| Exam Board | Edexcel |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2016 |
| Session | June |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Linear transformations |
| Type | Describe rotation from matrix |
| Difficulty | Standard +0.3 This is a standard FP1 matrix transformations question requiring recognition of a rotation matrix (from the orthogonal structure with 1/√2 entries), applying the transformation, writing down a standard reflection matrix, multiplying matrices, and verifying self-inverse property. All parts are routine textbook exercises with no novel problem-solving required, though slightly above average difficulty due to being Further Maths content and requiring multiple techniques. |
| Spec | 4.03d Linear transformations 2D: reflection, rotation, enlargement, shear4.03e Successive transformations: matrix products |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Rotation, 135 degrees or \(\frac{3\pi}{4}\) radians anticlockwise about \(O\), or 225 degrees or \(\frac{5\pi}{4}\) clockwise about \(O\) | M1, A1 | M1: Rotation only; A1: 135 degrees about \(O\); SC: 135 degrees about \(O\) only → M1A0 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(\begin{pmatrix}-\frac{1}{\sqrt{2}} & -\frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{2}} & -\frac{1}{\sqrt{2}}\end{pmatrix}\begin{pmatrix}p\\q\end{pmatrix} = \begin{pmatrix}6\sqrt{2}\\3\sqrt{2}\end{pmatrix}\) giving \(-p-q=12\) and \(p-q=6\) | M1, A1 | M1: Multiplies matrices in correct order to obtain two equations in \(p\) and \(q\); A1: Two correct equations |
| \(p = -3\) and \(q = -9\) or \(\begin{pmatrix}-3\\-9\end{pmatrix}\) | B1 cso | Both \(p\) and \(q\) correct; no errors in solution |
| ALT: Uses inverse matrix \(\mathbf{P}^{-1}\) with vector \(= \begin{pmatrix}-\frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}}\\-\frac{1}{\sqrt{2}} & -\frac{1}{\sqrt{2}}\end{pmatrix}\begin{pmatrix}6\sqrt{2}\\3\sqrt{2}\end{pmatrix}\) | M1, A1 | Attempt inverse matrix and pre-multiply; correct inverse matrix used |
| \(p=-3\) and \(q=-9\) | B1 cso |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(\mathbf{Q} = \begin{pmatrix}0 & 1\\1 & 0\end{pmatrix}\) | B1 | Accept \(\mathbf{T}\) if used instead of \(\mathbf{R}\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(\mathbf{R} = \mathbf{"Q"P} = \begin{pmatrix}0&1\\1&0\end{pmatrix}\begin{pmatrix}-\frac{1}{\sqrt{2}}&-\frac{1}{\sqrt{2}}\\\frac{1}{\sqrt{2}}&-\frac{1}{\sqrt{2}}\end{pmatrix} = \begin{pmatrix}\frac{1}{\sqrt{2}}&-\frac{1}{\sqrt{2}}\\-\frac{1}{\sqrt{2}}&-\frac{1}{\sqrt{2}}\end{pmatrix}\) | M1, A1, A1 | M1: Sets matrix product correct way; A1: Two correct terms from correct \(\mathbf{Q}\); A1: Completely correct matrix |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(\mathbf{R}^{-1} = \frac{1}{-1}\begin{pmatrix}-\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{2}}\\+\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{2}}\end{pmatrix} = \begin{pmatrix}\frac{1}{\sqrt{2}}&-\frac{1}{\sqrt{2}}\\-\frac{1}{\sqrt{2}}&-\frac{1}{\sqrt{2}}\end{pmatrix} = \mathbf{R}\), so matrix is self inverse and transformation is self inverse | B1 | Calculates \(\mathbf{R}^{-1}\) and shows \(\mathbf{R}^{-1}=\mathbf{R}\), or calculates \(\mathbf{R}^2=\mathbf{I}\), or states \(\mathbf{R}\) represents a reflection |
# Question 6:
## Part (a)
| Answer/Working | Mark | Guidance |
|---|---|---|
