## Inverse Trigonometric Functions Questions and Answers Part-1

1. The value of $\cot \left[\sum_{n=1}^{23}\cot^{-1}\left(1+\sum_{k=1}^{n}2k\right)\right]$
is
a) $\frac{23}{25}$
b) $\frac{25}{23}$
c) $\frac{23}{24}$
d) $\frac{24}{23}$

Explanation:

2. If $\sin^{-1}\sqrt{x^{2}+2x+1}+\sec^{-1}\sqrt{x^{2}+2x+1}=\pi/2,x\neq 0$
then the value of $2\sec^{-1}\left(\frac{x}{2}\right)+\sin^{-1}\left(\frac{x}{2}\right)$     is equal to
a) $-\frac{3\pi}{2}$
b) $\frac{3\pi}{2}$
c) $-\frac{\pi}{2}$
d) $\frac{\pi}{2}$

Explanation:

3. If 0 < x < 1, then$\sqrt{1+x^{2}}\left[\left\{x\cos\left(\cot^{-1}x\right)+\sin\left(\cot^{-1}x\right)\right\}^{2}-1\right]^{1/2}$
is equal to
a) $\frac{x}{\sqrt{1+x^{2}}}$
b) x
c) $x\sqrt{1+x^{2}}$
d) $\sqrt{1+x^{2}}$

Explanation:

4. If $\left(\tan^{-1}x\right)^{2}+\left(\cot^{-1}x\right)^{2}=5\pi^{2}/8$
then x equal
a) 0
b) -1
c) -2
d) -3

Explanation:

5. The number of solutions of the equation $\sin^{-1}\left(\frac{1+x^{2}}{2 x}\right)=\frac{\pi}{2}\left(\sec\left(x-1\right)\right)$
is/are
a) 0
b) 1
c) 2
d) 3

Explanation:

6. The number of positive integral solutions of $\tan^{-1}x+\cot^{-1}y=\tan^{-1}3$      is
a) 1
b) 2
c) 3
d) 4

Explanation:

7. In a triangle ABC, if $\cot A=\left(x^{3}+x^{2}+x\right)^{1/2} \cot B=\left(x+x^{-1}+1\right)^{1/2} and \cot C=\left(x^{-3}+x^{-2}+x^{-1}\right)^{-1/2}$
then the triangle is
a) equilateral
b) isosceles
c) right angled
d) obtuse angled

Explanation: cot (A + B)

8. The value of $\cos\left(\frac{1}{2}\cos^{-1}\frac{1}{8}\right)$     is equal to
a) 3/4
b) -3/4
c) 1/16
d) 1/4

Explanation:

9. The inequality $\sin^{-1}\left(\sin 5\right)> x^{2}-4x$     holds if
a) $x=2-\sqrt{9-2\pi}$
b) $x=2+\sqrt{9-2\pi}$
c) $x\epsilon\left(2-\sqrt{9-2\pi},2+\sqrt{9-2\pi}\right)$
d) $x > 2+\sqrt{9-2\pi}$

10. If $1< x<\sqrt{2}$  , the number of solutions of the equation
$\tan^{-1}\left(x-1\right)+\tan^{-1}x+\tan^{-1}\left(x+1\right)=\tan^{-1}3x$