Answer
a) Path $1$: Greatest.
Path $3$: Least.
b) Path $1$ and heat is absorbed by the system.
Work Step by Step
Work done is the area under the $P-V$ curve.
a) We see path $1$ has the greatest area. Thus it corresponds to maximum work done by the system.
For path $3$, area under the curve is the least. It corresponds to least work done.
b) From first law of Thermodynamics:
$Q=U_b-U_a +\Delta W$, where $\Delta W $ is the work done. $U_b>U_a$ and $\Delta W>0$ for all the paths. Therefore, $Q>0$. It implies $|Q|=Q$.
$U_b-U_a$ is fixed. For maximum $|Q|$ we have maximum $\Delta W $ which corresponds to path $1$.
$Q>0$ implies heat is absorbed by the system.
