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UPSC NDA II 2024 Mathematics Paper

Question Numbers: 99-100

Direction : Consider the following for the items that follow :

Let

\rm 2\int\frac{x^2-1}{\sqrt{x^2+1}}dx=U(x) V(x)-3\ln \{U(x)+V(x)\}+c

Question : 99 of 120

Marks: +1, -0

What is

\left| U^{2}(x) - V^{2}(x) \right|

equal to?

Solution:

Concept:

The expression simplifies to a constant after substituting the given squared functions.

Explanation:

From the problem, we have

U^2(x) = x^2

and

V^2(x) = x^2 + 1

Compute the difference:

U^2(x) - V^2(x) = x^2 - (x^2 + 1) = -1

Take the absolute value:

|U^2(x) - V^2(x)| = |-1| = 1

Answer:

Option B: 1.

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