| Exam Board | Edexcel |
|---|---|
| Module | CP AS (Core Pure AS) |
| Year | 2018 |
| Session | June |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Linear transformations |
| Type | Find invariant points |
| Difficulty | Standard +0.3 This is a straightforward multi-part question on linear transformations requiring standard techniques: recognizing a rotation matrix from its form, writing down a standard reflection matrix, multiplying two 2×2 matrices, and solving a simple invariant point equation. All parts are routine applications of Core Pure AS content with no novel problem-solving required, making it slightly easier than average. |
| Spec | 4.03d Linear transformations 2D: reflection, rotation, enlargement, shear4.03e Successive transformations: matrix products4.03g Invariant points and lines |
| V349 SIHI NI IMIMM ION OC | VJYV SIHIL NI LIIIM ION OO | VJYV SIHIL NI JIIYM ION OC |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Rotation | B1 | Identifies the transformation as a rotation |
| 120 degrees (anticlockwise) or \(\frac{2\pi}{3}\) radians (anticlockwise), or 240 degrees clockwise or \(\frac{4\pi}{3}\) radians clockwise | B1 | Correct angle; allow equivalents in degrees or radians |
| About (from) the origin. Allow \((0,0)\) or \(O\) for origin | B1 | Identifies the origin as centre of rotation. All three B marks only awarded as elements of a single transformation |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(\begin{pmatrix} 0 & -1 \\ -1 & 0 \end{pmatrix}\) | B1 | Shows the correct matrix in the correct form |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(\begin{pmatrix} 0 & -1 \\ -1 & 0 \end{pmatrix}\begin{pmatrix} \frac{1}{2} & -\frac{\sqrt{3}}{2} \\ \frac{\sqrt{3}}{2} & -\frac{1}{2} \end{pmatrix} = \begin{pmatrix} \cdots & \cdots \\ \cdots & \cdots \end{pmatrix}\) | M1 | Multiplies matrices in correct order; evidence of multiplication can be taken from 3 correct or 3 correct ft elements |
| \(= \begin{pmatrix} -\frac{\sqrt{3}}{2} & \frac{1}{2} \\ \frac{1}{2} & \frac{\sqrt{3}}{2} \end{pmatrix}\) | A1ft | Correct matrix (follow through from part (b)); a correct matrix or correct ft matrix implies both marks |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(\begin{pmatrix} -\frac{\sqrt{3}}{2} & \frac{1}{2} \\ \frac{1}{2} & \frac{\sqrt{3}}{2} \end{pmatrix}\begin{pmatrix} 1 \\ k \end{pmatrix} = \begin{pmatrix} 1 \\ k \end{pmatrix}\) or equivalent matrix equation giving at least one equation in \(k\) or in \(x\) and \(y\) | M1 | Translates the problem into a matrix multiplication to obtain at least one equation in \(k\) or in \(x\) and \(y\) |
| \(-\frac{\sqrt{3}}{2} + \frac{1}{2}k = 1\) or \(\frac{1}{2} + \frac{\sqrt{3}}{2}k = k\), or \(x = -\frac{\sqrt{3}}{2}x + \frac{1}{2}y\) or \(y = \frac{1}{2}x + \frac{\sqrt{3}}{2}y\) | A1ft | Obtains one correct equation (follow through from part (c)) |
| \(k = 2 + \sqrt{3}\) (or \(\frac{1}{2-\sqrt{3}}\)) from \(-\frac{\sqrt{3}}{2} + \frac{1}{2}k = 1\) | A1 | Correct value for \(k\) in any form |
| \(k = 2 + \sqrt{3}\) (or \(\frac{1}{2-\sqrt{3}}\)) from \(\frac{1}{2} + \frac{\sqrt{3}}{2}k = k\) | B1 | Checks answer by independently solving both equations correctly to obtain \(2+\sqrt{3}\) both times, or substitutes \(2+\sqrt{3}\) into the other equation to confirm validity |
