Answer
a) $\Delta S_{total}=0.000173\ kJ/K$
$\Delta S\geq 0 \rightarrow\mbox{possible}$
b) $\Delta S_{total}=-0.000101\ kJ/K$
$\Delta S\lt0 \rightarrow\mbox{impossible}$
Work Step by Step
Given: $Q_L=1\ kJ,\ T_H=303\ K,\ T_L=253\ K$
With $COP=4$:
$Q_H=Q_L(1+1/COP)=1.25\ kJ$
$\Delta S_{total}=\dfrac{Q_H}{T_H}-\dfrac{Q_L}{T_L}$
$\Delta S_{total}=0.000173\ kJ/K$
$\Delta S\geq 0 \rightarrow\mbox{possible}$
With $COP=6$:
$Q_H=Q_L(1+1/COP)=1.167\ kJ$
$\Delta S_{total}=\dfrac{Q_H}{T_H}-\dfrac{Q_L}{T_L}$
$\Delta S_{total}=-0.000101\ kJ/K$
$\Delta S\lt0 \rightarrow\mbox{impossible}$
