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)?

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

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. . . . . 

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

If $\frac{\sin ^4 x}{2}+\frac{\cos ^4 x}{3}=\frac{1}{5},$ then

$(A)$ $\tan ^2 x=\frac{2}{3}$ $(B)$ $\frac{\sin ^8 x}{8}+\frac{\cos ^8 x}{27}=\frac{1}{125}$

$(C)$ $\tan ^2 x=\frac{1}{3}$ $(D)$ $\frac{\sin ^8 x}{8}+\frac{\cos ^8 x}{27}=\frac{2}{125}$