Consider the following system of equationsαx+2y+z=12αx+3y+z=13x+αy+2z=βFor some α,β∈ℝ. Then which of the following is NOT correct.

Δ=α212α313α2=α(6−α)−2(4α−3)+12α2−9=α2−2α−3=(α−3)(α+1)Δ1=121131βα2=1(6−α)−2(2−β)+1(α−3β)=2−βΔ2=α112α113β2=α(2−β)−2α(2−β)+3(0)=−2α+αβ=α(−2+β)Δ3=α212α313αβ=α(3β−α)−2(2αβ−3)+12α2−9=α2−αβ−3α=−1,β≠2⇒Δ=0,Δ1≠0⇒ no solution α≠−1,β=2⇒Δ≠0⇒ unique solution α=3,β≠2⇒Δ=0,Δ1≠0⇒ no solution α=−1,β∈ℝ⇒ no solution or infinitely many sol.

Consider the following system of equationsαx+2y+z=12αx+3y+z=13x+αy+2z=βFor some α,β∈ℝ. Then which of the following is NOT correct.

Δ=α212α313α2=α(6−α)−2(4α−3)+12α2−9=α2−2α−3=(α−3)(α+1)Δ1=121131βα2=1(6−α)−2(2−β)+1(α−3β)=2−βΔ2=α112α113β2=α(2−β)−2α(2−β)+3(0)=−2α+αβ=α(−2+β)Δ3=α212α313αβ=α(3β−α)−2(2αβ−3)+12α2−9=α2−αβ−3α=−1,β≠2⇒Δ=0,Δ1≠0⇒ no solution α≠−1,β=2⇒Δ≠0⇒ unique solution α=3,β≠2⇒Δ=0,Δ1≠0⇒ no solution α=−1,β∈ℝ⇒ no solution or infinitely many sol.

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Consider the following system of equations
$\begin{array}{l} αx + 2 y + z = 1 \\ 2 αx + 3 y + z = 1 \\ 3 x + αy + 2 z = β \end{array}$
For some $α, β \in ℝ$ . Then which of the following is NOT correct.

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

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

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

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

answer is D.

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Detailed Solution

$\begin{array}{l} Δ = |\begin{array}{ccc} α & 2 & 1 \\ 2 α & 3 & 1 \\ 3 & α & 2 \end{array}| = α (6 - α) - 2 (4 α - 3) + 1 (2 α^{2} - 9) \\ = α^{2} - 2 α - 3 \\ = (α - 3) (α + 1) \\ Δ_{1} = |\begin{array}{lll} 1 & 2 & 1 \\ 1 & 3 & 1 \\ β & α & 2 \end{array}| = 1 (6 - α) - 2 (2 - β) + 1 (α - 3 β) \\ = 2 - β \\ Δ_{2} = |\begin{array}{ccc} α & 1 & 1 \\ 2 α & 1 & 1 \\ 3 & β & 2 \end{array}| = α (2 - β) - 2 α (2 - β) + 3 (0) \\ = - 2 α + αβ = α (- 2 + β) \\ Δ_{3} = |\begin{array}{ccc} α & 2 & 1 \\ 2 α & 3 & 1 \\ 3 & α & β \end{array}| = α (3 β - α) - 2 (2 αβ - 3) + 1 (2 α^{2} - 9) \\ = α^{2} - αβ - 3 \\ α = - 1, β \neq 2 \Rightarrow Δ = 0, Δ_{1} \neq 0 \\ \Rightarrow no solution \\ α \neq - 1, β = 2 \Rightarrow Δ \neq 0 \\ \Rightarrow unique solution \\ α = 3, β \neq 2 \Rightarrow Δ = 0, Δ_{1} \neq 0 \\ \Rightarrow no solution \\ α = - 1, β \in ℝ \Rightarrow no solution or infinitely many sol . \end{array}$