| Exam Board | OCR |
|---|---|
| Module | Further Pure Core AS (Further Pure Core AS) |
| Year | 2022 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Matrices |
| Type | Conditions for unique solution |
| Difficulty | Moderate -0.3 Part (a) involves routine matrix operations (addition, multiplication, squaring) requiring only direct application of definitions. Part (b)(i) is straightforward determinant calculation and solving a linear equation. Part (b)(ii) requires understanding that no unique solution means determinant equals zero, then solving—this is standard A-level Further Maths content with no novel insight required. Overall slightly easier than average due to being mostly procedural. |
| Spec | 4.03b Matrix operations: addition, multiplication, scalar4.03h Determinant 2x2: calculation4.03r Solve simultaneous equations: using inverse matrix |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(A+B=\begin{pmatrix}a-2 & 6\\-2 & 3\end{pmatrix}\) | B1 | Any double signs must be simplified correctly |
| \(AB=\begin{pmatrix}-2a-1 & 5a\\-1 & -5\end{pmatrix}\) | B1 | |
| \(A^2=\begin{pmatrix}a^2-1 & a+3\\-a-3 & 8\end{pmatrix}\) | B1 | |
| [3] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \((\det\mathbf{A}) = a\times3 - 1\times-1\) | M1 | Correct expansion of determinant of \(\mathbf{A}\) |
| \(3a+1=25 \Rightarrow a=8\) | A1 | |
| [2] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| System reduces to \(A\begin{pmatrix}x\\y\end{pmatrix}=\begin{pmatrix}-2\\-6\end{pmatrix}\) so no unique solution \(\Rightarrow (\det\mathbf{A})=3a+1=0\) | M1 | Setting their determinant to 0 if it is a linear function of \(a\). Answer only is ok here |
| \(\therefore a = -\frac{1}{3}\) | A1 | |
| Alternative: Multiplying the first equation by 3 gives \(3ax+3y=-6\) and \(-x+3y=-6\). These two equations are the same if \(3a=-1 \Rightarrow a=\dfrac{-1}{3}\) | M1, A1 | Or subtracting gives \((3a+1)x=0 \rightarrow a=\dfrac{-1}{3}\) |
| [2] |
# Question 2:
## Part (a):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $A+B=\begin{pmatrix}a-2 & 6\\-2 & 3\end{pmatrix}$ | B1 | Any double signs must be simplified correctly |
| $AB=\begin{pmatrix}-2a-1 & 5a\\-1 & -5\end{pmatrix}$ | B1 | |
| $A^2=\begin{pmatrix}a^2-1 & a+3\\-a-3 & 8\end{pmatrix}$ | B1 | |
| **[3]** | | |
## Part (b)(i):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $(\det\mathbf{A}) = a\times3 - 1\times-1$ | M1 | Correct expansion of determinant of $\mathbf{A}$ |
| $3a+1=25 \Rightarrow a=8$ | A1 | |
| **[2]** | | |
## Part (b)(ii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| System reduces to $A\begin{pmatrix}x\\y\end{pmatrix}=\begin{pmatrix}-2\\-6\end{pmatrix}$ so no unique solution $\Rightarrow (\det\mathbf{A})=3a+1=0$ | M1 | Setting their determinant to 0 if it is a linear function of $a$. Answer only is ok here |
| $\therefore a = -\frac{1}{3}$ | A1 | |
| **Alternative:** Multiplying the first equation by 3 gives $3ax+3y=-6$ and $-x+3y=-6$. These two equations are the same if $3a=-1 \Rightarrow a=\dfrac{-1}{3}$ | M1, A1 | Or subtracting gives $(3a+1)x=0 \rightarrow a=\dfrac{-1}{3}$ |
| **[2]** | | |
---2 Matrices $\mathbf { A }$ and $\mathbf { B }$ are given by $\mathbf { A } = \left( \begin{array} { r r } a & 1 \\ - 1 & 3 \end{array} \right)$ and $\mathbf { B } = \left( \begin{array} { l l } - 2 & 5 \\ - 1 & 0 \end{array} \right)$ where $a$ is a constant.
\begin{enumerate}[label=(\alph*)]
\item Find the following matrices.
\begin{itemize}
\item $\mathbf { A } + \mathbf { B }$
\item AB
\item $\mathbf { A } ^ { 2 }$
\item \begin{enumerate}[label=(\roman*)]
\item Given that the determinant of $\mathbf { A }$ is 25 find the value of $a$.
\item You are given instead that the following system of equations does not have a unique solution.
\end{itemize}
$$\begin{array} { r }
a x + y = - 2 \\
- x + 3 y = - 6
\end{array}$$
Determine the value of $a$.
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{OCR Further Pure Core AS 2022 Q2 [7]}}