Let $\mathrm{A}=\left[\mathrm{a}_{\mathrm{ij}}\right]$ be a $2 \times 2$ matrix such that $\mathrm{a}_{\mathrm{ij}} \in\{0,1\}$ for all i and j . Let the random variable X denote the possible values of the determinant of the matrix A . Then, the variance of X is:

JEE Main · 2025 · 29 Jan 2025 · Shift 2 · Probability

Let

A = [a_{ij}]

be a

2 \times 2

matrix such that

a_{ij} \in {0, 1}

for all i and j . Let the random variable X denote the possible values of the determinant of the matrix A . Then, the variance of X is:

Answer: (B)

Each entry of

A = (a_{11} a_{21} a_{12} a_{22})

0

1

, giving

16

equally likely matrices, and

X = a_{11} a_{22} - a_{21} a_{12}

E (X) = \frac{3 ( 1 ) + 3 ( - 1 ) + 10 ( 0 )}{16} = 0, E (X^{2}) = \frac{3 ( 1 ) + 3 ( 1 ) + 10 ( 0 )}{16} = \frac{3}{8}

Var (X) = E (X^{2}) - [E (X)]^{2} = \frac{3}{8}

JEE Main · 2025 · 29 Jan 2025 · Shift 2 · Single correct option