In a process, temperature and volume of one mole of an ideal monoatomic gas are varied according to the relation VT=K, where K is a constant. In this process the temperature of the gas is increased by ΔT. The amount of heat absorbed by gas is?

VT=K ⇒V(PV/nR)=k ⇒PV²=K (Since n=1 and R is constant)
For a polytropic process, PVx = constant. Comparing this with PV²=K, we get x=2.
The molar heat capacity for a polytropic process is given by:
C = R/(1-x) + Cv
For a monoatomic gas, Cv = (3/2)R
C = R/(1-2) + (3/2)R = -R + (3/2)R = (1/2)R
Therefore, ΔQ = nCΔT = 1 × (1/2)R × ΔT = (1/2)RΔT