Question

If $0\le x\le\pi$ and $81^{\sin^{2}x}+81^{\cos^{2}x}=30$ Then $x$ takes the value}

• A. $\pi/6$
• B. $\pi/4$
• C. $3\pi/6$
• D. $2\pi/3$

Hint:

Substitute $a^{\sin^2 x}$ to convert the trigonometric equation into a quadratic one.

Answer 1

The Correct Option is A

Step 1: Understanding the Question:
We solve a trigonometric exponential equation using substitution.

Step 2: Key Formula or Approach:
Use $\cos^2 x = 1 - \sin^2 x$. Let $t = 81^{\sin^2 x}$.

Step 3: Detailed Explanation:
$81^{\sin^2 x} + 81^{1 - \sin^2 x} = 30$
$81^{\sin^2 x} + \frac{81}{81^{\sin^2 x}} = 30$
Let $t = 81^{\sin^2 x}$:
$t + \frac{81}{t} = 30 \implies t^2 - 30t + 81 = 0$.
Factorizing: $(t - 27)(t - 3) = 0$.
Case 1: $t = 3 \implies 81^{\sin^2 x} = 3^1 \implies (3^4)^{\sin^2 x} = 3^1 \implies 4\sin^2 x = 1$.
$\sin^2 x = 1/4 \implies \sin x = 1/2$ (as $\sin x \ge 0$ for $x \in [0, \pi]$).
$x = \pi/6$ or $5\pi/6$.
Case 2: $t = 27 \implies 81^{\sin^2 x} = 3^3 \implies (3^4)^{\sin^2 x} = 3^3 \implies 4\sin^2 x = 3$.
$\sin^2 x = 3/4 \implies \sin x = \sqrt{3}/2$.
$x = \pi/3$ or $2\pi/3$.
Comparing with options, (A) contains both $\pi/6$ and $\pi/3$.

Step 4: Final Answer:
The values are $\pi/6$ and $\pi/3$.