| **Rotation**, 135 degrees or $\frac{3\pi}{4}$ radians anticlockwise about $O$, or 225 degrees or $\frac{5\pi}{4}$ clockwise about $O$ | M1, A1 | M1: Rotation only; A1: 135 degrees about $O$; SC: 135 degrees about $O$ only → M1A0 |
## Part (b)
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\begin{pmatrix}-\frac{1}{\sqrt{2}} & -\frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{2}} & -\frac{1}{\sqrt{2}}\end{pmatrix}\begin{pmatrix}p\\q\end{pmatrix} = \begin{pmatrix}6\sqrt{2}\\3\sqrt{2}\end{pmatrix}$ giving $-p-q=12$ and $p-q=6$ | M1, A1 | M1: Multiplies matrices in correct order to obtain two equations in $p$ and $q$; A1: Two correct equations |
| $p = -3$ and $q = -9$ or $\begin{pmatrix}-3\\-9\end{pmatrix}$ | B1 cso | Both $p$ and $q$ correct; no errors in solution |
| ALT: Uses inverse matrix $\mathbf{P}^{-1}$ with vector $= \begin{pmatrix}-\frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}}\\-\frac{1}{\sqrt{2}} & -\frac{1}{\sqrt{2}}\end{pmatrix}\begin{pmatrix}6\sqrt{2}\\3\sqrt{2}\end{pmatrix}$ | M1, A1 | Attempt inverse matrix and pre-multiply; correct inverse matrix used |
| $p=-3$ and $q=-9$ | B1 cso | |
## Part (c)
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\mathbf{Q} = \begin{pmatrix}0 & 1\\1 & 0\end{pmatrix}$ | B1 | Accept $\mathbf{T}$ if used instead of $\mathbf{R}$ |
## Part (d)
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\mathbf{R} = \mathbf{"Q"P} = \begin{pmatrix}0&1\\1&0\end{pmatrix}\begin{pmatrix}-\frac{1}{\sqrt{2}}&-\frac{1}{\sqrt{2}}\\\frac{1}{\sqrt{2}}&-\frac{1}{\sqrt{2}}\end{pmatrix} = \begin{pmatrix}\frac{1}{\sqrt{2}}&-\frac{1}{\sqrt{2}}\\-\frac{1}{\sqrt{2}}&-\frac{1}{\sqrt{2}}\end{pmatrix}$ | M1, A1, A1 | M1: Sets matrix product correct way; A1: Two correct terms from correct $\mathbf{Q}$; A1: Completely correct matrix |
## Part (e)
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\mathbf{R}^{-1} = \frac{1}{-1}\begin{pmatrix}-\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{2}}\\+\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{2}}\end{pmatrix} = \begin{pmatrix}\frac{1}{\sqrt{2}}&-\frac{1}{\sqrt{2}}\\-\frac{1}{\sqrt{2}}&-\frac{1}{\sqrt{2}}\end{pmatrix} = \mathbf{R}$, so matrix is self inverse and transformation is self inverse | B1 | Calculates $\mathbf{R}^{-1}$ and shows $\mathbf{R}^{-1}=\mathbf{R}$, or calculates $\mathbf{R}^2=\mathbf{I}$, or states $\mathbf{R}$ represents a reflection |
---6.
$$\mathbf { P } = \left( \begin{array} { c c }
- \frac { 1 } { \sqrt { } 2 } & - \frac { 1 } { \sqrt { } 2 } \\
\frac { 1 } { \sqrt { } 2 } & - \frac { 1 } { \sqrt { } 2 }
\end{array} \right)$$
\begin{enumerate}[label=(\alph*)]
\item Describe fully the single geometrical transformation $U$ represented by the matrix $\mathbf { P }$.
The transformation $U$ maps the point $A$, with coordinates $( p , q )$, onto the point $B$, with coordinates $( 6 \sqrt { } 2,3 \sqrt { } 2 )$.
\item Find the value of $p$ and the value of $q$.
The transformation $V$, represented by the $2 \times 2$ matrix $\mathbf { Q }$, is a reflection in the line with equation $y = x$.
\item Write down the matrix $\mathbf { Q }$.
The transformation $U$ followed by the transformation $V$ is the transformation $T$. The transformation $T$ is represented by the matrix $\mathbf { R }$.
\item Find the matrix $\mathbf { R }$.
\item Deduce that the transformation $T$ is self-inverse.
\end{enumerate}
\hfill \mbox{\textit{Edexcel FP1 2016 Q6 [10]}}