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UPSC NDA I 2025 Mathematics Paper

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Question Numbers: 41-43

Consider the following for the three (03) items that follow:

Let

p = tan

2α - tanα and

q =

cotα - cot 2α

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Question : 41 of 120

Marks: +1, -0

What is

(p/q)

equal to?

$−tanα⋅tan2α$
$−cotα⋅cot2α$
$tanα⋅tan2α$
$cotα⋅cot2α$

Solution:

Concept:

Use the identity

\cot \theta = \frac{1}{\tan \theta}

to rewrite

q

in terms of

\tan

, then simplify the ratio

\frac{p}{q}

.

Explanation:

Given

p = \tan 2\alpha - \tan \alpha

and

q = \cot \alpha - \cot 2\alpha

.

Rewrite

q

:

\cot \alpha = \frac{1}{\tan \alpha}

,

\cot 2\alpha = \frac{1}{\tan 2\alpha}

.

So

q = \frac{1}{\tan \alpha} - \frac{1}{\tan 2\alpha}

.

Combine the fractions:

q = \frac{\tan 2\alpha - \tan \alpha}{\tan \alpha \cdot \tan 2\alpha}

.

Now

\frac{p}{q} = \frac{\tan 2\alpha - \tan \alpha}{\frac{\tan 2\alpha - \tan \alpha}{\tan \alpha \cdot \tan 2\alpha}}

.

Cancel the common term

\tan 2\alpha - \tan \alpha

to get

\frac{p}{q} = \tan \alpha \cdot \tan 2\alpha

.

Answer:

Option C:

\tan\alpha \cdot \tan2\alpha

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