## Question 5:
### Part (a):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Rotation | B1 | Identifies the transformation as a rotation |
| 120 degrees (anticlockwise) or $\frac{2\pi}{3}$ radians (anticlockwise), or 240 degrees clockwise or $\frac{4\pi}{3}$ radians clockwise | B1 | Correct angle; allow equivalents in degrees or radians |
| About (from) the origin. Allow $(0,0)$ or $O$ for origin | B1 | Identifies the origin as centre of rotation. All three B marks only awarded as elements of a **single transformation** |
### Part (b):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\begin{pmatrix} 0 & -1 \\ -1 & 0 \end{pmatrix}$ | B1 | Shows the correct matrix in the correct form |
### Part (c):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\begin{pmatrix} 0 & -1 \\ -1 & 0 \end{pmatrix}\begin{pmatrix} \frac{1}{2} & -\frac{\sqrt{3}}{2} \\ \frac{\sqrt{3}}{2} & -\frac{1}{2} \end{pmatrix} = \begin{pmatrix} \cdots & \cdots \\ \cdots & \cdots \end{pmatrix}$ | M1 | Multiplies matrices in correct order; evidence of multiplication can be taken from 3 correct or 3 correct ft elements |
| $= \begin{pmatrix} -\frac{\sqrt{3}}{2} & \frac{1}{2} \\ \frac{1}{2} & \frac{\sqrt{3}}{2} \end{pmatrix}$ | A1ft | Correct matrix (follow through from part (b)); a correct matrix or correct ft matrix implies both marks |
### Part (d):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\begin{pmatrix} -\frac{\sqrt{3}}{2} & \frac{1}{2} \\ \frac{1}{2} & \frac{\sqrt{3}}{2} \end{pmatrix}\begin{pmatrix} 1 \\ k \end{pmatrix} = \begin{pmatrix} 1 \\ k \end{pmatrix}$ or equivalent matrix equation giving at least one equation in $k$ or in $x$ and $y$ | M1 | Translates the problem into a matrix multiplication to obtain at least one equation in $k$ or in $x$ and $y$ |
| $-\frac{\sqrt{3}}{2} + \frac{1}{2}k = 1$ or $\frac{1}{2} + \frac{\sqrt{3}}{2}k = k$, or $x = -\frac{\sqrt{3}}{2}x + \frac{1}{2}y$ or $y = \frac{1}{2}x + \frac{\sqrt{3}}{2}y$ | A1ft | Obtains one correct equation (follow through from part (c)) |
| $k = 2 + \sqrt{3}$ (or $\frac{1}{2-\sqrt{3}}$) from $-\frac{\sqrt{3}}{2} + \frac{1}{2}k = 1$ | A1 | Correct value for $k$ in any form |
| $k = 2 + \sqrt{3}$ (or $\frac{1}{2-\sqrt{3}}$) from $\frac{1}{2} + \frac{\sqrt{3}}{2}k = k$ | B1 | Checks answer by independently solving both equations correctly to obtain $2+\sqrt{3}$ both times, or substitutes $2+\sqrt{3}$ into the other equation to confirm validity |
---5.
$$\mathbf { A } = \left( \begin{array} { r r }
- \frac { 1 } { 2 } & - \frac { \sqrt { 3 } } { 2 } \\
\frac { \sqrt { 3 } } { 2 } & - \frac { 1 } { 2 }
\end{array} \right)$$
\begin{enumerate}[label=(\alph*)]
\item Describe fully the single geometrical transformation $U$ represented by the matrix $\mathbf { A }$.
The transformation $V$, represented by the $2 \times 2$ matrix $\mathbf { B }$, is a reflection in the line $y = - x$
\item Write down the matrix $\mathbf { B }$.
Given that $U$ followed by $V$ is the transformation $T$, which is represented by the matrix $\mathbf { C }$, (c) find the matrix $\mathbf { C }$.\\
(d) Show that there is a real number $k$ for which the point $( 1 , k )$ is invariant under $T$.
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\hfill \mbox{\textit{Edexcel CP AS 2018 Q5 [10]}}