If $\sin 2x + \sin 4x = 2\sin 3x,$ then $x =$

  • A

    $\frac{{n\pi }}{3}$

  • B

    $n\pi + \frac{\pi }{3}$

  • C

    $2n\pi \pm \frac{\pi }{3}$

  • D

    None of these

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  • [KVPY 2018]

In a triangle $P Q R, P$ is the largest angle and $\cos P=\frac{1}{3}$. Further the incircle of the triangle touches the sides $P Q, Q R$ and $R P$ at $N, L$ and $M$ respectively, such that the lengths of $P N, Q L$ and $R M$ are consecutive even integers. Then possible length$(s)$ of the side$(s)$ of the triangle is (are)

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  • [IIT 2013]