For the system of equations x + y + z = 6 , x + 2 y + z = 10 , x + 3 y + 5 z , which one of the following is not true? [2023]

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Solution

Q.
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 [2023]

1 System has infinitely many solutions for $α = 3, β = 14 .$

2 System has no solution for $α = 3, β = 24 .$

3 System has a unique solution for $α = 3, β \neq 14 .$

4 System has a unique solution for $α = - 3, β = 14 .$

Ans.
(3)

Let $Δ =$ $| \begin{matrix} 1 & 1 & 1 \\ 1 & 2 & α \\ 1 & 3 & 5 \end{matrix} |$ $= 1 (10 - 3 α) - 1 (5 - α) + 1$ $= 6 - 2 α$

For a unique solution, $Δ \neq 0 \Rightarrow α \neq 3$

Now, if $α = 3$ , then

$Δ_{1} =$ $| \begin{matrix} 6 & 1 & 1 \\ 10 & 2 & 3 \\ β & 3 & 5 \end{matrix} |$ $= 6 (10 - 9) - 1 (50 - 3 β) + 1 (30 - 2 β)$ $= β - 14$

$Δ_{2} =$ $| \begin{matrix} 1 & 6 & 1 \\ 1 & 10 & 3 \\ 1 & β & 5 \end{matrix} |$ $= 1 (50 - 3 β) - 6 (5 - 3) + 1 (β - 10)$ $= 28 - 2 β = - 2 (β - 14)$

$Δ_{3} =$ $| \begin{matrix} 1 & 1 & 6 \\ 1 & 2 & 10 \\ 1 & 3 & β \end{matrix} |$ $= 1 (2 β - 30) - 1 (β - 10) + 6 (3 - 2)$ $= β - 14$

Clearly, at $β = 14$ , $Δ_{i} = 0 ∀ i = 1, 2, 3$

$∴$ $At α = 3, β = 14, the system of equations has infinitely many solutions.$

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