| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9795/1 (Pre-U Further Mathematics Paper 1) |
| Year | 2014 |
| Session | June |
| Marks | 8 |
| Topic | 3x3 Matrices |
| Type | Inverse given/derived then solve system |
| Difficulty | Standard +0.3 This is a structured multi-part question that guides students through finding a parameter value, setting up a matrix equation, and using matrix inverses to solve a system. While it involves 3×3 matrices (which are computationally heavier than 2×2), the question provides clear scaffolding and the connection between parts makes it easier than average. The matrix multiplication in part (i) is routine but tedious, and recognizing that C = B^T in part (ii) requires some observation, but overall this is a standard Further Maths exercise with no novel insights required. |
| Spec | 4.03b Matrix operations: addition, multiplication, scalar4.03c Matrix multiplication: properties (associative, not commutative)4.03o Inverse 3x3 matrix4.03r Solve simultaneous equations: using inverse matrix |
(i) $\mathbf{AB} = \begin{bmatrix} 1 & 2 & 1 \\ t & 1 & -t \\ 3 & 2 & 1 \end{bmatrix}\begin{bmatrix} t-2 & 0 & 5 \\ 12 & -2 & -6 \\ 3t & 4 & 7 \end{bmatrix} = \begin{bmatrix} 4t+22 & 0 & 0 \\ 12-2t-2t^2 & -2-4t & -2t-6 \\ 6t+18 & 0 & 10 \end{bmatrix}$ **M1**
M1 good effort; A1 all correct **A1**
Give M0 (A0) B1 B1 for **BA** found ($k$, $t$ $\checkmark$) or $k$, $t$ found from 1 or 2 elements of **AB** only
Give M1 (A0) B1 B1 for $k$, $t$ found from most (but not all) elements of **AB**
$= 10\mathbf{I}$ when $t = -3$ Allow these correct from most of **AB** correct **A1 A1** [4]
(ii) $\begin{bmatrix} -5 & 0 & 5 \\ 12 & -2 & -6 \\ -9 & 4 & 7 \end{bmatrix}\begin{bmatrix} x \\ y \\ z \end{bmatrix} = \begin{bmatrix} 8 \\ 12 \\ 22 \end{bmatrix}$ **B1** [1]
(iii) $\begin{bmatrix} x \\ y \\ z \end{bmatrix} = \frac{1}{10}\begin{bmatrix} 1 & 2 & 1 \\ -3 & 1 & 3 \\ 3 & 2 & 1 \end{bmatrix}\begin{bmatrix} 8 \\ 12 \\ 22 \end{bmatrix}$ M1 for $\mathbf{x} = C^{-1}\mathbf{u}$ B1 for $C^{-1}$ correct **M1 B1**
$x = 5.4,\ y = 5.4,\ z = 7$ **A1** [3]
**B1** SC for correct $x$, $y$, $z$ without inverse matrix method seen2 (i) Show that there is a value of $t$ for which $\mathbf { A B }$ is an integer multiple of the $3 \times 3$ identity matrix $\mathbf { I }$, where
$$\mathbf { A } = \left( \begin{array} { r r r }
1 & 2 & 1 \\
t & 1 & - t \\
3 & 2 & 1
\end{array} \right) \quad \text { and } \quad \mathbf { B } = \left( \begin{array} { c r r }
t - 2 & 0 & 5 \\
12 & - 2 & - 6 \\
3 t & 4 & 7
\end{array} \right) .$$
(ii) Express the system of equations
$$\begin{aligned}
- 5 x + 5 z & = 8 \\
12 x - 2 y - 6 z & = 12 \\
- 9 x + 4 y + 7 z & = 22
\end{aligned}$$
in the form $\mathbf { C x } = \mathbf { u }$, where $\mathbf { C }$ is a $3 \times 3$ matrix, and $\mathbf { x }$ and $\mathbf { u }$ are suitable column vectors.\\
(iii) Use the result of part (i) to solve the system of equations given in part (ii).
\hfill \mbox{\textit{Pre-U Pre-U 9795/1 2014 Q2 [8]}}