NIMCET 2018 — Mathematics PYQ

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Question

NIMCET 2018 — Mathematics PYQ

NIMCET | Mathematics | 2018

The number of the solutions of the equation $s in x + s in 5 x = s in 3 x$ , lying in the interval $[0, π]$ , is:

Choose the correct answer:

Explanation

Concept:
$sin (A \pm B) = sin A cos B \pm sin B cos A .$
$sin 2 A + sin 2 B = 2 sin (A + B) cos (A - B) .$
$cos (2 nπ + θ) = cos θ .$

Calculation:

$sin x + sin 5 x = sin 3 x$
Using sin 2A+ sin 2B= 2 sin ( A+ B) cos ( A- B) , we get:
$2 sin 3 x cos 2 x = sin 3 x$
$\Rightarrow sin 3$ x $(2 cos 2$ x-1)=0
$\Rightarrow sin 3$ x=0 OR 2 cos 2x-1=0 CASE 1: $sin$ 3x= 0= $sin$ n $π$ , n $\in$ Z. $\Rightarrow x =$ $\frac{n π}{3}$
$\Rightarrow x = \dots, 0, \frac{π}{3}, \frac{2 π}{3}, π, \dots$ CASE 2: 2 cos 2x- 1= 0 $\Rightarrow cos 2 x = \frac{1}{2} = cos (2 n π \pm \frac{π}{2})$ ,n $\in Z$
$\Rightarrow x =$ n $π \pm \frac{π}{6} = (6$ n $\pm 1) \frac{π}{6}$
$\Rightarrow x = \dots, \frac{π}{6}, \frac{5 π}{6}, \dots$
Therefore, there are 6 possible values of x in the interval [0, π].

Choose the correct answer:

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