This question is difficult for who do not know perturbation theory. However, you can eliminate (D) and (E) since they are not famous for correct answers (no thinking, no relation with symmetry of perturbed potential). (A) is also wrong since scaling old ground state wave function yields same ground state energy - not the new perturbed ground state energy. Now, you can guess between (B) and (C).
Actually, the coefficient

is equal to

. Since

is even function and

, the ground state of a particle in a infinite square well, is even,

needs to be odd so that

becomes zero. As a result, we needs

to be odd to make

.
The correct answer is therefore (B). 23 of 100 people answered this question correctly.