Question

If $m\le 3{\sin }^{-1}x+2{\cos }^{-1}x\le n,$ then the value of $m$ and $n$ are, respectively,

• A. $\frac{\pi }{2},\frac{3\pi }{2}$
• B. $\frac{-\pi }{2},\frac{3\pi }{2}$
• C. $\frac{\pi }{2},\frac{-3\pi }{2}$
• D. $-\pi ,\pi$

Answer 1

The correct option is A $\frac{\pi }{2},\frac{3\pi }{2}$

$m\le 3{\sin }^{-1}x+2{\cos }^{-1}x\le n,$
We know ${\sin }^{-1}x+{\cos }^{-1}x=\frac{\pi }{2}$
$\Rightarrow 2{\sin }^{-1}x+2{\cos }^{-1}x=\pi \cdots \left(1\right)$
$\text{Also,}\frac{-\pi }{2}\le {\sin }^{1}x\le \frac{\pi }{2}\cdots \left(2\right)$
$\text{Adding}\left(1\right)\&\left(2\right),$ we get
$\frac{\pi }{2}\le 3{\sin }^{-1}x+2{\cos }^{-1}x\le \frac{3\pi }{2}$