The system of linear equation $x+y+z=2,2 x+3 y+2 z=5,2 x+3 y+\left(a^{2}-1\right) z=a+1$ then

(D) inconsistent when $|\mathrm{a}|=\sqrt{3}$

(A) is inconsistent when $\mathrm{a}=4$

(B) has a unique solution for $|\mathrm{a}|=\sqrt{3}$

(C) has infinitely many solutions for $\mathrm{a}=4$

JEE Main · 2019 · 09 Jan 2019 · Shift 1 · Matrices

The system of linear equation

x + y + z = 2, 2 x + 3 y + 2 z = 5, 2 x + 3 y + (a^{2} - 1) z = a + 1

then

Answer: (D)

The coefficient determinant works out to

a^{2} - 3

, so for

∣ a ∣ \neq = 3

there is a unique solution. When

∣ a ∣ = 3

the determinant vanishes; the third equation then reads

2 x + 3 y + 2 z = a + 1

, but the second already fixes

2 x + 3 y + 2 z = 5

, and

a + 1 \neq = 5

for

a = \pm 3

. The two demands clash, so the system is inconsistent when

∣ a ∣ = 3

JEE Main · 2019 · 09 Jan 2019 · Shift 1 · Single correct option