Question

If $0\le x\le \frac{\pi }{2}$ and ${81}^{si{n}^{2}x}+{81}^{co{s}^{2}x}=30$ then x is equal to

• A.
• B.
• C.
• D.

Answer 1

The correct option is A

Let ${81}^{si{n}^{2}x}=y$ then ${81}^{co{s}^{2}x}={81}^{1-si{n}^{2}x}=\frac{81}{y}$

So, $y^2-30y+81=0$ y = 3 or y = 27

$\Rightarrow$ ${81}^{si{n}^{2}x}=3or27$ $si{n}^{2}x=\frac{1}{4}or\frac{3}{4}$

sinx = $\pm \frac{1}{2}$ or $\pm \frac{\sqrt{3}}{2}$ $(neglecting\frac{-1}{2},\frac{\sqrt{3}}{2}as0\le x\le \frac{\pi }{2})$

$x=\frac{\pi }{6}or\frac{\pi }{3}$