The solution of x + 5x = 3x in ≤ft(0, π/2) are

Correct Answer is : π6,π3π/6, π/36π,3π

Correct Answer is : π6,π3π/6, π/36π,3π

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MHTCET 13 May 2023 Shift 2 Solved Paper

Section: Mathematics

Question : 90 of 150

Marks: +1, -0

The solution of

\sin x + \sin 5x = \sin 3x

\left(0, \frac{\pi}{2}\right)

are

Solution:

Here,

\sin x + \sin 5x = \sin 3x

We know that,

\sin A + \sin B = 2 \sin \frac{A+B}{2} \cos \frac{A-B}{2}

\therefore \sin x + \sin 5x = \sin 3x

\Rightarrow 2 \sin \left( \frac{5x+x}{2} \right) \cos \left( \frac{5x-x}{2} \right) = \sin 3x

\Rightarrow 2 \sin 3x \cos 2x = \sin 3x

\therefore

Either,

\sin 3x = 0

2 \cos 2x - 1 = 0

\sin 3x = \sin 0 \text{ or } \cos 2x = \frac{1}{2} = \cos \frac{\pi}{3}

Comparing obtained equation with standard equation, we have

3x = n\pi \text{ or } 2x = 2m\pi \pm \frac{\pi}{3}

\therefore x = \frac{n\pi}{3} \text{ or } x = m\pi + \frac{\pi}{6}, \text{ where } m, n \in \mathbb{Z}

x = \frac{\pi}{6}, \frac{\pi}{3}

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