If f(x) = x(4x² - 3), then what is f(θ) equal to ?

Correct Answer is : −sin⁡3θ- 3θ−sin3θ

Correct Answer is : −sin⁡3θ- 3θ−sin3θ

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UPSC NDA I 2023 Mathematics Solved Paper

Question : 98 of 120

Marks: +1, -0

f(x) = x(4x^{2} - 3)

f(\sin\theta)

equal to ?

Solution:

Concept:

The function is

f(x) = x(4x^2 - 3)

We need to evaluate

f(\sin\theta)

and simplify using a trigonometric identity.

Explanation:

Substitute

x = \sin\theta

into

f(x)

f(\sin\theta) = \sin\theta \,\bigl(4(\sin\theta)^2 - 3\bigr)

This simplifies to

4\sin^3\theta - 3\sin\theta

Recall the triple-angle identity for sine:

\sin 3\theta = 3\sin\theta - 4\sin^3\theta

Rearrange this identity to express

4\sin^3\theta - 3\sin\theta

4\sin^3\theta - 3\sin\theta = -(3\sin\theta - 4\sin^3\theta) = -\sin 3\theta

Thus,

f(\sin\theta) = -\sin 3\theta

Answer:

-\sin 3\theta

(option A).

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