Q32 / 90 · JEE Main 2022 (26 Jun Shift 2) · Mathematics

If the system of equations $α x + y + z = 5, x + 2 y + 3 z = 4, x + 3 y + 5 z = β$ . Has infinitely many solutions, then the ordered pair $(α, β)$ is equal to

Solution

We know that, for infinitely many solutions,

$Δ = 0 = Δ_{x} = Δ_{y} = Δ_{z}$

So, $Δ = |\begin{array}{l} α & 1 & 1 \\ 1 & 2 & 3 \\ 1 & 3 & 5 \end{array}| = 0$

$\Rightarrow α (10 - 9) - 1 (5 - 3) + 1 (3 - 2) = 0$

$\Rightarrow α - 2 + 1 = 0$

$\Rightarrow α = 1$

Now checking for $Δ_{x} = |\begin{array}{l} 5 & 1 & 1 \\ 4 & 2 & 3 \\ β & 3 & 5 \end{array}| = 0$

$\Rightarrow 5 (10 - 9) - 1 (20 - 3 β) + 1 (12 - 2 β)$

$\Rightarrow 5 - 20 + 3 β + 12 - 2 β$

$\Rightarrow - 3 + β = 0$

$\Rightarrow β = 3$

Hence $(α, β) = (1, 3)$

Hence, the correct answer is option: 'C'.