$1.$ A house is to be maintained at a temperature of $200$C by means of a heat pump pumping heat from the atmosphere.$$ Heat through the walls of the house is estimated at $0\mathrm{.}65$ kW per unit of temperature difference between the inside of the house and the atmosphere.$($a$)$ If the atmospheric temperature is $-100$C$,$ what is the minimum power required to drive the pump? $($b$)$ It is proposed to use the same heat pump to cool the house in summer.$$ For the same room temperature, the same heat loss rate, and the same power input to the pump, what is the maximum permissible atmospheric temperature?