Step 1: From Gopal’s information.
Birthday is between 5th and 6th December (after 4th, before 7th).
Step 2: From sister’s information.
Her birthday is on 8th December.
Two days before 8th → 6th December
Two days after 8th → 10th December
But 10th December is not between 5th and 6th.
Thus, 6th December matches.
Step 3: Conclusion.
Rajesh's birthday is on 6th December.