JEE 2023 — Mathematics PYQ

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Question

JEE 2023 — Mathematics PYQ

JEE | Mathematics | 2023

For the system of equations

$x + y + z = 6$

$x + 2 y + α z = 10$

$x + 3 y + 5 z = β$ ,

which one of the following is NOT true :

Choose the correct answer:

Explanation

For unique solution $Δ \neq = 0$
For infinitely many solutions $Δ = Δ_{1} = Δ_{2} = Δ_{3} = 0$

$Δ_{1} = 6 10 β am p; 1 am p; 2 am p; 3 am p; 1 am p; α am p; 5 = 6 (10 - 3 α) - (50 - α β) + (30 - 2 β) = 40 - 18 α + α β - 2 β$

$Δ_{2} = 1 1 1 am p; 6 am p; 10 am p; β am p; 1 am p; α am p; 5 = (50 - α β) - 6 (5 - α) + (β - 10) = 10 + 6 α + β - α β$

$Δ_{3} = 1 1 1 am p; 1 am p; 2 am p; 3 am p; 6 am p; 10 am p; β = (2 β - 30) - (β - 10) + 6 = β - 14$

$Δ = 1 1 1 am p; 1 am p; 2 am p; 3 am p; 1 am p; α am p; 5 = 1 (10 - 3 α) - (5 - α) + (3 - 2) = 6 - 2 α$

For infinitely many solutions
$Δ = 0, Δ_{1} = 0, Δ_{2} = 0, Δ_{3} = 0$

$α = 3, β = 14$

For unique solution α ≠ 3