| Exam Board | Edexcel |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2018 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Matrices |
| Type | Solving matrix equations for unknown matrix |
| Difficulty | Standard +0.3 This is a straightforward FP1 question testing standard matrix operations: finding a 2×2 inverse using the formula, multiplying matrices, and recognizing a rotation matrix with its periodic properties. All parts are routine applications of learned techniques with no problem-solving insight required, making it slightly easier than average. |
| Spec | 4.03d Linear transformations 2D: reflection, rotation, enlargement, shear4.03e Successive transformations: matrix products4.03n Inverse 2x2 matrix |
| Answer | Marks | Guidance |
|---|---|---|
| \(A^{-1} = \frac{1}{-2-3}\begin{pmatrix} 1 & -3 \\ -1 & -2 \end{pmatrix}\) | M1 | Either \(\frac{1}{-2-3}\) or \(\frac{1}{-5}\) or \(\begin{pmatrix} 1 & -3 \\ -1 & -2 \end{pmatrix}\) |
| A1 | Correct expression for \(A^{-1}\) |
| Answer | Marks | Guidance |
|---|---|---|
| \(\{B = A^{-1}(AB)\}\) | ||
| \(B = -\frac{1}{5}\begin{pmatrix} 1 & -3 \\ -1 & -2 \end{pmatrix}\begin{pmatrix} -1 & 5 & 12 \\ 3 & -5 & -1 \end{pmatrix}\) | M1 | Writing down their \(A^{-1}\) multiplied by \(AB\) |
| \(= -\frac{1}{5}\begin{pmatrix} -10 & 20 & 15 \\ -5 & 5 & -10 \end{pmatrix}\) | A1 | At least one correct row or at least two correct columns of \(\begin{pmatrix} ... \\ ... \end{pmatrix}\). (Ignore \(-\frac{1}{5}\)). |
| \(= \begin{pmatrix} 2 & -4 & -3 \\ 1 & -1 & 2 \end{pmatrix}\) | A1 | Correct simplified matrix for B |
| Answer | Marks | Guidance |
|---|---|---|
| 90° clockwise about the origin | M1 | Rotation only. |
| A1 | \(90°\) or \(\left(\frac{\pi}{2}\right)\) clockwise about the origin or \(270°\) or \(\left(\frac{3\pi}{2}\right)\) (anti-clockwise) about the origin or \(-90°\) or \(\left(-\frac{\pi}{2}\right)\) (anti-clockwise) about the origin. Origin can be written as \((0, 0)\) or O. |
| Answer | Marks | Guidance |
|---|---|---|
| \(\{C^{-90}\} = C^{-1}\) or \(C' = \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}\) | M1 | For stating \(C^{-1}\) or \(C'\) or 'rotation of 270° clockwise o.e. about the origin. Can be implied by correct matrix. |
| A1 | \(\begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}\) | |
| M1A1 | Correct answer with no working award M1A1 |
**Part (i)(a)**
| $A^{-1} = \frac{1}{-2-3}\begin{pmatrix} 1 & -3 \\ -1 & -2 \end{pmatrix}$ | M1 | Either $\frac{1}{-2-3}$ or $\frac{1}{-5}$ or $\begin{pmatrix} 1 & -3 \\ -1 & -2 \end{pmatrix}$ |
| | A1 | Correct expression for $A^{-1}$ |
**Part (i)(b)**
| $\{B = A^{-1}(AB)\}$ | | |
| $B = -\frac{1}{5}\begin{pmatrix} 1 & -3 \\ -1 & -2 \end{pmatrix}\begin{pmatrix} -1 & 5 & 12 \\ 3 & -5 & -1 \end{pmatrix}$ | M1 | Writing down their $A^{-1}$ multiplied by $AB$ |
| $= -\frac{1}{5}\begin{pmatrix} -10 & 20 & 15 \\ -5 & 5 & -10 \end{pmatrix}$ | A1 | At least one correct row or at least two correct columns of $\begin{pmatrix} ... \\ ... \end{pmatrix}$. (Ignore $-\frac{1}{5}$). |
| $= \begin{pmatrix} 2 & -4 & -3 \\ 1 & -1 & 2 \end{pmatrix}$ | A1 | Correct simplified matrix for B |
**Part (ii)(a)**
| 90° clockwise about the origin | M1 | Rotation only. |
| | A1 | $90°$ or $\left(\frac{\pi}{2}\right)$ clockwise about the origin or $270°$ or $\left(\frac{3\pi}{2}\right)$ (anti-clockwise) about the origin or $-90°$ or $\left(-\frac{\pi}{2}\right)$ (anti-clockwise) about the origin. Origin can be written as $(0, 0)$ or O. |
**Part (ii)(b)**
| $\{C^{-90}\} = C^{-1}$ or $C' = \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}$ | M1 | For stating $C^{-1}$ or $C'$ or 'rotation of 270° clockwise o.e. about the origin. Can be implied by correct matrix. |
| | A1 | $\begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}$ |
| | M1A1 | Correct answer with no working award M1A1 |
**Total: 9 marks**
---3. (i) Given that
$$\mathbf { A } = \left( \begin{array} { r r }
- 2 & 3 \\
1 & 1
\end{array} \right) , \quad \mathbf { A } \mathbf { B } = \left( \begin{array} { r r r }
- 1 & 5 & 12 \\
3 & - 5 & - 1
\end{array} \right)$$
\begin{enumerate}[label=(\alph*)]
\item find $\mathbf { A } ^ { - 1 }$
\item Hence, or otherwise, find the matrix $\mathbf { B }$, giving your answer in its simplest form.\\
(ii) Given that
$$\mathbf { C } = \left( \begin{array} { r r }
0 & 1 \\
- 1 & 0
\end{array} \right)$$
(a) describe fully the single geometrical transformation represented by the matrix $\mathbf { C }$.\\
(b) Hence find the matrix $\mathbf { C } ^ { 39 }$
\end{enumerate}
\hfill \mbox{\textit{Edexcel FP1 2018 Q3 [9]}}