JEE Main 2025 — 29 January, Evening Shift — Question 56

Let

$$A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}, \quad a,b,c,d \in \{0,1\}.$$

Let the random variable $X = \det(A) = ad - bc$.

Count possibilities $X = 1$ if $ad = 1$ and $bc = 0$ $\Rightarrow$ 3 cases $X = -1$ if $ad = 0$ and $bc = 1$ $\Rightarrow$ 3 cases $X = 0$ otherwise $\Rightarrow$ 10 cases

Total matrices: $2^4 = 16$.

Hence, the probabilities are:

$$P(X=1) = \frac{3}{16}, \quad P(X=-1) = \frac{3}{16}, \quad P(X=0) = \frac{10}{16} = \frac{5}{8}.$$

Compute expectation and variance

$$E(X) = 1\cdot \frac{3}{16} + (-1)\cdot \frac{3}{16} + 0 \cdot \frac{10}{16} = 0$$ $$E(X^2) = 1^2 \cdot \frac{3}{16} + (-1)^2 \cdot \frac{3}{16} + 0^2 \cdot \frac{10}{16} = \frac{6}{16} = \frac{3}{8}$$ $$\operatorname{Var}(X) = E(X^2) - [E(X)]^2 = \frac{3}{8} - 0 = \frac{3}{8}$$ $$\boxed{\operatorname{Var}(X) = \frac{3}{8}}$$