The value of $\cos ^{-1} \left[ \cot \left\{ \sin ^{-1} \sqrt{\frac{2-\sqrt{3}}{4}} + \cos ^{-1} \frac{\sqrt{12}}{4} + \sec ^{-1} \sqrt{2} \right\} \right]$ is

$A$ transformer is employed to reduce $220\, V$ to $11\, V$. The primary draws a current of $5\, A$ and the secondary $90\, A$. The efficiency of the transformer is......$\%$

Match the following (Take the relative strength of the strongest fundamental forces in nature as $1$):
| List-$I$ (Fundamental forces in nature) | List-$II$ (Relative strength) |
| :--- | :--- |
| $(A)$ Strong nuclear force | $(e)$ $10^{-2}$ |
| $(B)$ Weak nuclear force | $(f)$ $1$ |
| $(C)$ Electromagnetic force | $(g)$ $10^{10}$ |
| $(D)$ Gravitational force | $(h)$ $10^{-13}$ |
| | $(i)$ $10^{-39}$ |
The correct match is:

The ratio of most probable velocity $(u_{mp})$,average velocity $(u_{av})$,and root mean square velocity $(u_{rms})$ is .......

In which of the following is the gametophytic phase reduced,short-lived,and limited to the formation of gametes?

Charges $+q$ and $-q$ are placed at points $A$ and $B$ respectively,which are at a distance $2L$ apart. $C$ is the midpoint between $A$ and $B$. The work done in moving a charge $+Q$ along the semicircle $CRD$ is: