Consider the system of linear equations x + y + z = 5 , x + 2 y + 2 z = 9 , x + 3 y + z = , where. Then which of the following statement is NOT correct? [2024]

Question

Consider the system of linear equations $x + y + z = 5,$ $x + 2 y + λ^{2} z = 9,$ $x + 3 y + λ z = μ,$ where $λ, μ \in R .$ Then which of the following statement is NOT correct? [2024]

Smart Achievers · Accepted Answer

(1)

$| \begin{matrix} 1 & 1 & 1 \\ 1 & 2 & λ^{2} \\ 1 & 3 & λ \end{matrix} |$ $= 0 \Rightarrow 2 λ^{2} - λ - 1 = 0$

$\Rightarrow λ = 1, - \frac{1}{2} and$ $| \begin{matrix} 1 & 1 & 5 \\ 2 & λ^{2} & 9 \\ 3 & λ & μ \end{matrix} |$ $= 0 \Rightarrow μ = 13$

For infinite solution, $λ = 1$ and $μ = 13$

For unique solution $λ \neq 1$

For no solution, $λ = 1$ and $μ \neq 13$

If $λ \neq 1$ and $μ \neq 13$

Considering the case when $λ = - \frac{1}{2}$ and $μ \neq 13$ this will generate no solution case.

Solution

Q.
Consider the system of linear equations $x + y + z = 5,$ $x + 2 y + λ^{2} z = 9,$ $x + 3 y + λ z = μ,$ where $λ, μ \in R .$ Then which of the following statement is NOT correct [2024]

1 System has unique solution if $λ \neq 1$ and $μ \neq 13$ .

2 System is inconsistent if $λ = 1$ and $μ \neq 13 .$

3 System is consistent if $λ \neq 1$ and $μ = 13 .$

4 System has infinite number of solutions if $λ = 1$ and $μ = 13 .$

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Solution

Q. Consider the system of linear equations x+y+z=5,x+2y+λ2z=9,x+3y+λz=μ, where λ,μ∈R. Then which of the following statement is NOT correct [2024]

1 System has unique solution if λ≠1 and μ≠13.

2 System is inconsistent if λ=1 and μ≠13.

3 System is consistent if λ≠1 and μ=13.

4 System has infinite number of solutions if λ=1 and μ=13.