| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9795 (Pre-U Further Mathematics) |
| Session | Specimen |
| Topic | 3x3 Matrices |
| Type | Find inverse then solve system |
| Difficulty | Standard +0.8 Part (a) is a standard 3×3 matrix inversion and system solving, requiring careful calculation but routine technique. Part (b) elevates this significantly by requiring recognition of when a system is consistent (determinant = 0), finding the parameter k, solving an underdetermined system, and geometric interpretation—this combination of algebraic manipulation with conceptual understanding of linear dependence places it moderately above average difficulty. |
| Spec | 4.03o Inverse 3x3 matrix4.03r Solve simultaneous equations: using inverse matrix4.03s Consistent/inconsistent: systems of equations |
9
\begin{enumerate}[label=(\alph*)]
\item Find the inverse of the matrix $\left( \begin{array} { r r r } 1 & 3 & 4 \\ 2 & 5 & - 1 \\ 3 & 8 & 2 \end{array} \right)$, and hence solve the set of equations
$$\begin{aligned}
x + 3 y + 4 z & = - 5 \\
2 x + 5 y - z & = 10 \\
3 x + 8 y + 2 z & = 8
\end{aligned}$$
\item Find the value of $k$ for which the set of equations
$$\begin{aligned}
x + 3 y + 4 z & = - 5 \\
2 x + 5 y - z & = 15 \\
3 x + 8 y + 3 z & = k
\end{aligned}$$
is consistent. Find the solution in this case and interpret it geometrically.
\end{enumerate}
\hfill \mbox{\textit{Pre-U Pre-U 9795 Q9}}