A coal based thermal power plant producing electricity operates between the temperatures $27°\mathrm{C}$ and $227°\mathrm{C}.$ The plant works at $80\%$ of its maximum theoretical efficiency. Complete burning of $1 \mathrm{kg}$ of coal yields $36000 \mathrm{KJ}$ of heat. A house needs$10$ units of electricity each day. Coal used for supplying the amount of energy for the house in one year is

In the given figure, an ideal gas is trapped between a mercury column and the closed end of a uniform vertical tube. The upper end of the tube is open to the atmosphere. Initially, the lengths of the mercury column and the trapped air column are $12 \mathrm{cm}$ and $50 \mathrm{cm}$, respectively. When the tube is tilted slowly in a vertical plane through an angle of $30°$ with horizontal, then the new length of air column is $\frac{x }{41} \mathrm{m}$. Find $x$ (assume the temperature to remain constant $(P_{\mathrm{atm}}=76 \mathrm{cm} \text{of} \mathrm{Hg})$).

Two moles of an ideal monoatomic gas undergo a cyclic process which is indicated on a $P-U$ diagram where $U$ is the internal energy of the gas. The work done by the gas in the cycle is $k\times {10}^2\ln 2$. Find $k$.