Inverse Function

Q. Find the value of $\sin \frac{31\pi }{3}$.

Q. Which of the following is the principal value branch of ${\tan }^{-1}x$

1. $(-\frac{\pi }{2},\frac{\pi }{2})$
2. $[-\frac{\pi }{2},\frac{\pi }{2}]$
3. $(-\frac{\pi }{2},\frac{\pi }{2})-\left\{0\right\}$
4. $(0,\pi )$

Q. $e^{({\cos }^{2}x+{\cos }^{4}x+{\cos }^{6}x+\cdots \infty ){\log }_{e}2}$ satisfies the equation $t^2–9t+8=0,$ then the value of $\frac{2\sin x}{\sin x+\sqrt{3}\cos x},(0<x<\frac{\pi }{2})$ is :

1. $\frac{3}{2}$
2. $2\sqrt{3}$
3. $\frac{1}{2}$
4. $\sqrt{3}$

Q. Complete solution set $\left[{\cot }^{-1}x\right]+2\left[{\tan }^{-1}x\right]=0$, where $[.]$ denotes the greatest integer function, is

1. $(0,\cot 1)$
2. $(0,\tan 1)$
3. $(\tan 1,\infty )$
4. $(\cot 1,\tan 1)$

Q. If ${\cot }^{-1}\frac{n^2-10n+21.6}{\pi }>\frac{\pi }{6},n\in N$, then $n$ can be

1. $3$
2. $2$
3. $4$
4. $8$

Q.

If 8x=π, show that cos7x + cosx =0

Q. The range of the function, $f\left(x\right)={\cot }^{-1}x+{\sec }^{-1}x+{\text{cosec}}^{-1}x$ is

1. $(\frac{\pi }{2},\frac{3\pi }{2})$
2. $(\frac{\pi }{2},\frac{3\pi }{4}]\cup [\frac{5\pi }{4},\frac{3\pi }{2})$
3. $[\frac{\pi }{2},\pi )\cup (\pi ,\frac{3\pi }{2}]$
4. $(\frac{\pi }{2},\pi )\cup (\pi ,\frac{3\pi }{2})$

Q. If $g\left(x\right)={(4{\cos }^{4}x-2\cos 2x-\frac{\cos 4x}{2}-x^7)}^{1/7}$ for $x\in R,$ then the value of $g\left(g\left(100\right)\right)$ is equal to

1. value of $50\left(\frac{{\cos }^2{1}^{\circ }-{\cos }^2{2}^{\circ }}{\sin 3^{\circ }\sin 1^{\circ }}\right)$
2. value of $\frac{200}{3}\sum _{n=1}^{\infty }\left(\frac{5^n-2^n}{{10}^n}\right)$
3. value of $\frac{400}{3}\sum _{n=1}^{\infty }\left(\frac{5^n-2^n}{{10}^n}\right)$
4. value of $50\left({12}^{\left(\frac{1-a-b}{2(1-b)}\right)}\right)$ if ${60}^a=3$ and ${60}^b=5$

Q.

Prove that cos4x = cos2x

Q. If $f\left(x\right)=\frac{2{\sin }^{2}x+2\sin x+3}{{\sin }^{2}x+\sin x+1},$ then the number of integer(s) in the range of $f$ is

1. $7$
2. $3$
3. $1$
4. $4$

Q. The domain of the function $f\left(x\right)={\sin }^{-1}\left(5x\right)$ is

1. $[-\frac{\pi }{5},\frac{\pi }{5}]$
2. $[-\frac{\pi }{10},\frac{\pi }{10}]$
3. $R$
4. $[-\frac{1}{5},\frac{1}{5}]$

Q. If ${\text{cot}}^{-1}\left[{\left(\text{cos}\alpha \right)}^{1/2}\right]-{\text{tan}}^{-1}\left[{\left(\text{cos}\alpha \right)}^{1/2}\right]=x,$ then $\sin x$ equals

1. ${\tan }^2\left(\frac{\alpha }{2}\right)$
2. $\text{tan}\alpha$
3. $\text{cot}\left(\frac{\alpha }{2}\right)$
4. ${\text{cot}}^2\left(\frac{\alpha }{2}\right)$

