At what time between $10\,\,O'clock$ and $11\,\,O 'clock$ are the two hands of a clock symmetric with respect to the vertical line (give the answer to the nearest second)?

  • [KVPY 2009]
  • A

    $10h\,9m \,13s$

  • B

    $10h \,9m \,14s$

  • C

    $10h \,9m \,22s$

  • D

    $10h \,9m \,50s$

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If $\sin x+\sin ^2 x=1, x \in\left(0, \frac{\pi}{2}\right)$, then $\left(\cos ^{12} x+\tan ^{12} x\right)+3\left(\cos ^{10} x+\tan ^{10} x+\cos ^8 x+\tan ^8 x\right)$ $+\left(\cos ^6 x+\tan ^6 x\right)$ is equal to

  • [JEE MAIN 2025]

The value of $\cot \frac{\pi}{24}$ is :

  • [JEE MAIN 2021]

Let the function $:(0, \pi) \rightarrow R$ be defined by

$f (\theta)=(\sin \theta+\cos \theta)^2+(\sin \theta-\cos \theta)^4$

Suppose the function $f$ has a local minimum at $\theta$ precisely when $\theta \in\left\{\lambda_1 \pi, \ldots, \lambda_{ T } \pi\right\}$, where $0<\lambda_1<\cdots<\lambda_r<1$. Then the value of $\lambda_1+\cdots+\lambda_r$ is. . . . . 

  • [IIT 2020]

If $x = \sec \,\phi - \tan \phi ,y = {\rm{cosec}}\phi + \cot \phi ,$ then

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  • [IIT 2009]