$A$ system goes from $P$ to $Q$ by two different paths in the $P-V$ diagram as shown in the figure. Heat given to the system in path $1$ is $1000 \, J$. The work done by the system along path $1$ is more than path $2$ by $100 \, J$. What is the heat exchanged by the system in path $2$?

(N/A) For path $1$: Heat given $Q_1 = 1000 \, J$,Work done $= W_1$.
For path $2$: Work done $W_2 = W_1 - 100 \, J$.
Since the change in internal energy $\Delta U$ between two states is a state function,it is the same for both paths.
According to the first law of thermodynamics,$\Delta U = Q - W$.
Therefore,$\Delta U = Q_1 - W_1 = Q_2 - W_2$.
Substituting the given values: $1000 - W_1 = Q_2 - (W_1 - 100)$.
$1000 - W_1 = Q_2 - W_1 + 100$.
$Q_2 = 1000 - 100 = 900 \, J$.
Thus,the heat exchanged by the system in path $2$ is $900 \, J$.