| Exam Board | Edexcel |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2009 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Matrices |
| Type | Matrix powers and patterns |
| Difficulty | Standard +0.3 This is a straightforward Further Maths question requiring matrix multiplication to find R², then using the property that an enlargement matrix is a scalar multiple of the identity matrix (15I). The algebra is routine: equating off-diagonal elements to zero gives a=0, contradicting a>0, so students must recognize the diagonal must equal 15, leading to simple simultaneous equations. While it's Further Maths content, the execution is mechanical with no conceptual subtlety. |
| Spec | 4.03b Matrix operations: addition, multiplication, scalar4.03d Linear transformations 2D: reflection, rotation, enlargement, shear |
| Answer | Marks | Guidance |
|---|---|---|
| Working/Answer | Marks | Notes |
| \(\mathbf{R}^2 = \begin{pmatrix} a^2+2a & 2a+2b \\ a^2+ab & 2a+b^2 \end{pmatrix}\) | M1 A1 A1 | 1 term correct: M1 A0 A0; 2 or 3 terms correct: M1 A1 A0 |
| Answer | Marks | Guidance |
|---|---|---|
| Working/Answer | Marks | Notes |
| Puts \(a^2 + 2a = 15\) or \(2a + b^2 = 15\) | M1 | |
| or \((a^2+2a)(2a+b^2)-(a^2+ab)(2a+2b) = 225\) | ||
| Puts \(a^2 + ab = 0\) or \(2a + 2b = 0\) | M1 | |
| Solve to find either \(a\) or \(b\) | M1 | M1 requires solving equations to find \(a\) and/or \(b\) |
| \(a = 3,\quad b = -3\) | A1, A1 | A1 A1 for correct answers only; any extra answers given e.g. \(a=-5, b=5\) deduct last A1 awarded; answer with no working is 0 marks |
| Alternative: Uses \(\mathbf{R}^2 \times\) column vector \(= 15 \times\) column vector, equates rows to give two equations in \(a\) and \(b\) only; solves to find either \(a\) or \(b\) | M1, M1, M1 A1 A1 |
## Question 5:
### Part (a)
| Working/Answer | Marks | Notes |
|---|---|---|
| $\mathbf{R}^2 = \begin{pmatrix} a^2+2a & 2a+2b \\ a^2+ab & 2a+b^2 \end{pmatrix}$ | M1 A1 A1 | 1 term correct: M1 A0 A0; 2 or 3 terms correct: M1 A1 A0 |
### Part (b)
| Working/Answer | Marks | Notes |
|---|---|---|
| Puts $a^2 + 2a = 15$ or $2a + b^2 = 15$ | M1 | |
| or $(a^2+2a)(2a+b^2)-(a^2+ab)(2a+2b) = 225$ | | |
| Puts $a^2 + ab = 0$ or $2a + 2b = 0$ | M1 | |
| Solve to find either $a$ or $b$ | M1 | M1 requires solving equations to find $a$ and/or $b$ |
| $a = 3,\quad b = -3$ | A1, A1 | A1 A1 for correct answers only; any extra answers given e.g. $a=-5, b=5$ deduct last A1 awarded; answer with no working is 0 marks |
| Alternative: Uses $\mathbf{R}^2 \times$ column vector $= 15 \times$ column vector, equates rows to give two equations in $a$ and $b$ only; solves to find either $a$ or $b$ | M1, M1, M1 A1 A1 | |
---5. $\mathbf { R } = \left( \begin{array} { l l } a & 2 \\ a & b \end{array} \right)$, where $a$ and $b$ are constants and $a > 0$.
\begin{enumerate}[label=(\alph*)]
\item Find $\mathbf { R } ^ { 2 }$ in terms of $a$ and $b$.
Given that $\mathbf { R } ^ { 2 }$ represents an enlargement with centre ( 0,0 ) and scale factor 15 ,
\item find the value of $a$ and the value of $b$.
\end{enumerate}
\hfill \mbox{\textit{Edexcel FP1 2009 Q5 [8]}}