34
MCQ (Single Correct Answer)
If $|x| \leq 1$, then $2 \tan ^{-1} x+\sin ^{-1}\left(\frac{2 x}{1+x^2}\right)$ is equal to
A
$4 \tan ^{-1} x$
B
0
C
$\frac{\pi}{2}$
D
$\pi$
35
MCQ (Single Correct Answer)
If $\cos ^{-1} \alpha+\cos ^{-1} \beta+\cos ^{-1} \gamma=3 \pi$, then $\alpha(\beta+\gamma)+\beta(\gamma+\alpha)+\gamma(\alpha+\beta)$ equals to
A
0
B
1
C
6
D
12
36
MCQ (Single Correct Answer)
The number of real solutions of the equation $\sqrt{1+\cos 2 x}=\sqrt{2} \cos ^{-1}(\cos x)$ in $\left[\frac{\pi}{2}, \pi\right]$ is
A
0
B
1
C
2
D
$\infty$
37
MCQ (Single Correct Answer)
If $\cos ^{-1} x>\sin ^{-1} x$, then
A
$\frac{1}{\sqrt{2}}< x \leq 1$
B
$0 \leq x<\frac{1}{\sqrt{2}}$
C
$-1 \leq x<\frac{1}{\sqrt{2}}$
D
$x>0$
38
The principal value of $\cos ^{-1}\left(-\frac{1}{2}\right)$ is ........... .
Explanation
$$\begin{gathered} \because\quad 0 \leq \cos ^{-1} x \leq \pi \\ \cos ^{-1}\left(-\frac{1}{2}\right)=\cos ^{-1}\left(\cos \frac{2 \pi}{3}\right)=\frac{2 \pi}{3} \end{gathered}$$