The following simultaneous equations
$x + y + z = 3$
$x + 2 y + 3 z = 4$
$x + 4 y + k z = 6$
will not have a unique solution for k equal to

zero

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Solution

The correct option is D 7
$[A : B] = \begin{matrix} ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ \begin{matrix} 1 & 1 & 1 & ⋮ & 3 1 & 2 & 3 & ⋮ & 4 1 & 4 & k & ⋮ & 6 \end{matrix} ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ \end{matrix}$

$\begin{matrix} R_{2} \to R_{2} - R_{1} R_{3} \to R_{3} - R_{1} \end{matrix} \sim \begin{matrix} ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ \begin{matrix} 1 & 1 & 1 & ⋮ & 3 0 & 1 & 2 & ⋮ & 1 1 & 3 & k - 1 & ⋮ & 3 \end{matrix} ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ \end{matrix}$

$\begin{matrix} R_{3} \to R_{3} - 3 R_{2} \end{matrix} \sim \begin{matrix} ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ \begin{matrix} 1 & 1 & 1 & ⋮ & 3 0 & 1 & 2 & ⋮ & 1 0 & 0 & k - 7 & ⋮ & 0 \end{matrix} ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ \end{matrix}$
For infinite number of solutions,

$ρ (A) = ρ (A : B) < 3 \Rightarrow K - 7 = 0$
$\Rightarrow K = 7$

Suggest Corrections

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x + y + z = 3; x + 2 y + 3 z = 4

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