When alternating current is passed through $\mathrm{L}-\mathrm{R}$ series circuit, the power factor is $\frac{\sqrt{3}}{2}$ and $\mathrm{R}=50 \Omega$, then the value of L is

$$ \left[\cos \frac{\pi}{6}=\frac{\sqrt{3}}{2}, \sin \frac{\pi}{6}=\frac{1}{2}, \tan \frac{\pi}{6}=\frac{1}{\sqrt{3}}\right] $$

Time period of a simple pendulum on earth's surface is ' T '. It time period becomes ' xT ' when taken to a height ' $2 R$ ' above earth's surface. The value of $x$ will be $(R=$ radius of earth)

The wire loop PQRSP formed by joining two semicircular wire of radii $R_1$ and $R_2$ carries a current $I$ as shown. The magnitude of the magnetic field at the centre ' O ' is

A body of mass 1 kg begins to move under the action of a time dependent force $\overrightarrow{\mathrm{F}}=\left(\hat{\mathrm{t}}+2 t^2 \hat{\mathrm{j}}\right) \mathrm{N}$, where $\hat{i}$ and $\hat{j}$ are unit vectors along $x$ and $y$ axis. The power developed by above force at time $\mathrm{t}=3$ second will be