| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9795/1 (Pre-U Further Mathematics Paper 1) |
| Year | 2013 |
| Session | November |
| Marks | 10 |
| Topic | 3x3 Matrices |
| Type | Consistency conditions for systems |
| Difficulty | Challenging +1.2 This is a standard linear algebra problem on systems of equations requiring determinant calculation to find when the system is singular, then using consistency conditions. While it involves multiple steps (finding determinant, setting to zero, solving for k, then finding m via row reduction or substitution), these are well-practiced techniques at Further Maths level. The geometric interpretation part is straightforward recall. More routine than problems requiring novel insight, but above average due to the algebraic manipulation and multi-part nature. |
| Spec | 4.03l Singular/non-singular matrices4.03r Solve simultaneous equations: using inverse matrix4.03t Plane intersection: geometric interpretation |
\begin{enumerate}[label=(\roman*)]
\item Show that there is exactly one value of $k$ for which the system of equations
\begin{align}
kx + 2y + kz &= 4\\
3x + 10y + 2z &= m\\
(k - 1)x - 4y + z &= k
\end{align}
does not have a unique solution. [4]
\item Given that the system of equations is consistent for this value of $k$, find the value of $m$. [4]
\item Explain the geometrical significance of a non-unique solution to a $3 \times 3$ system of linear equations. [2]
\end{enumerate}
\hfill \mbox{\textit{Pre-U Pre-U 9795/1 2013 Q9 [10]}}