Consider the following system of equations: \[ \begin{aligned} & x+2 y-3 z=a \\ & 2 x+6 y-11 z=b \\ & x-2 y+7 z=c \end{aligned} \] where $a, b$ and $c$ are real constants. Then the system of equations :

(A) has infinite number of solutions when $5 a=2 b+c$

(B) has no solution for all $\mathrm{a}, \mathrm{b}$, and c

(C) has a unique solution when $5 a=2 b+c$

(D) has a unique solution for all $\mathrm{a}, \mathrm{b}$ and c

Matrices — JEE Main · 2021 · 26 Feb 2021 · Shift 2 Question with Solution

Consider the following system of equations:

x + 2 y - 3 z = a 2 x + 6 y - 11 z = b x - 2 y + 7 z = c

where

a, b

and

c

are real constants. Then the system of equations :

Answer: (A)

The determinant of the coefficient matrix is

121 26 - 2 - 3 - 11 7 = 1 (42 - 22) - 2 (14 + 11) - 3 (- 4 - 6) = 20 - 50 + 30 = 0

so a unique solution is impossible and the system is either inconsistent or has infinitely many solutions. Looking for

λ, μ

with

R_{3} = λ R_{1} + μ R_{2}

among the coefficients,

λ + 2 μ = 1

and

2 λ + 6 μ = - 2

give

μ = - 2

and

λ = 5

, and these also satisfy

- 3 λ - 11 μ = 7

. The third left side is therefore

5

times the first minus

2

times the second, and consistency demands the same combination of the right sides:

c = 5 a - 2 b

When

5 a = 2 b + c

the three equations are consistent with a vanishing determinant and the system has an infinite number of solutions, while any other choice of

a, b, c

makes it inconsistent.

JEE Main · 2021 · 26 Feb 2021 · Shift 2 · Single correct option