Let N denote the number that turns up when a fair die is rolled. If the probability that the system of equations $\mathrm{x}+\mathrm{y}+\mathrm{z}=1$ $2 \mathrm{x}+\mathrm{Ny}+2 \mathrm{z}=2$ $3 \mathrm{x}+3 \mathrm{y}+\mathrm{Nz}=3$ has unique solution is $\frac{k}{6}$, then the sum of value of k and all possible values of N is

JEE Main · 2023 · 24 Jan 2023 · Shift 2 · Determinants

Let N denote the number that turns up when a fair die is rolled. If the probability that the system of equations

x + y + z = 1

2 x + Ny + 2 z = 2

3 x + 3 y + Nz = 3

has unique solution is

\frac{k}{6}

, then the sum of value of k and all possible values of N is

Answer: (C)

The system has a unique solution precisely when its coefficient determinant is non-zero. Expanding,

Δ = 123 1 N 3 12 N = N^{2} - 5 N + 6 = (N - 2) (N - 3)

, so uniqueness holds whenever

N \neq = 2

and

N \neq = 3

. On a fair die the favourable values are

N \in {1, 4, 5, 6}

, four outcomes, giving probability

\frac{4}{6}

, hence

k = 4

. Adding

k

to these admissible values of

N

4 + (1 + 4 + 5 + 6) = 20.

JEE Main · 2023 · 24 Jan 2023 · Shift 2 · Single correct option