Question

If $0\le x\le \pi$ and ${81}^{{\sin }^{2}x}+{81}^{{\cos }^{2}x}=30$, then x is equal to.

• A. $\pi /6$
• B. $\pi /3$
• C. $5\pi /6$
• D. $2\pi /3$
• E. All correct

Answer 1

Answer: (E)

All correct

Put ${\cos }^{2}x=1-{\sin }^{2}x$
If $y={81}^{{\sin }^{2}x}$ then $y+\frac{81}{y}=30$
$y^2-30y+81=0$, $y=27,3$
$3^{4{\sin }^{2}x}=3^3,3^1$
$\therefore {\sin }^{2}x=\frac{3}{4},\frac{1}{4}$
$\therefore \sin x=\frac{\sqrt{3}}{2},\frac{1}{2}$
as $\sin x$ is $+$ive in the given interval $0<x<\pi$
$\therefore x=\frac{\pi }{3},\pi -\frac{\pi }{3}=\frac{2\pi }{3},\frac{\pi }{6},\pi -\frac{\pi }{6}=\frac{5\pi }{6}$.