| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9795/1 (Pre-U Further Mathematics Paper 1) |
| Year | 2019 |
| Session | Specimen |
| Marks | 2 |
| Topic | 3x3 Matrices |
| Type | Determinant calculation and singularity |
| Difficulty | Standard +0.3 This is a straightforward multi-part question on 3×3 determinants and linear systems. Part (a) requires routine determinant calculation using cofactor expansion or row operations. Part (b) is direct application of the singularity condition (det=0). Part (c) involves solving a 3×3 system, which is standard but slightly tedious. The connection between parts is transparent, requiring no novel insight—slightly easier than average due to its mechanical nature. |
| Spec | 4.03j Determinant 3x3: calculation4.03r Solve simultaneous equations: using inverse matrix |
**(a)** Method for Sarrus' Rule, or expanding by $R_1$ for example **M1**
$\det = 11k - 66$ **A1**
**Total: 2**
**(b)** $k = 6$ **ft** from their $\det = 0$ **B1** (B1ft)
**Total: 1**
**(c) EITHER**
e.g. $\circled{3} - 6\times\circled{1} \Rightarrow z = 7$
e.g. $\circled{3} + 2\times\circled{2} \Rightarrow 22y + 23z = 73 \Rightarrow 22y = -88 \Rightarrow y = -4$
M1 for a complete solution strategy **M1**
e.g. $x = 4 - 2y - z = 5$ A1 for first correct **A1**
$x = 5, y = -4, z = 7$ A1 for all 3 correct **A1**
**OR**
$\frac{1}{11}\begin{pmatrix}-61 & -2 & 11 \\ 69 & 1 & -11 \\ -66 & 0 & 11\end{pmatrix}\begin{pmatrix}4\\21\\31\end{pmatrix} = \begin{pmatrix}5\\-4\\7\end{pmatrix}$
M1 for complete method **M1**
B1 for correct inverse of the matrix of coefficients **B1**
A1 for correct answer **A1**
**Total: 3**3
\begin{enumerate}[label=(\alph*)]
\item Evaluate, in terms of $k$, the determinant of the matrix $\left( \begin{array} { c c c } 1 & 2 & 1 \\ - 3 & 5 & 8 \\ 6 & 12 & k \end{array} \right)$.
Three planes have equations $x + 2 y + z = 4 , - 3 x + 5 y + 8 z = 21$ and $6 x + 12 y + k z = 31$.
\item State the value of $k$ for which these three planes do not meet at a single point.
\item Find the coordinates of the point of intersection of the three planes when $k = 7$.
\end{enumerate}
\hfill \mbox{\textit{Pre-U Pre-U 9795/1 2019 Q3 [2]}}