| Exam Board | OCR |
|---|---|
| Module | Further Pure Core 2 (Further Pure Core 2) |
| Year | 2018 |
| Session | September |
| Marks | 10 |
| Topic | 3x3 Matrices |
| Type | General solution with parameters |
| Difficulty | Challenging +1.2 This is a structured Further Maths question on matrices and linear systems. Part (i) is routine matrix multiplication given the inverse. Part (ii) requires finding when det(A)=0 and applying consistency conditions, which is standard FM material. Part (iii) asks for geometric interpretation. While it spans multiple concepts, each step follows predictable FM techniques without requiring novel insight—slightly above average due to the multi-part nature and FM content. |
| Spec | 4.03r Solve simultaneous equations: using inverse matrix4.03s Consistent/inconsistent: systems of equations |
| Answer | Marks | Guidance |
|---|---|---|
| \(a = 1\) | B1 | soi |
| \(\frac{1}{-24} \begin{pmatrix} 24 & -12 & 0 \\ -48 & 15 & 6 \\ 16 & -6 & -4 \end{pmatrix} \begin{pmatrix} 6 \\ 8 \\ k \end{pmatrix}\) | M1 | |
| \(x = -2, y = 7 - \frac{1}{4}k, z = -2 + \frac{1}{4}k\) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| \(a = 2\) \(b = 5\) | B1 B1 | soi soi |
| \(4x - 6c = 18\) | M1 | Use of \(2x + 2y + 3z = 5\) and \(-2x + 2y + 9z = -13\) to eliminate \(y\) or \(z\) |
| \(z = \frac{2}{3}x - 3\) \(y = 7 - 2x\) | A1 A1 |
| Answer | Marks |
|---|---|
| The solution represents a straight line E.g. Two of the planes are identical and the third intersects it/them (in a straight line) | B1 B1 |
## (i)
$a = 1$ | B1 | soi
$\frac{1}{-24} \begin{pmatrix} 24 & -12 & 0 \\ -48 & 15 & 6 \\ 16 & -6 & -4 \end{pmatrix} \begin{pmatrix} 6 \\ 8 \\ k \end{pmatrix}$ | M1 |
$x = -2, y = 7 - \frac{1}{4}k, z = -2 + \frac{1}{4}k$ | A1 |
[3]
## (ii)
$a = 2$ $b = 5$ | B1 B1 | soi soi
$4x - 6c = 18$ | M1 | Use of $2x + 2y + 3z = 5$ and $-2x + 2y + 9z = -13$ to eliminate $y$ or $z$
$z = \frac{2}{3}x - 3$ $y = 7 - 2x$ | A1 A1 |
[5]
## (iii)
The solution represents a straight line E.g. Two of the planes are identical and the third intersects it/them (in a straight line) | B1 B1 |
[2]
---The matrix $\mathbf{A}$ is given by $\mathbf{A} = \begin{pmatrix} a & 2 & 3 \\ 4 & 4 & 6 \\ -2 & 2 & 9 \end{pmatrix}$ where $a$ is a constant. It is given that if $\mathbf{A}$ is not singular then
$$\mathbf{A}^{-1} = \frac{1}{24a-48} \begin{pmatrix} 24 & -12 & 0 \\ -48 & 9a+6 & 12-6a \\ 16 & -2a-4 & 4a-8 \end{pmatrix}.$$
\begin{enumerate}[label=(\roman*)]
\item Use $\mathbf{A}^{-1}$ to solve the simultaneous equations below, giving your answer in terms of $k$.
\begin{align}
x + 2y + 3z &= 6\\
4x + 4y + 6z &= 8\\
-2x + 2y + 9z &= k
\end{align} [3]
\item Consider the equations below where $a$ takes the value which makes $\mathbf{A}$ singular.
\begin{align}
ax + 2y + 3z &= b\\
4x + 4y + 6z &= 10\\
-2x + 2y + 9z &= -13
\end{align}
$b$ takes the value for which the equations have an infinite number of solutions.
\begin{itemize}
\item Determine the value of $b$.
\item Find the solutions for $y$ and $z$ in terms of $x$. [5]
\end{itemize}
\item For the equations in part (ii) with the values of $a$ and $b$ found in part (ii) describe fully the geometrical arrangement of the planes represented by the equations. [2]
\end{enumerate}
\hfill \mbox{\textit{OCR Further Pure Core 2 2018 Q4 [10]}}