Let n be the number obtained on rolling a fair die. If the probability that the system x - ny + z =6 , x +( n -2) y +( n +1) z =8 , (n-1) y+z=1 Has a unique solution is frac k 6 , then the sum of k and all possible values of n is :

Mathematics · Matrices

JEE Main 2026 — 22 January, Evening Shift — Question 1

Let $n$ be the number obtained on rolling a fair die. If the probability that the system $\mathrm{x}-\mathrm{ny}+\mathrm{z}=6$ , $\mathrm{x}+(\mathrm{n}-2) \mathrm{y}+(\mathrm{n}+1) \mathrm{z}=8$ , $(n-1) y+z=1$ Has a unique solution is $\frac{\mathrm{k}}{6}$ , then the sum of k and all possible values of $n$ is :

Option A:
$21$
Option B:
$24$
Option C:
$20$
Option D:
$22$
Correct

Answer: D

Step-by-step solution

x - ny + z = 6

x + (n-2)y + (n+1)z = 8

(n-1)y + z = 1

For a unique solution,

\begin{vmatrix} 1 & -n & 1 \\ 1 & n-2 & n+1 \\ 0 & n-1 & 1 \end{vmatrix} \neq 0

\Rightarrow n^2 - 3n + 2 \neq 0

\Rightarrow (n-1)(n-2) \neq 0

\Rightarrow n \neq 1,\,2

Hence for unique solution,

n = 3,4,5,6

$P$ (system has unique solution) $= \frac{4}{6}$

\therefore k = 4

Required sum $= 4 + (3+4+5+6) = 22$

Answer key and solution verified before publishing.

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Exam: JEE Main 2026
Paper: 22 January, Evening Shift
Subject: Mathematics
Chapter: Matrices
Topic: solving System of Linear Equations using Matrices

If X =beginbmatrixx \y \zendbmatrix is a solution of the system of equations AX = B where adj(A) = beginbmatrix 4 & 2 & 2 \\ -5 & 0 & 5 \\…

Answer: D

Step-by-step solution

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