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UPSC CDS 1 2021 Math Paper

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Question : 56 of 100

Marks: +1, -0

What is the least value of

3\sin^{2}\theta + 4\cos^{2}\theta

?

5
4
3
2

Solution:

Concept:

Use the identity

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

to rewrite the expression.

Explanation:

Rewrite

3\sin^2\theta + 4\cos^2\theta

as

3\sin^2\theta + 3\cos^2\theta + \cos^2\theta

.

Factor to get

3(\sin^2\theta + \cos^2\theta) + \cos^2\theta

.

Since

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

, this simplifies to

3 + \cos^2\theta

.

The minimum value of

\cos^2\theta

is

0

(because

\cos^2\theta \ge 0

).

Therefore, the least value of the expression is

3 + 0 = 3

.

Answer:

Option C:

3

.

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