If the system of equations $2x+3y-z=5$ $x+\alpha y+3z=-4$ $3x-y+\beta z=7$ has infinitely many solutions, then $13 \alpha \beta$ is equal to

Matrices — JEE Main · 2024 · 01 Feb 2024 · Shift 1 Question with Solution

If the system of equations

2 x + 3 y - z = 5

x + α y + 3 z = - 4

3 x - y + β z = 7

has infinitely many solutions, then

13 α β

is equal to

Answer: (B)

For the system

2 x + 3 y - z = 5, x + α y + 3 z = - 4, 3 x - y + β z = 7

to have infinitely many solutions, the third equation must be a linear combination of the first two:

R_{3} = m R_{1} + n R_{2}

for some scalars

m, n

, matching coefficients of

x

and the constant term first:

3 = 2 m + n, 7 = 5 m - 4 n

Solving: from the first,

n = 3 - 2 m

; substituting,

7 = 5 m - 4 (3 - 2 m) = 13 m - 12 ⟹ m = \frac{19}{13}, n = \frac{1}{13}

.
Matching the

y

-coefficient:

- 1 = 3 m + α n ⟹ - 1 = \frac{57}{13} + \frac{α}{13} ⟹ α = - 70

.
Matching the

z

-coefficient:

β = - m + 3 n = - \frac{19}{13} + \frac{3}{13} = - \frac{16}{13}

13 α β = 13 \times (- 70) \times (- \frac{16}{13}) = 70 \times 16 = 1120

JEE Main · 2024 · 01 Feb 2024 · Shift 1 · Single correct option