An a.c. source of $$15 \mathrm{~V}, 50 \mathrm{~Hz}$$ is connected across an inductor (L) and resistance (R) in series R.M.S. current of $$0.5 \mathrm{~A}$$ flows in the circuit. The phase difference between applied voltage and current is $$\left(\frac{\pi}{3}\right)$$ radian. The value of resistance $$(\mathrm{R})$$ is $$\left(\tan 60^{\circ}=\sqrt{3}\right)$$

A ball is projected vertically upwards from ground. It reaches a height '$$h$$' in time $$t_1$$, continues its motion and then takes a time $$t_2$$ to reach ground. The height $h$ in terms of $$g, t_1$$ and $$\mathrm{t}_2$$ is $$(\mathrm{g}=$$ acceleration due to gravity)

The volume of a metal block increases by $$0.225 \%$$ when its temperature is increased by $$30^{\circ} \mathrm{C}$$. Hence coefficient of linear expansion of the material of metal block is

A tuning fork gives 3 beats with $$50 \mathrm{~cm}$$ length of sonometer wire. If the length of the wire is shortened by $$1 \mathrm{~cm}$$, the number of beats is still the same. The frequency of the fork is