What is the current through R₂?

ANS:A - 177 mA

Even using the super-position theorem, the result remains the same. But we need to apply it with care.
Lets, take the first source, 12V alone and put a short across the 6V supply.
Now, calculate the effective resistance of the circuit.
Eff Res. = 49 + (80*24)/(80+24)
= 67.17 ohm.
Now total current supplied by the source 12V = 12/67.17
= 0.178A.

The current through the 24-ohm resistor (using current divider rule) = main current * resistance of other branch / total resistance.

ie. = 0.178 * 80/(80+24).
= 0.137A ----------------- current through R2 by source 12V.

Now take the same using 2nd source, 6V.

Eff. res. = 49*24/(49+24) + 80.
= 96.109 ohm.

Total current supplied by the 6V source = 6/96.109.
= 0.06 A.

current through R2 resistor = 0.06*49/(49+24).
= 0.040A ---- current through R2 by the source 6 V.

Now add the individual source currents to obtain the resultant current through the R2 resistor = 0.137 + 0.40.
= 0.177A.

@Nawal Kumar.

You made some mistakes. Please refer the above derivation.