Find the value of \(k\) for which the set of linear equations
$$\begin{aligned}
x + 3 y + k z & = 4 \\
4 x - 2 y - 10 z & = - 5 \\
x + y + 2 z & = 1
\end{aligned}$$
has no unique solution.
For this value of \(k\), find the set of possible solutions, giving your answer in the form
$$\left( \begin{array} { c }
x \\
y \\
z
\end{array} \right) = \mathbf { a } + t \mathbf { b } ,$$
where \(\mathbf { a }\) and \(\mathbf { b }\) are vectors and \(t\) is a scalar.
4 (i) Find the value of $k$ for which the set of linear equations
$$\begin{aligned}
x + 3 y + k z & = 4 \\
4 x - 2 y - 10 z & = - 5 \\
x + y + 2 z & = 1
\end{aligned}$$
has no unique solution.\\
(ii) For this value of $k$, find the set of possible solutions, giving your answer in the form
$$\left( \begin{array} { c }
x \\
y \\
z
\end{array} \right) = \mathbf { a } + t \mathbf { b } ,$$
where $\mathbf { a }$ and $\mathbf { b }$ are vectors and $t$ is a scalar.\\
\hfill \mbox{\textit{CAIE FP1 2017 Q4}}