| Exam Board | Edexcel |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2011 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Linear transformations |
| Type | Singular matrix conditions |
| Difficulty | Moderate -0.3 This is a straightforward Further Maths question testing standard matrix operations and transformations. Part (a) requires routine matrix multiplication and recognizing a rotation (A² gives a 90° rotation), part (b) is identifying a reflection from a standard matrix form, and part (c) uses the basic condition det(C)=0 for singularity. While it's Further Maths content, these are all textbook exercises requiring recall and direct application rather than problem-solving, making it slightly easier than average overall. |
| Spec | 4.03b Matrix operations: addition, multiplication, scalar4.03d Linear transformations 2D: reflection, rotation, enlargement, shear4.03l Singular/non-singular matrices |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(\mathbf{A}^2 = \begin{pmatrix}1 & \sqrt{2}\\ \sqrt{2} & -1\end{pmatrix}\begin{pmatrix}1 & \sqrt{2}\\ \sqrt{2} & -1\end{pmatrix}\), correct multiplication method | M1 | A correct method to multiply two matrices. Can be implied by two out of four correct elements |
| \(= \begin{pmatrix}3 & 0\\ 0 & 3\end{pmatrix}\) | A1 | Correct answer |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Enlargement; scale factor 3, centre \((0,0)\) | B1; B1 | Enlargement; scale factor 3, centre (0, 0). Allow 'from' or 'about' for centre and 'O' or 'origin' for \((0,0)\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(\mathbf{B} = \begin{pmatrix}0 & -1\\ -1 & 0\end{pmatrix}\); Reflection in the line \(y = -x\) | B1; B1 | Reflection; \(y = -x\). Allow 'in the axis' 'about the line' \(y=-x\) etc. |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(\mathbf{C}\) is singular \(\Rightarrow \det\mathbf{C} = 0\) | B1 | \(\det\mathbf{C} = 0\) |
| \(9(k+1) - 12k = 0\) | M1 | Applies \(9(k+1) - 12k\) |
| \(9k + 9 = 12k \Rightarrow 9 = 3k\) | ||
| \(k = 3\) | A1 | \(k = 3\) |
# Question 3:
## Part (a)(i)
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\mathbf{A}^2 = \begin{pmatrix}1 & \sqrt{2}\\ \sqrt{2} & -1\end{pmatrix}\begin{pmatrix}1 & \sqrt{2}\\ \sqrt{2} & -1\end{pmatrix}$, correct multiplication method | M1 | A correct method to multiply two matrices. Can be implied by two out of four correct elements |
| $= \begin{pmatrix}3 & 0\\ 0 & 3\end{pmatrix}$ | A1 | Correct answer |
## Part (a)(ii)
| Answer/Working | Mark | Guidance |
|---|---|---|
| **Enlargement**; scale factor 3, centre $(0,0)$ | B1; B1 | Enlargement; scale factor **3**, centre **(0, 0)**. Allow 'from' or 'about' for centre and 'O' or 'origin' for $(0,0)$ |
## Part (b)
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\mathbf{B} = \begin{pmatrix}0 & -1\\ -1 & 0\end{pmatrix}$; Reflection in the line $y = -x$ | B1; B1 | **Reflection**; $y = -x$. Allow 'in the axis' 'about the line' $y=-x$ etc. |
> **Note:** The question does not specify a single transformation so combinations that are completely correct are scored as B2 (no part marks).
## Part (c)
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\mathbf{C}$ is singular $\Rightarrow \det\mathbf{C} = 0$ | B1 | $\det\mathbf{C} = 0$ |
| $9(k+1) - 12k = 0$ | M1 | Applies $9(k+1) - 12k$ |
| $9k + 9 = 12k \Rightarrow 9 = 3k$ | | |
| $k = 3$ | A1 | $k = 3$ |
> **Note:** $k = 3$ with no working can score full marks.3. (a) Given that
$$\mathbf { A } = \left( \begin{array} { c c }
1 & \sqrt { } 2 \\
\sqrt { } 2 & - 1
\end{array} \right)$$
\begin{enumerate}[label=(\roman*)]
\item find $\mathbf { A } ^ { 2 }$,
\item describe fully the geometrical transformation represented by $\mathbf { A } ^ { 2 }$.\\
(b) Given that
$$\mathbf { B } = \left( \begin{array} { r r }
0 & - 1 \\
- 1 & 0
\end{array} \right)$$
describe fully the geometrical transformation represented by $\mathbf { B }$.\\
(c) Given that
$$\mathbf { C } = \left( \begin{array} { c c }
k + 1 & 12 \\
k & 9
\end{array} \right)$$
where $k$ is a constant, find the value of $k$ for which the matrix $\mathbf { C }$ is singular.
\end{enumerate}
\hfill \mbox{\textit{Edexcel FP1 2011 Q3 [9]}}