If S is the set of distinct values of ' b ' for which the following system of linear equations \[ \begin{aligned} & x+y+z=1 \\ & x+\mathrm{a} y+z=1 \\ & \mathrm{a} x+\mathrm{b} y+z=0 \end{aligned} \] has no solution, then S is :

(D) a finite set containing two or more elements

JEE Main · 2017 · 02 Apr 2017 · Offline · Matrices

If S is the set of distinct values of ' b ' for which the following system of linear equations

x + y + z = 1 x + a y + z = 1 a x + b y + z = 0

has no solution, then S is :

Answer: (A)

No solution demands a vanishing coefficient determinant together with inconsistent equations. That determinant works out to

- (a - 1)^{2}

, zero only at

a = 1

. With

a = 1

the first two equations become identical, and the third,

a x + b y + z = 0

, then clashes with

x + y + z = 1

precisely when

b = 1

— parallel but not coincident. So

b = 1

alone kills the system, and

S = {1}

is a singleton.

JEE Main · 2017 · 02 Apr 2017 · Offline · Single correct option