Exhaustive set to values of x satisfying $| cos 3 x + sin 3 x | = | cos 3 x | + | sin 3 x |$ in $[0, \frac{π}{2}]$ is :

[0, \frac{π}{6}]

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[0, \frac{π}{2}]

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[0, \frac{π}{2}] - (\frac{π}{6}, \frac{π}{3})

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[0, \frac{π}{2}] - (\frac{π}{4}, \frac{π}{3})

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Solution

The correct option is C $[0, \frac{π}{2}] - (\frac{π}{6}, \frac{π}{3})$
$∣ sin 3 x + cos 3 x ∣=∣ sin 3 x ∣ + ∣ cos 3 x ∣$
we know, $∣ a + b ∣=∣ a ∣ + ∣ b ∣$ is only true if ,
$a b \geq 0$
$\Rightarrow sin 3 x . cos 3 x \geq 0 \Rightarrow sin 6 x \geq 0$
thus solution in the given interval is,
$6 x \in [0, π] \cup [2 π, 3 π]$

$\Rightarrow x \in [0, \frac{π}{3}] \cup [\frac{π}{3}, \frac{π}{2}] = [0, \frac{π}{2}] - (\frac{π}{6}, \frac{π}{3})$
Hence, option 'C' is correct.

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Exhaustive set to values of x satisfying |cos3x+sin3x|=|cos3x|+|sin3x| in [0,π2] is :

Exhaustive set to values of x satisfying $| cos 3 x + sin 3 x | = | cos 3 x | + | sin 3 x |$ in $[0, \frac{π}{2}]$ is :