The relation between efficiency $(\eta)$ of Carnot engine and coefficient of performance $\left(\eta_1\right)$ of refrigerator is

A radioactive element having half-life 30 min . is undergoing beta decay. The fraction of radioactive element remains undecayed after 90 min . will be

500 gram of a diatomic gas is enclosed at a pressure of $10^5 \mathrm{Nm}^{-2}$. The density of the gas is $5 \mathrm{kgm}^{-3}$. The energy of one mole of the gas due to its thermal motion is [consider the gas molecule as a rigid rotator]

In metre-bridge experiment the balance point is obtained if the gaps are closed by $2 \Omega$ and $3 \Omega$. A shunt of $\mathrm{x} \Omega$ is added to $3 \Omega$ resistor to shift the balance point by 22.5 cm . The value of x is