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UPSC NDA I 2026 Mathematics Paper

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Question Numbers: 91-92

For the next two (02) items that follow:

Let

\begin{aligned}& \int \frac{\sin \theta d \theta}{(2+\cos \theta)(3+4 \cos \theta)}= \\& \quad A \ln |2+\cos \theta|+B \ln |3+4 \cos \theta|\end{aligned}

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Question : 91 of 120

Marks: +1, -0

What is the value of

A

?

$-\frac{2}{5}$
$-\frac{1}{5}$
$\frac{1}{5}$
$\frac{2}{5}$

Solution:

Concept:

Substitute

u=\cos\theta

and compare coefficients after partial fractions.

Explanation:

Let

u=\cos\theta

, so

du=-\sin\theta\,d\theta

. Then

I=-\int \frac{du}{(u+2)(4u+3)}.

Now decompose:

\frac{1}{(u+2)(4u+3)}=\frac{P}{u+2}+\frac{Q}{4u+3}.

Multiplying both sides:

1=P(4u+3)+Q(u+2).

Put

u=-2

:

1=P(-5) \Rightarrow P=-\frac15.

Since the integral gives

-P\ln|u+2|-\frac{Q}{4}\ln|4u+3|

,

the coefficient of

\ln|2+\cos\theta|

is

A=-P=\frac15.

Answer:

A=\frac{1}{5}

(Option C).

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