The values of $a$ and $b$, for which the system of equations $2 x+3 y+6 z=8$ $x+2 y+a z=5$ $3 \mathrm{x}+5 \mathrm{y}+9 \mathrm{z}=\mathrm{b}$ Has no solution, are :

(D) $\mathrm{a}=3, \mathrm{~b}=13$

JEE Main · 2021 · 25 Jul 2021 · Shift 1 · Matrices

The values of

a

and

b

, for which the system of equations

2 x + 3 y + 6 z = 8

x + 2 y + a z = 5

3 x + 5 y + 9 z = b

Has no solution, are :

Answer: (C)

Adding the first two equations gives

3 x + 5 y + (6 + a) z = 13

while the third is

3 x + 5 y + 9 z = b

. The two left sides coincide exactly when

6 + a = 9

, that is

a = 3

; the system then contains

3 x + 5 y + 9 z = 13

alongside

3 x + 5 y + 9 z = b

, which is contradictory precisely when

b \neq = 13

.
If

a \neq = 3

the coefficient determinant

213325 6 a 9 = 2 (18 - 5 a) - 3 (9 - 3 a) + 6 (5 - 6) = 3 - a

is non-zero, so a unique solution exists for every

b

. The system therefore has no solution when

a = 3

and

b \neq = 13

JEE Main · 2021 · 25 Jul 2021 · Shift 1 · Single correct option