The number of solutions $x$ of the equation $\sin \left(x+x^2\right)-\sin \left(x^2\right)=\sin x$ in the interval $[2,3]$ is

  • [KVPY 2018]
  • A

    $0$

  • B

    $1$

  • C

    $2$

  • D

    $3$

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For $0<\theta<\frac{\pi}{2}$, the solution(s) of $\sum_{m=1}^6 \operatorname{cosec}\left(\theta+\frac{(m-1) \pi}{4}\right) \operatorname{cosec}\left(\theta+\frac{m \pi}{4}\right)=4 \sqrt{2}$ is(are)

$(A)$ $\frac{\pi}{4}$ $(B)$ $\frac{\pi}{6}$ $(C)$ $\frac{\pi}{12}$ $(D)$ $\frac{5 \pi}{12}$

  • [IIT 2009]

Let $A=\left\{\theta \in R \mid \cos ^2(\sin \theta)+\sin ^2(\cos \theta)=1\right\}$ and $B=\{\theta \in R \mid \cos (\sin \theta) \sin (\cos \theta)=0\}$. Then, $A \cap B$ 

  • [KVPY 2011]

If $\sin \theta  + 2\sin \phi  + 3\sin \psi  = 0$ and $\cos \theta  + 2\cos \phi  + 3\cos \psi  = 0$ , then the value of $\cos 3\theta  + 8\cos 3\phi  + 27\cos 3\psi  = $ 

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The number of solutions to $\sin x=\frac{6}{x}$ with $0 \leq x \leq 12 \pi$ is

  • [KVPY 2009]