| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9795/1 (Pre-U Further Mathematics Paper 1) |
| Year | 2010 |
| Session | June |
| Marks | 4 |
| Topic | 3x3 Matrices |
| Type | Find inverse then solve system |
| Difficulty | Moderate -0.8 This is a straightforward application of matrix methods to solve simultaneous equations. Students are guided through each step: forming equations from coordinate substitution, writing in matrix form, and finding the inverse. The 3×3 inverse calculation is routine, and the context (fitting a quadratic) is familiar. Easier than average due to the scaffolding and standard technique. |
| Spec | 4.03o Inverse 3x3 matrix4.03r Solve simultaneous equations: using inverse matrix |
**(i)**
$3 = p + q + r$
$36 = p + 4q + 16r$
$151 = p + 9q + 81r$ **B1** [1]
**(ii)**
$\begin{pmatrix}1 & 1 & 1\\1 & 4 & 16\\1 & 9 & 81\end{pmatrix}\begin{pmatrix}p\\q\\r\end{pmatrix} = \begin{pmatrix}3\\36\\151\end{pmatrix}$ **B1** ft (i) and condone the odd copying error [1]
**(iii)**
$\mathbf{C}^{-1} = \frac{1}{120}\begin{pmatrix}180 & -72 & 12\\-65 & 80 & -15\\5 & -8 & 3\end{pmatrix}$
**M1** Attempt at $\mathbf{C}^{-1}$ involving either 1/det or transposed matrix of co-factors
**M1** Both
**M1** for $\begin{pmatrix}p\\q\\r\end{pmatrix} = \mathbf{C}^{-1}\begin{pmatrix}3\\36\\151\end{pmatrix}$ **A1** for $p = -2$, $q = 3.5$, $r = 1.5$ **[4]**3 The points $A ( 1,3 ) , B ( 4,36 )$ and $C ( 9,151 )$ lie on the curve with equation $y = p + q x + r x ^ { 2 }$.\\
(i) Using this information, write down three simultaneous equations in $p , q$ and $r$.\\
(ii) Re-write this system of equations in the matrix form $\mathbf { C x } = \mathbf { a }$, where $\mathbf { C }$ is a $3 \times 3$ matrix, $\mathbf { x }$ is an unknown vector, and $\mathbf { a }$ is a fixed vector.\\
(iii) By finding $\mathbf { C } ^ { - 1 }$, determine the values of $p , q$ and $r$.
\hfill \mbox{\textit{Pre-U Pre-U 9795/1 2010 Q3 [4]}}