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UPSC NDA Math Model Paper 2

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Question : 3 of 120

Marks: +1, -0

The maximum value of

4 \sin^{2} + 3 \cos^{2} x + \sin\frac{x}{2} + \cos\frac{x}{2}

is

4
$3+\sqrt{2}$
9
$4+\sqrt{2}$

Solution:

Let,

f(x)=4 \sin^{2} x+3 \cos^{2} x+\sin\frac{x}{2}

+\cos\frac{x}{2}

f(x)=4 \sin^{2} x+3(1-\sin^{2} x)+\sin\frac{x}{2}

+\cos\frac{x}{2}

=4 \sin^{2} x+3-3 \sin^{2} x+\sin\frac{x}{2}+\cos\frac{x}{2}

=\sin^{2} x+3+\sin\frac{x}{2}+\cos\frac{x}{2}

=\sin^{2} x+3+\sqrt{2}\left[\frac{1}{\sqrt{2}} \sin\frac{x}{2}+\cos\frac{x}{2}\right]

\left.\cdot \frac{1}{\sqrt{2}}\right]

=3+\sin^{2} x+\sqrt{2} \sin\left(\frac{\pi}{4}+\frac{x}{2}\right)

=3+1+\sqrt{2}(1)

(\because

Max value of

\sin^{2} x=1

and

\sin\left(\frac{\pi}{4}+\frac{x}{2}\right)=1)

So the maximum value of

4 \sin^{2} x+3 \cos^{2} x+\sin\frac{x}{2}+\cos\frac{x}{2}

is

4+\sqrt{2}

Hence, option (4) is correct.

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