Question

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

• A. $4$
• B. $12$
• C. $3$
• D. None of these

Answer 1

The correct option is A $4$

R.E.F.Image.

$lo{g}_{1/\sqrt{2}}sinx>0$

$lo{g}_{1/\sqrt{2}}sinx>lo{g}_{1/\sqrt{2}}1$

$\because$ If base is less then $1$, equality sign will be change.

$sinx<1$

In logarithm , $\sin x$ must be the,

$\Rightarrow sinx>0$

$xϵ[0,\pi ]$

then 0 ,to $\pi$ check $sinx<1$

$\frac{\pi }{4},\frac{3\pi }{4}$ are the value where we get $sinx<1$

from $\pi$ to$2\pi$ to $3\pi$

$2\pi +\frac{\pi }{4},2\pi +\frac{3\pi }{4}$ $\sin x$ will less then $1$.

& $3\pi$ to $4\pi$ will be neglected due to negative

we have only $4$ values.