Question

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

• A. $\frac{\pi }{6}$
• B. $\frac{\pi }{3}$
• C. $\frac{5\pi }{6}$
• D. $\frac{2\pi }{3}$

Answer 1

Answer: (A), (B), (C), (D)

The correct options are
A $\frac{5\pi }{6}$
B $\frac{\pi }{6}$
C $\frac{\pi }{3}$
D $\frac{2\pi }{3}$

$0\le x\le \pi$
${81}^{{\sin }^{2}x}+{81}^{{\cos }^{2}x}=30\Rightarrow 3^{4{\sin }^{2}x}+\frac{81}{3^{4{\sin }^{2}x}}=30$
Let $3^{4{\sin }^{2}x}=y$
$\Rightarrow y^2-30y+81=0\Rightarrow y=27,3$
$\therefore 3^{4{\sin }^{2}x}=3^3\Rightarrow {\sin }^{2}x=\frac{3}{4}\Rightarrow x=\frac{\pi }{3},\frac{2\pi }{3}$
Or $3^{4{\sin }^{2}x}=3\Rightarrow {\sin }^{2}x=\frac{1}{4}\Rightarrow x=\frac{\pi }{6},\frac{5\pi }{6}$
Hence, options 'A', 'B', 'C' and 'D' are correct.