Q109 / 180 · COMEDK 2013 · Mathematics
If $0 \leq x \leq \pi$ and $81^{\sin ^{2} x}+81^{\cos ^{2} x}=30$ then $x$ is equal to
- Correct answerA$\frac{\pi}{6}$
- B$\frac{\pi}{2}$
- C$\frac{\pi}{4}$
- D$\frac{3 \pi}{4}$
Solution
We have,
$$
\begin{gathered}
81^{\sin ^{2} x}+81^{\cos ^{2} x}=30 \\
\Rightarrow 81^{\sin ^{2} x}+81^{\cos ^{2} x}=3+27 \\
=81^{\sin ^{2} x}+81^{\cos ^{2} x} \\
=(81)^{\frac{1}{4}}+(81)^{\frac{3}{4}} \\
\Rightarrow \sin ^{2} x=\frac{1}{4} \text { and } \cos ^{2} x=\frac{3}{4} \\
\Rightarrow \sin x=\frac{\pm 1}{2} \text { and } \cos x=\frac{\pm \sqrt{3}}{2} \\
\text { Since, } \quad 0 < x < \pi \\
\Rightarrow \sin x=\frac{1}{2} \text { and } \cos x=\frac{\sqrt{3}}{2} \\
\Rightarrow \quad x=\frac{\pi}{6}
\end{gathered}
$$
Hence, the correct answer is option: 'A'.