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Equations of the Form acosθ+bsinθ = c

If x1 and ...

Question

If $x_{1}$ and $x_{2}$ are the two solutions of the equation $2 cos x sin 3 x = sin 4 x + 1$ lying in the interval $[0, 2 π]$ then the value of $| x_{1} - x_{2} |$ is

\frac{π}{4}

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\frac{π}{2}

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π

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2 π

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Solution

The correct option is C $π$
If $x_{1} & x_{2} a x () o f (1) i n [0, 2 π]$
$2 cos x sin 3 x = sin 4 x + 1$
$sin 4 x + sin 2 x = sin 4 x + 1$
$∵ 2 cos x sin y = sin (x + y) - sin (x - y)$
$sin 2 x = 1$
$sin 2 x = sin \frac{π}{2}$
$2 x = \frac{π}{2} + 2 n π$
$x = \frac{π}{4} + n π$
$x = \frac{π}{4} + n π$
$x_{1} = \frac{π}{4} + π$
$x_{2} = \frac{π}{4}$
$| x_{2} - x_{1} | = ∣ ∣ ∣ \frac{π}{4} + π - \frac{π}{4} ∣ ∣ ∣ = π$

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