Consider the following system of equations α x+2 y+z=1 2 α x+3 y+z=1 3 x+α y+2 z=β for some α, β ∈ R. Then which of the following is NOT correct.

Question

Consider the following system of equations α x+2 y+z=1 2 α x+3 y+z=1 3 x+α y+2 z=β for some α, β ∈ R. Then which of the following is NOT correct. (A) It has no solution if α=-1 and β ≠ 2 (B) It has no solution for α=-1 and for all β ∈ R (C) It has a solution for all α ≠-1 and β=2 (D) It has no solution for α=3 and for all β ≠ 2

Tardigrade Admin · Accepted Answer

D=|α 2 1 2 α 3 1 3 α 2|=0 ⇒ α=-1,3 Dx=|2 1 1 3 1 1 α 2 β|=0 ⇒ β=2 Dy= |α 1 1 2 α 1 1 3 2 β|=0 Dz= | α 2 1 2 α 3 1 3 α β|=0 β=2, α=-1 α=-1, β=2 Infinite solution

Q. Consider the following system of equations
$αx + 2 y + z = 1$
$2 αx + 3 y + z = 1$
$3 x + α y + 2 z = β$
for some $α, β \in R$ . Then which of the following is NOT correct.

It has no solution if $α = - 1$ and $β \neq = 2$

It has no solution for $α = - 1$ and for all $β \in R$

It has a solution for all $α \neq = - 1$ and $β = 2$

It has no solution for $α = 3$ and for all $β \neq = 2$

$D = ∣ ∣ α 2 α 3 23 α 112 ∣ ∣ = 0 \Rightarrow α = - 1, 3$
$D_{x} = ∣ ∣ 23 α 112 11 β ∣ ∣ = 0 \Rightarrow β = 2$
$D_{y} = ∣ ∣ α 2 α 3 112 11 β ∣ ∣ = 0$
$D_{z} = ∣ ∣ α 2 α 3 23 α 11 β ∣ ∣ = 0$
$β = 2, α = - 1$
$α = - 1, β = 2$ Infinite solution

Q. Consider the following system of equations αx+2y+z=1 2αx+3y+z=1 3x+αy+2z=β for some α,β∈R. Then which of the following is NOT correct.

It has no solution if α=−1 and β=2

It has no solution for α=−1 and for all β∈R

It has a solution for all α=−1 and β=2

It has no solution for α=3 and for all β=2