \(+\tan ^{-1}\left(\sqrt{\frac{c(a+b+c)}{a b}}\right)=\pi\), where \(a, b, c>0 .\)

Solve: \(\theta=\tan ^{-1}\left(2 \tan ^{2} \theta\right)-\frac{1}{2} \sin ^{-1}\left(\frac{3 \sin 2 \theta}{5+4 \cos 2 \theta}\right)\).

Simplify: \(\tan ^{-1}\left(\frac{x \cos \theta}{1-x \sin \theta}\right)-\cot ^{-1}\left(\frac{\cos \theta}{x-\sin \theta}\right)\)

Solve: \(\cos ^{-1}\left(\frac{x^{2}-1}{x^{2}+1}\right)+\sin ^{-1}\left(\frac{2 x}{x^{2}+1}\right)+\tan ^{-1}\left(\frac{2 x}{x^{2}-1}\right)=\frac{2 \pi}{3}\)

Prove that \(\tan ^{-1}\left(\frac{y z}{x r}\right)+\tan ^{-1}\left(\frac{x z}{y r}\right)+\tan ^{-1}\left(\frac{x y}{z r}\right)=\frac{\pi}{2} .\) where \(x^{2}+y^{2}+z^{2}=r^{2}\).

If \(\sum_{r=1}^{10} \tan ^{-1}\left(\frac{3}{9 r^{2}+3 r-1}\right)=\cot ^{-1}\left(\frac{m}{n}\right)\) where \(m\) and \(n\) are co-prime, find the value of \((2 m+n+4)\).

Let \(f(x)=\frac{1}{\pi}\left(\sin ^{-1} x+\cos ^{-1} x+\tan ^{-1} x\right)+\frac{(x+1)}{x^{2}+2 x+10}\) such that the maximum value of \(f(x)\) is \(m\), then find the value of \((104 m-90)\).

Let \(m\) be the number of solutions of \(\sin (2 x)+\cos (2 x)+\cos x+1=0\) in \(0

Let \(f(n)=\sum_{k=-n}^{n}\left(\cot ^{-1}\left(\frac{1}{k}\right)-\tan ^{-1}(k)\right)\) such that \(\sum_{n=2}^{10}(f(n)+f(n-1))=a \pi\) then find the value of \((a+1)\).