Standard +0.8 This is a two-part Further Maths question requiring understanding of when systems lack unique solutions (determinant = 0), finding k algebraically, then solving the resulting dependent system with a parameter. It combines matrix theory with systematic algebraic manipulation, going beyond routine A-level exercises but using standard Further Maths techniques.
2 Find the value of the constant \(k\) for which the system of equations
$$\begin{aligned}
2 x - 3 y + 4 z & = 1 \\
3 x - y & = 2 \\
x + 2 y + k z & = 1
\end{aligned}$$
does not have a unique solution.
For this value of \(k\), solve the system of equations.
2 Find the value of the constant $k$ for which the system of equations
$$\begin{aligned}
2 x - 3 y + 4 z & = 1 \\
3 x - y & = 2 \\
x + 2 y + k z & = 1
\end{aligned}$$
does not have a unique solution.
For this value of $k$, solve the system of equations.
\hfill \mbox{\textit{CAIE FP1 2015 Q2 [6]}}