Consider the system of linear equations x + y + z = 4 , x + 2 y + 2 z = 10, x + 3 y + 4 2 z = 2 + 15 , where ? R . Which one of the following statements is NOT correct? [2024]

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Solution

Q.
Consider the system of linear equations $x + y + z = 4 μ,$ $x + 2 y + 2 λ z = 10 μ,$ $x + 3 y + 4 λ^{2} z = μ^{2} + 15,$ where $λ, μ \in R .$ Which one of the following statements is NOT correct [2024]

1 The system is inconsistent if $λ = \frac{1}{2}$ and $μ \neq 1$

2 The system has unique solution if $λ \neq \frac{1}{2}$ and $μ \neq 1, 15$

3 The system has infinite number of solutions if $λ = \frac{1}{2}$ and $μ = 15$

4 The system is consistent if $λ \neq \frac{1}{2}$

Ans.
(1)

We have, $[\begin{matrix} 1 & 1 & 1 \\ 1 & 2 & 2 λ \\ 1 & 3 & 4 λ^{2} \end{matrix}] [\begin{matrix} x \\ y \\ z \end{matrix}] = [\begin{matrix} 4 μ \\ 10 μ \\ μ^{2} + 15 \end{matrix}]$

For $λ = \frac{1}{2}$ and $μ = 15,$ we get $[\begin{matrix} 1 & 1 & 1 \\ 1 & 2 & 1 \\ 1 & 3 & 1 \end{matrix}] [\begin{matrix} x \\ y \\ z \end{matrix}] = [\begin{matrix} 60 \\ 150 \\ 240 \end{matrix}]$

System is consistent but does not have unique solution as matrix have zero determinant because two columns are same.

Also, let $λ \neq \frac{1}{2}$ and $λ = μ = 1,$ then we have

$[\begin{matrix} 1 & 1 & 1 \\ 1 & 2 & 2 \\ 1 & 3 & 4 \end{matrix}] [\begin{matrix} x \\ y \\ z \end{matrix}] = [\begin{matrix} 4 \\ 10 \\ 16 \end{matrix}]$

$\Rightarrow x + y + z = 4; x + 2 y + 2 z = 10; x + 3 y + 4 z = 16$

On solving these equations, we get $(- 2, 6, 0)$ as unique solution.

Clearly, for $λ = \frac{1}{2}$ and $μ \neq 1$ i.e., $μ = 15$ the system is consistent and have infinite solution.

So, statement (a) is not correct.

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