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UPSC CDS 1 2023 Math Paper

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Question : 50 of 100

Marks: +1, -0

What is the value of

x

4 \cos^{2} 30^{\circ} + 2x \sin 30^{\circ} - \cot^{2} 30^{\circ} - 6 \tan 15^{\circ} \tan 75^{\circ} = 0

?

1
2
3
6

Solution:

Concept:

Use standard trigonometric values and the identity

\tan \theta \cdot \cot \theta = 1

.

Explanation:

Step 1: Substitute known values:

\cos 30^{\circ} = \frac{\sqrt{3}}{2}

,

\sin 30^{\circ} = \frac{1}{2}

,

\cot 30^{\circ} = \sqrt{3}

.

Step 2: Simplify

\tan 15^{\circ} \tan 75^{\circ}

. Since

\tan(90^{\circ} - \theta) = \cot \theta

, we get

\tan 75^{\circ} = \cot 15^{\circ}

.

Step 3: Then

\tan 15^{\circ} \tan 75^{\circ} = \tan 15^{\circ} \cot 15^{\circ} = 1

.

Step 4: Substitute all into the equation:

4\left(\frac{\sqrt{3}}{2}\right)^2 + 2x\left(\frac{1}{2}\right) - (\sqrt{3})^2 - 6 \times 1 = 0

.

Step 5: Simplify each term:

4 \cdot \frac{3}{4} = 3

,

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

,

(\sqrt{3})^2 = 3

. So

3 + x - 3 - 6 = 0

.

Step 6: This gives

x - 6 = 0

, so

x = 6

.

Answer:

x = 6

(Option D).

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