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UPSC CDS I 2025 Mathematics Paper

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Question : 35 of 100

Marks: +1, -0

What is the value of x, where

0 ≤ x ≤ 30^{∘}

tan3x tan 6x = 1

?

$0^{∘}$
$10^{∘}$
$12^{∘}$
$15^{∘}$

Solution:

Concept:

If

\tan A \times \tan B = 1

, then

\tan A = \cot B = \tan(90° - B + n \times 180°)

.

Explanation:

Given

\tan 3x \times \tan 6x = 1

with

0 \le x \le 30°

.

Using the concept:

\tan 3x = \tan(90° - 6x + n \times 180°)

.

Equate the angles:

3x = 90° - 6x + n \times 180°

.

Simplify:

9x = 90° + n \times 180°

.

Thus

x = 10° + n \times 20°

.

For

n = 0

:

x = 10°

(valid).

For

n = 1

:

x = 30°

, but then

3x = 90°

making

\tan 3x

undefined, so reject.

For

n = -1

:

x = -10°

(outside range).

Hence the only valid solution is

x = 10°

.

Answer:

B.

10°

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