The probability that a man lives after ten years is $\frac{1}{4}$ and that his wife being alive after ten years is $\frac{1}{3}$. Then probability that neither of them is alive after 10 years is

$\frac{1}{2}$

Let P(A) = the probability that the man is alive after ten years.

Then P(A) = $\frac{1}{4}$ and $P(\bar{A})=\frac{3}{4}$.

Let P(B) = the probability that his wife is alive after 10 years.

Then P(B) = $\frac{1}{3}$ and $P(\bar{B})=\frac{2}{3}$.

Hence $P(\bar{A}∩\bar{B})=P(\bar{A}).P(\bar{B})=\frac{3}{4}.\frac{2}{3}=\frac{1}{2}$

Hence (A) is the correct answer.