NIMCET 2017 — Mathematics PYQ

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Question

NIMCET 2017 — Mathematics PYQ

NIMCET | Mathematics | 2017

The maximum value of $4 sin^{2} x + 3 cos^{2} x + sin \frac{x}{2} + cos \frac{x}{2}$ is

Choose the correct answer:

Explanation

Concept:
$sin^{2} x + cos^{2} x = 1$

Calculation:
Let, $f (x) = 4 sin^{2} x + 3 cos^{2} x + sin \frac{x}{2} + cos \frac{x}{2}$

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

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

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

$= sin^{2} x + 3 + 2 [\frac{1}{2} sin \frac{x}{2} + cos \frac{x}{2} \cdot \frac{1}{2}]$

$= 3 + sin^{2} x + 2 sin (\frac{π}{4} + \frac{x}{2})$

$= 3 + 1 + 2 (1)$
(Max value of $sin^{2} x = 1$ and $sin (\frac{π}{4} + \frac{x}{2}) = 1$ )

So the maximum value of
$4 sin^{2} x + 3 cos^{2} x + sin \frac{x}{2} + cos \frac{x}{2}$
is $4 + 2$ .

Hence, option (4) is correct.