What is (dy/dx)² equal to?

Correct Answer is : 4(y2+4)(x2+4){4(y²+4)}{(x²+4)}(x2+4)4(y2+4)

Correct Answer is : 4(y2+4)(x2+4){4(y²+4)}{(x²+4)}(x2+4)4(y2+4)

Test Index

UPSC NDA I 2025 Mathematics Paper

Question Numbers: 71-72

Let

x=sec\theta -cos\theta andy=sec^{4}\theta -cos^{4}\theta

Question : 71 of 120

Marks: +1, -0

What is

(\frac{dy}{dx})^{2}

equal to?

Solution:

Calculation:

Given,

x=sec\theta -cos\theta

y=sec^{4}\theta -cos^{4}\theta

Compute derivatives w.r.t.

\theta\theta

\frac{dx}{d\theta }=sec\theta tan\theta +sin\theta

\frac{dy}{d\theta }=4sec^{4}\theta tan\theta + 4cos^{3}\theta sin\theta

From the ratio

\frac{dy}{dx}=\frac{\frac{dy}{d\theta }}{\frac{dx}{d\theta }}

\frac{dy}{dx}=\frac{4(sec^{4}\theta tan\theta +cos^{3}\theta sin\theta )}{sec\theta tan\theta +sin\theta }

Using identities

x^{2}=sec^{2}\theta +cos^{2}\theta -2,sec^{3}\theta +cos^{5}\theta )^{2}=y^{2}+4

, and

(1+cos^{2}\theta )^{2}=x^{2}+4,

(\frac{dy}{dx})^{2}=16 \frac{ y^{2}+4 }{ x^{2}+4 }

∴

(\frac{dy}{dx})^{2}=16 \frac{y^{2}+4}{x^{2}+4}

Hence, the correct answer is Option 3.

Go to Question:

Prev Question

Next Question