Q.
The value of $\frac{3+\cot {76}^{\circ }\cot {16}^{\circ }}{\cot {76}^{\circ }+\cot {16}^{\circ }}$ is

1. $\tan {44}^{\circ }$
2. $\cot {46}^{\circ }$
3. $\tan 2^{\circ }$
4. $\tan {46}^{\circ }$

Q. Evaluate each of the following:

(i) ${\sec }^{-1}\left(\sec \frac{\mathrm{\pi }}{3}\right)$
(ii) ${\sec }^{-1}\left(\sec \frac{2\mathrm{\pi }}{3}\right)$
(iii) ${\sec }^{-1}\left(\sec \frac{5\mathrm{\pi }}{4}\right)$
(iv) ${\sec }^{-1}\left(\sec \frac{7\mathrm{\pi }}{3}\right)$

(v) ${\sec }^{-1}\left(\sec \frac{9\mathrm{\pi }}{5}\right)$
(vi) ${\sec }^{-1}\left\{\sec \left(-\frac{7\mathrm{\pi }}{3}\right)\right\}$
(vii) ${\sec }^{-1}\left(\sec \frac{13\mathrm{\pi }}{4}\right)$
(viii) ${\sec }^{-1}\left(\sec \frac{25\mathrm{\pi }}{6}\right)$

Q. If $co{s}^{-1}x+co{s}^{-1}y+co{s}^{-1}z=\pi$, then $x^2+y^2+z^2+2xyz$ is equal to

1. 0
2. -1
3. 1
4. 5

Q. $\begin{array}{ccccc}& \text{List - 1}& & \text{List - 2}& \\ \text{(I)}& \text{If}f\left(x\right)=e^{\left[x\right]}\text{and}g\left(x\right)=\frac{x^2-4x+3}{x^2-2x+3},& \text{(P)}& 0& \\ & \text{then number of integer(s) in the range of}(f\circ g)\left(x\right)\text{is}& & & \\ & \text{(where [.] represents the greatest integer function)}& & & \\ \text{(II)}& \underset{0}{\overset{4}{\int }}\frac{z^{18}-1}{\sum _{n=0}^{17}{z}^n}dz& \left(Q\right)& 1& \\ \text{(III)}& \text{In a}\Delta XYZ,y^2\sin \left(2Z\right)+z^2\sin \left(2Y\right)=2yz,& \text{(R)}& 2& \\ & \text{where}y=15,z=8\text{. Then the length of inradius is}& & & \\ \text{(IV)}& \text{The number of integers in the range of the function}& \text{(S)}& 3& \\ & f\left(x\right)=\sqrt{{\sin }^{-1}x-{\cos }^{-1}x}+\sqrt{{\tan }^{-1}x-{\cot }^{-1}x}\text{is}& & & \\ & & \text{(T)}& 4& \\ & & \text{(U)}& 5& \end{array}$

Which of the following has CORRECT pair of combination?

1. $\left(I\right)\to \left(T\right)$, $\left(III\right)\to \left(P\right)$
2. $\left(I\right)\to \left(Q\right)$, $\left(II\right)\to \left(T\right)$
3. $\left(II\right)\to \left(U\right)$, $\left(IV\right)\to \left(R\right)$
4. $\left(III\right)\to \left(S\right)$, $\left(IV\right)\to \left(Q\right)$

Q. The principal value of ${\sin }^{-1}(-\frac{\sqrt{3}}{2})$ is

1. $\frac{\pi }{3}$
2. $-\frac{\pi }{3}$
3. $\frac{\pi }{6}$
4. $-\frac{\pi }{6}$

Q. The values of $x$ for which ${\tan }^{-1}x>{\cot }^{-1}x$ is

1. $x\in (0,1)$
2. $x\in (-1,1)$
3. $x\in (-\infty ,-1)$
4. $x\in (1,\infty )$

Q.

