Question

Pls pls pls help.solve 70th que

Answer 1

$\cos(3x) + \cos(2x) = \sin(1.5x) + \sin(0.5x) \\ \\ \Rightarrow 2 \cos( \frac{3x + 2x}{2} ) \cos( \frac{3x - 2x}{2} ) = 2 \sin( \frac{1.5x + 0.5x}{2} ) \cos( \frac{1.5x - 0.5x}{2} )$

$\Rightarrow 2 \cos( \frac{5x}{2} ) \cos( \frac{x}{2} ) = 2 \sin(x) \cos( \frac{x}{2} ) \\ \\ \Rightarrow 2 \cos( \frac{5x}{2} ) \cos( \frac{x}{2} ) - 2 \sin(x) \cos( \frac{x}{2} ) = 0$

$\Rightarrow 2 \cos( \frac{x}{2} ) [( \cos( \frac{5x}{2} ) - \sin(x)] = 0$
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For the equation to be true, either cos(x/2) = 0 or cos(5x/2) - sin(x) = 0
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case-i : Take cos(x/2) = 0

from 0 to 2π, only ONE value of x satisfies the condition: x = π
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case ii : For cos(5x/2) - sin(x) = 0

=> cos(5x/2) = sin(x)

Plot sin(x) and cos (5x/2) graph. It will be easier to find the number of solutions.

See the attachment for graph. There are FIVE values of x satisfying the condition.
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But π is a solution for both. So there are only 5 solutions for x.

Answer is D.