A source supplies heat to a system at the rate of 1000 W. If the system performs work at a rate of 200 W. The rate at which internal energy of the system increases is

By the first law of thermodynamics, the rate of heat supply equals the rate of change of internal energy plus the rate of work done by the system:

$$\frac{dQ}{dt} = \frac{dU}{dt} + \frac{dW}{dt}$$

We are given $$\frac{dQ}{dt} = 1000\;\text{W}$$ and $$\frac{dW}{dt} = 200\;\text{W}$$. Substituting:

$$1000 = \frac{dU}{dt} + 200$$

So $$\frac{dU}{dt} = 1000 - 200 = 800\;\text{W}$$. Hence, the rate at which internal energy increases is 800 W.