If $cos(si{n}^{-1}\frac{2}{5}+co{s}^{-1}x)=0,$ then x is equal to

(a) $\frac{1}{5}$ (b) $\frac{2}{5}$ (c) 0 (d) 1

Q. Consider two functions $f$ and $g$ defined by $f\left(x\right)={\sin }^{-1}x+{\tan }^{-1}x$ and $g\left(x\right)={\cos }^{-1}x+{\cot }^{-1}x.$ Let $A$ and $B$ be sets of non-negative integers in $R_f$ and $R_g$ respectively, where $R_f$ is the range of $f$ and $R_g$ is the range of $g.$ Then

1. number of real solutions of $f\left(x\right)=g\left(x\right)$ is one.
2. if $R_f$ is $[a\pi ,b\pi ]$ and $R_g$ is $[c\pi ,d\pi ],$ then $\frac{c+d}{b-a}=\frac{4}{3}.$
3. number of one-one functions from $A$ to $B$ is $60.$
4. number of values of $x$ satisfying $|f\left(x\right)|+|g\left(x\right)|=\pi$ is infinitely many.

Q. If $co{s}^{-1}x+co{s}^{-1}y+co{s}^{-1}z=\pi$, then

1. $x^2+y^2+z^2+xyz=0$
2. $x^2+y^2+z^2+2xyz=0$
3. $x^2+y^2+z^2+2xyz=1$
4. $x^2+y^2+z^2+xyz=1$

Q. If ${\cos }^{-1}x+{\cos }^{-1}y+{\cos }^{-1}z=\pi$, prove that $x^2+y^2+z^2+2xyz=1$

Q. If the range of the function $f\left(x\right)=8({\sin }^{4}x+{\cos }^{4}x-\sin x\cos x)\forall x\in R$ is $[a,b],$ then the value of ${\left(\underset{x\to a}{lim}\frac{{(3x+b-1)}^{1/3}-{(b-1)}^{1/3}}{x-a}\right)}^{-1}$ is

Q. If $\int \frac{\cos 2x+\sin 2x}{{(2\cos x-\sin x)}^2}dx=\frac{-A}{1725}x-\frac{2}{5}\log |2\cos x-\sin x|+2\frac{1}{2-\tan x}+C$ then A is equal to

Q. The principal value of ${\sin }^{-1}(-\frac{\sqrt{3}}{2})$ is

1. $\frac{\pi }{3}$
2. $-\frac{\pi }{3}$
3. $\frac{\pi }{6}$
4. $-\frac{\pi }{6}$

Q. If $y={\tan }^{-1}\left(\frac{\cos x-\sin x}{\cos x+\sin x}\right)$, then $\frac{dy}{dx}=$

1. $-1$
2. $\sin 2\pi$
3. $\cos 2x$
4. Zero

Q. The sum of three positive numbers $\alpha ,\beta ,\gamma$ is equal to $\frac{\pi }{2}.$ If $e^{\cot \alpha },e^{\cot \beta }$ and $e^{\cot \gamma }$ form a geometric progression, then which of the following hold(s) good?

1. $\sum \cot \alpha =\prod \cot \alpha$
2. $2\cot \beta =\cot \alpha +\cot \gamma$
3. $\cot \alpha \cot \gamma =3$
4. $4\prod \cos \alpha =\sum \sin 2\alpha$

Q. The number of positive integral solutions of ${\tan }^{-1}x+{\cos }^{-1}\frac{y}{\sqrt{1+y^2}}={\sin }^{-1}\frac{3}{\sqrt{10}}$ is

1. $1$
2. $2$
3. $3$
4. $4$

Q. If $S_n={\cot }^{-1}\left(3\right)+{\cot }^{-1}\left(7\right)+{\cot }^{-1}\left(13\right)+{\cot }^{-1}\left(21\right)+\dots n\text{terms}$, then

1. $S_{10}={\tan }^{-1}\frac{5}{6}$
2. $S_{\infty }=\frac{\pi }{4}$
3. $S_6={\sin }^{-1}\frac{3}{5}$
4. $S_{20}={\cot }^{-1}\left(1.1\right)$

Q. If the value of integral $\int {\sin }^{4}x{\cos }^{2}xdx=\frac{x}{p}+\frac{\sin 2x}{q}+\frac{\sin 4x}{r}+\frac{\sin 6x}{s}+C,$ for fixed constants $p,q,r$ and $s.$ Then the value of $\frac{p+q+r+s}{4}=$
(where $C$ is integration constant)