In an $A.C.$ circuit,$V$ and $I$ are given by $V = 100 \sin(100t) \ V$ and $I = 100 \sin(100t + \frac{\pi}{3}) \ mA$. The power dissipated in the circuit is ...... $W$.

The limiting points of the co-axial system containing the two circles $x^2+y^2+2x-2y+2=0$ and $25(x^2+y^2)-10x-80y+65=0$ are

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:

Which relation is incorrect?

$A$ proton enters a magnetic field of flux density $1.5 \, Wb/m^2$ with a velocity of $2 \times 10^7 \, m/s$ at an angle of $30^\circ$ with the field. The force on the proton will be

An ideal gas expands isothermally from a volume $V_1$ to $V_2$ and then is compressed to the original volume $V_1$ adiabatically. The initial pressure is $P_1$ and the final pressure is $P_3$. The total work done is $W$. Then: