| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9795/1 (Pre-U Further Mathematics Paper 1) |
| Session | Specimen |
| Marks | 3 |
| Topic | 3x3 Matrices |
| Type | Determinant calculation and singularity |
| Difficulty | Standard +0.3 This is a straightforward multi-part question requiring determinant calculation (standard 3×3 formula), recognizing that singular matrices correspond to non-unique solutions, and solving a 3×3 system. All techniques are routine for Further Maths students with no novel insight required, making it slightly easier than average. |
| Spec | 4.03j Determinant 3x3: calculation4.03r Solve simultaneous equations: using inverse matrix4.03s Consistent/inconsistent: systems of equations |
**(i)** Method for Sarrus' Rule, or expanding by $R_1$ e.g.: M1
$\text{Det} = 11k - 66$: A1 **[2]**
**(ii)** $k = 6$ **ft** from their Det $= 0$: B1 **[1]**
**(iii)** *EITHER:* e.g. $\circled{3} - 6 \times \circled{1} \Rightarrow z = 7$
e.g. $\circled{3} + 2 \times \circled{2} \Rightarrow 22y + 23z = 73 \Rightarrow 22y = -88 \Rightarrow y = -4$
For a complete solution strategy: M1
e.g. $x = 4 - 2y - z = 5$, for first correct: A1
$x = 5,\ y = -4,\ z = 7$, for all 3 correct: A1
*OR:* $\frac{1}{11}\begin{pmatrix}-61 & -2 & 11 \\ 69 & 1 & -11 \\ -66 & 0 & 11\end{pmatrix}\begin{pmatrix}4\\21\\31\end{pmatrix} = \begin{pmatrix}5\\-4\\7\end{pmatrix}$
For complete method: M1; For correct inverse of the matrix of coefficients: B1; For correct answer: A1 **[3]**3 (i) Evaluate, in terms of $k$, the determinant of the matrix $\left( \begin{array} { r r r } 1 & 2 & 1 \\ - 3 & 5 & 8 \\ 6 & 12 & k \end{array} \right)$.
Three planes have equations $x + 2 y + z = 4 , - 3 x + 5 y + 8 z = 21$ and $6 x + 12 y + k z = 31$.\\
(ii) State the value of $k$ for which these three planes do not meet at a single point.\\
(iii) Find the coordinates of the point of intersection of the three planes when $k = 7$.
\hfill \mbox{\textit{Pre-U Pre-U 9795/1 Q3 [3]}}