JEE Main · 2024 · 05 Apr 2024 · Shift 2 · Matrices

The values of

m, n

, for which the system of equations

x + y + z = 4

2 x + 5 y + 5 z = 17

x + 2 y + m z = n

has infinitely many solutions, satisfy the equation :

Answer: (B)

Solution

From the first two equations, eliminate

z

2 (x + y + z) - (2 x + 5 y + 5 z) = 2 (4) - 17 ⟹ - 3 y - 3 z = - 9 ⟹ y + z = 3

, and then

x = 4 - (y + z) = 1

.
For the system to have infinitely many solutions, the third equation must be automatically satisfied for every value of the free parameter

y

(with

z = 3 - y

1 + 2 y + m (3 - y) = n ⟹ (2 - m) y + (1 + 3 m) = n for all y

This requires

2 - m = 0 ⟹ m = 2

, and then

1 + 3 (2) = n ⟹ n = 7

m^{2} + n^{2} - mn = 4 + 49 - 14 = 39

JEE Main · 2024 · 05 Apr 2024 · Shift 2 · Single correct option