The sum of all $x \in[0, \pi]$ which satisfy the equation $\sin x+\frac{1}{2} \cos x=\sin ^2\left(x+\frac{\pi}{4}\right)$ is

  • [KVPY 2012]
  • A

    $\frac{\pi}{6}$

  • B

    $\frac{5 \pi}{6}$

  • C

    $\pi$

  • D

    $2 \pi$

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  • [JEE MAIN 2022]