| Exam Board | Edexcel |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2014 |
| Session | January |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Matrices |
| Type | Area transformation under matrices |
| Difficulty | Standard +0.3 Part (i) requires knowing that area scales by |det(A)| and solving a quadratic equation from |det(A)| = 2, which is a standard FP1 application. Part (ii) is routine matrix multiplication. Both parts are straightforward applications of core Further Maths techniques with no novel problem-solving required, making this slightly easier than an average A-level question overall. |
| Spec | 4.03b Matrix operations: addition, multiplication, scalar4.03i Determinant: area scale factor and orientation |
| Answer | Marks | Guidance |
|---|---|---|
| \(\det \mathbf{A} = (-4)(k) - (-3)(10) \Rightarrow -4k + 30 = 2\) or \(-4k + 30 = -2\) | M1, dM1 | Applies "\(ad \pm bc\)" to A; Equates their det A to either 2 or -2 |
| \(\Rightarrow k = 7\) or \(k = 8\) | A1 | Either \(k = 8\) or \(k = 7\) |
| Both \(k = 8\) and \(k = 7\) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| \(\mathbf{B} = \begin{pmatrix} 1 & -2 & 3 \\ -2 & 5 & 1 \end{pmatrix}\), \(\mathbf{C} = \begin{pmatrix} 2 & 8 \\ 0 & 2 \\ 1 & -2 \end{pmatrix}\) | ||
| \(\mathbf{BC} = \begin{pmatrix} 1 & -2 & 3 \\ -2 & 5 & 1 \end{pmatrix} \begin{pmatrix} 2 & 8 \\ 0 & 2 \\ 1 & -2 \end{pmatrix} = \begin{pmatrix} 5 & -2 \\ -3 & -8 \end{pmatrix}\) | M1, A1, A1 | Writes down a complete \(2 \times 2\) matrix; Any 3 out of 4 elements correct; Correct answer |
**(i)**
| $\det \mathbf{A} = (-4)(k) - (-3)(10) \Rightarrow -4k + 30 = 2$ or $-4k + 30 = -2$ | M1, dM1 | Applies "$ad \pm bc$" to A; Equates their det A to either 2 or -2 |
| $\Rightarrow k = 7$ or $k = 8$ | A1 | Either $k = 8$ or $k = 7$ |
| Both $k = 8$ and $k = 7$ | A1 | |
**(ii)**
| $\mathbf{B} = \begin{pmatrix} 1 & -2 & 3 \\ -2 & 5 & 1 \end{pmatrix}$, $\mathbf{C} = \begin{pmatrix} 2 & 8 \\ 0 & 2 \\ 1 & -2 \end{pmatrix}$ | | |
| $\mathbf{BC} = \begin{pmatrix} 1 & -2 & 3 \\ -2 & 5 & 1 \end{pmatrix} \begin{pmatrix} 2 & 8 \\ 0 & 2 \\ 1 & -2 \end{pmatrix} = \begin{pmatrix} 5 & -2 \\ -3 & -8 \end{pmatrix}$ | M1, A1, A1 | Writes down a complete $2 \times 2$ matrix; Any 3 out of 4 elements correct; Correct answer |
---2.\\
(i)
$$\mathbf { A } = \left( \begin{array} { c c }
- 4 & 10 \\
- 3 & k
\end{array} \right) , \quad \text { where } k \text { is a constant. }$$
The triangle $T$ is transformed to the triangle $T ^ { \prime }$ by the transformation represented by $\mathbf { A }$.
Given that the area of triangle $T ^ { \prime }$ is twice the area of triangle $T$, find the possible values of $k$.\\
(ii) Given that
$$\mathbf { B } = \left( \begin{array} { r r r }
1 & - 2 & 3 \\
- 2 & 5 & 1
\end{array} \right) , \quad \mathbf { C } = \left( \begin{array} { r r }
2 & 8 \\
0 & 2 \\
1 & - 2
\end{array} \right)$$
find $\mathbf { B C }$.\\
\includegraphics[max width=\textwidth, alt={}, center]{9093bb1d-4f32-44e7-b0e7-b8c4f8a844e1-05_124_42_2608_1902}\\
\hfill \mbox{\textit{Edexcel FP1 2014 Q2 [7]}}