JEE Main · 2021 · 20 Jul 2021 · Shift 2 · Matrices

The value of

k \in R

, for which the following system of linear equations

3 x - y + 4 z = 3

x + 2 y - 3 z = - 2

6 x + 5 y + k z = - 3

,
has infinitely many solutions, is :

Answer: (C)

Solution

For infinitely many solutions the third equation must be a linear combination of the first two. Writing

λ (3 x - y + 4 z) + μ (x + 2 y - 3 z) = 6 x + 5 y + k z

and matching the coefficients of

x

and

y

3 λ + μ = 6, - λ + 2 μ = 5 ⟹ λ = 1, μ = 3

The same combination of the right-hand sides is

1 (3) + 3 (- 2) = - 3

, which is exactly the third right-hand side, so the system is consistent. Matching the coefficient of

z

k = 4 λ - 3 μ = 4 - 9 = - 5

With

k = - 5

the third equation carries no new information and the system has infinitely many solutions, so

k = - 5

JEE Main · 2021 · 20 Jul 2021 · Shift 2 · Single correct option