Question

If ${\log }_{\frac{1}{\sqrt{2}}}\sin x\ge 0$, $x\in [0,4\pi ]$, then the number of values of $x$ which are integral multiples of $\frac{\pi }{4}$ is/are

• A. $6$
• B. $12$
• C. $3$
• D. none of these

Answer 1

The correct option is A $6$

${\log }_{\frac{1}{\sqrt{2}}}\sin x\ge 0$, where $x\in [0,4\pi ]$
Above equation is valid when $\sin x>0$
$\Rightarrow x\in (0,\pi )\cup (2\pi ,3\pi )$
${\log }_{\frac{1}{\sqrt{2}}}sinx\ge 0$
$\Rightarrow \sin x\le 1$
For, $x\in (0,\pi )\cup (2\pi ,3\pi )$
Values of $x$ which are integral multiples of $\frac{\pi }{4}$ are $\{\frac{\pi }{4},\frac{2\pi }{4},\frac{3\pi }{4},\frac{9\pi }{4},\frac{10\pi }{4},\frac{11\pi }{4}\}$
Ans: A