$\Rightarrow \mathrm{Q}_{\mathrm{ABCA}}=\mathrm{W}_{\mathrm{ABCA}}$
$\Rightarrow \mathrm{Q}_{\mathrm{A} \rightarrow \mathrm{B}}+\mathrm{Q}_{\mathrm{B} \rightarrow \mathrm{C}}+\mathrm{Q}_{\mathrm{C} \rightarrow \mathrm{A}}$
$=\mathrm{W}_{\mathrm{A} \rightarrow \mathrm{B}}+\mathrm{W}_{\mathrm{B} \rightarrow \mathrm{C}}+\mathrm{W}_{\mathrm{C} \rightarrow \mathrm{A}}$ $.(1)$
Given $Q_{B \rightarrow C}+Q_{C \rightarrow A}=W_{A \rightarrow B}+W_{B \rightarrow C}$ $(2)$
Subtracting $( 2)$ from $( 1)$
$\mathrm{Q}_{\mathrm{A} \rightarrow \mathrm{B}}=\mathrm{W}_{\mathrm{C} \rightarrow \mathrm{A}}=0$
(as process $\mathrm{C} \rightarrow \mathrm{A}$ is isochoric)
process $A \rightarrow B$ is adiabatic.
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$(A)$ The average energy per mole of the gas mixture is $2RT$.
$(B)$ The ratio of speed of sound in the gas mixture to that in helium gas is $\sqrt{6 / 5}$.
$(C)$ The ratio of the rms speed of helium atoms to that of hydrogen molecules is $1 / 2$.
$(D)$ The ratio of the rms speed of helium atoms to that of hydrogen molecules is $1 / \sqrt{2}$.