If 0 ≤ x ≤ π and 81^{ ² x}+81^{ ² x}=30 Then x takes the value

Correct Answer is : π6,π3π/6, π/36π,3π

Correct Answer is : π6,π3π/6, π/36π,3π

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MHTCET 26 Apr 2025 Shift 2 Paper

Question : 148 of 150

Marks: +1, -0

0 \leq x \leq \pi

81^{\sin ^{2} x}+81^{\cos ^{2} x}=30

x

takes the value

Solution:

Concept:

Using

\sin^2 x + \cos^2 x = 1

, the equation is reduced to a quadratic in

81^{\sin^2 x}

Explanation:

Let

y = 81^{\sin^2 x}

Then

81^{\cos^2 x} = 81^{1-\sin^2 x} = \frac{81}{y}

The equation becomes

y + \frac{81}{y} = 30

Multiplying by

y

y^2 - 30y + 81 = 0

Factorising:

(y-27)(y-3)=0

, so

y=27

y=3

Thus

81^{\sin^2 x}=27

81^{\sin^2 x}=3

Since

81 = 3^4

, we get

3^{4\sin^2 x}=3^3

3^{4\sin^2 x}=3^1

4\sin^2 x = 3

4\sin^2 x = 1

Hence

\sin^2 x = \frac{3}{4}

\sin^2 x = \frac{1}{4}

For

0 \le x \le \pi

\sin x \ge 0

, so

\sin x = \frac{\sqrt{3}}{2}

\sin x = \frac{1}{2}

Therefore

x = \frac{\pi}{3}, \frac{2\pi}{3}, \frac{\pi}{6}, \frac{5\pi}{6}

Answer:

The values of

x

are

\frac{\pi}{6}, \frac{\pi}{3}, \frac{2\pi}{3}, \frac{5\pi}{6}

. Among the given options, option A is the correct choice.

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