First slide

System of linear equations

Question

If a, b, c are in G.P. with common ratio r₁ and $α, β, γ$ are in G.P. with common ratio r₂, and equations ax + $α$ y + z = 0, bx + $β$ y + z = 0, cx + $γ$ y + z = 0 have only zero solution, then which of the following is not true?

Moderate

A

$r_{1} \neq 1$
B

$r_{2} \neq 1$
C

$r_{1} \neq r_{2}$
D

none of these

Solution

As a, b, c are in G.P. with common ratio r₁ and $α, β, γ$ are in G.P. having common ratio $r_{2}, a \neq 0, α \neq 0, b = {ar}_{1}, c = {ar}_{1}^{2}, β = {ar}_{2}, γ = {ar}_{2}^{2}$ .
Also the system of equations has only zero (trivial) solution.
$Δ = |\begin{matrix} a & α & 1 \\ b & β & 1 \\ c & γ & 1 \end{matrix}| \neq 0$
$\begin{array}{l} \Rightarrow aα |\begin{array}{ccc} 1 & 1 & 1 \\ r_{1} & r_{2} & 1 \\ r_{1}^{2} & r_{2}^{2} & 1 \end{array}| \neq 0 \\ \Rightarrow aα (r_{1} - 1) (r_{2} - 1) (r_{1} - r_{2}) \neq 0 \\ \Rightarrow r_{1} \neq 1, r_{2} \neq 1, and r_{1} \neq r_{2} \end{array}$

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