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UPSC NDA I 2023 Mathematics Solved Paper

Question Numbers: 31-32

Consider the following for the next two (02) items that follow :

Let sin β be the GM of sin α and cos α; tan γ be the AM of sin α and cos α.

Question : 31 of 120

Marks: +1, -0

What is cos 2β equal to ?

Solution:

Concept:

\sin \beta

is the geometric mean of

\sin \alpha

and

\cos \alpha

, then

\sin \beta = \sqrt{\sin \alpha \cos \alpha}

Explanation:

Given

\sin \beta = \sqrt{\sin \alpha \cos \alpha}

Square both sides:

\sin^2 \beta = \sin \alpha \cos \alpha

Multiply by

2

2 \sin^2 \beta = 2 \sin \alpha \cos \alpha

Rewrite

1 - 2 \sin^2 \beta = 1 - 2 \sin \alpha \cos \alpha

Use identity

\cos 2\beta = 1 - 2 \sin^2 \beta

. So

\cos 2\beta = 1 - 2 \sin \alpha \cos \alpha

Since

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

, substitute:

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

Thus

\cos 2\beta = \sin^2 \alpha + \cos^2 \alpha - 2 \sin \alpha \cos \alpha = (\cos \alpha - \sin \alpha)^2

Answer:

Option A:

(\cos \alpha - \sin \alpha)^2

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