| Exam Board | Edexcel |
|---|---|
| Module | CP AS (Core Pure AS) |
| Year | 2023 |
| Session | June |
| Marks | 4 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Linear transformations |
| Type | Combined transformation matrix product |
| Difficulty | Moderate -0.3 This is a straightforward Core Pure AS question on 3D transformations requiring: (a) recognition of a rotation matrix about the x-axis by 30°, (b) identifying the common error of multiplying matrices in the wrong order (should be BA not AB), and (c) performing a 3×3 matrix multiplication with some surds. All parts are standard bookwork with no novel problem-solving required, making it slightly easier than average. |
| Spec | 4.03b Matrix operations: addition, multiplication, scalar4.03d Linear transformations 2D: reflection, rotation, enlargement, shear |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Rotation | B1 | 1.1b — identifies single transformation as rotation only |
| \(30\) degrees or \(\frac{\pi}{6}\) about the \(x\)-axis; ignore any reference to direction | B1 | 1.1b — correct angle and axis; note: \(x\)-plane, \(zy\)-plane and \(x=0\) are 2nd B0; any additional incorrect statements is 2nd B0 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| They have found \(\mathbf{AB}\) when they should find \(\mathbf{BA}\); multiplication is the wrong way round; it should be \(\mathbf{BA}\); matrix \(B\) should be on the left; student has done transformation \(B\) followed by transformation \(A\). It should be \(\begin{pmatrix}1&3&0\\\sqrt{3}&0&5\sqrt{3}\\1&2&0\end{pmatrix}\begin{pmatrix}1&0&0\\0&\frac{\sqrt{3}}{2}&-\frac{1}{2}\\0&\frac{1}{2}&\frac{\sqrt{3}}{2}\end{pmatrix}\) | B1 | 2.3 — explains matrices should be multiplied the other way around |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(\begin{pmatrix}1&3&0\\\sqrt{3}&0&5\sqrt{3}\\1&2&0\end{pmatrix}\begin{pmatrix}1&0&0\\0&\frac{\sqrt{3}}{2}&-\frac{1}{2}\\0&\frac{1}{2}&\frac{\sqrt{3}}{2}\end{pmatrix} = \begin{pmatrix}1&\frac{3\sqrt{3}}{2}&-\frac{3}{2}\\\sqrt{3}&\frac{5\sqrt{3}}{2}&\frac{15}{2}\\1&\sqrt{3}&-1\end{pmatrix}\) | B1 | 1.1b — correct exact matrix; note \(5\sqrt{3}\times\frac{\sqrt{3}}{2}\) must simplify to \(\frac{15}{2}\); condone \(\frac{2\sqrt{3}}{2}\) not simplified |
## Question 3(a):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Rotation | B1 | 1.1b — identifies single transformation as rotation only |
| $30$ degrees or $\frac{\pi}{6}$ about the $x$-axis; ignore any reference to direction | B1 | 1.1b — correct angle and axis; note: $x$-plane, $zy$-plane and $x=0$ are 2nd B0; any additional incorrect statements is 2nd B0 |
---
## Question 3(b):
| Answer/Working | Mark | Guidance |
|---|---|---|
| They have found $\mathbf{AB}$ when they should find $\mathbf{BA}$; multiplication is the wrong way round; it should be $\mathbf{BA}$; matrix $B$ should be on the left; student has done transformation $B$ followed by transformation $A$. It should be $\begin{pmatrix}1&3&0\\\sqrt{3}&0&5\sqrt{3}\\1&2&0\end{pmatrix}\begin{pmatrix}1&0&0\\0&\frac{\sqrt{3}}{2}&-\frac{1}{2}\\0&\frac{1}{2}&\frac{\sqrt{3}}{2}\end{pmatrix}$ | B1 | 2.3 — explains matrices should be multiplied the other way around |
---
## Question 3(c):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\begin{pmatrix}1&3&0\\\sqrt{3}&0&5\sqrt{3}\\1&2&0\end{pmatrix}\begin{pmatrix}1&0&0\\0&\frac{\sqrt{3}}{2}&-\frac{1}{2}\\0&\frac{1}{2}&\frac{\sqrt{3}}{2}\end{pmatrix} = \begin{pmatrix}1&\frac{3\sqrt{3}}{2}&-\frac{3}{2}\\\sqrt{3}&\frac{5\sqrt{3}}{2}&\frac{15}{2}\\1&\sqrt{3}&-1\end{pmatrix}$ | B1 | 1.1b — correct exact matrix; note $5\sqrt{3}\times\frac{\sqrt{3}}{2}$ must simplify to $\frac{15}{2}$; condone $\frac{2\sqrt{3}}{2}$ not simplified |
---3.
$$\mathbf { A } = \left( \begin{array} { c c c }
1 & 0 & 0 \\
0 & \frac { \sqrt { 3 } } { 2 } & - \frac { 1 } { 2 } \\
0 & \frac { 1 } { 2 } & \frac { \sqrt { 3 } } { 2 }
\end{array} \right)$$
\begin{enumerate}[label=(\alph*)]
\item Describe fully the single geometric transformation $A$ represented by the matrix $\mathbf { A }$.
$$\mathbf { B } = \left( \begin{array} { c c c }
1 & 3 & 0 \\
\sqrt { 3 } & 0 & 5 \sqrt { 3 } \\
1 & 2 & 0
\end{array} \right)$$
The transformation $B$ is represented by the matrix $\mathbf { B }$.\\
The transformation $A$ followed by the transformation $B$ is the transformation $C$, which is represented by the matrix $\mathbf { C }$.
To determine matrix $\mathbf { C }$, a student attempts the following matrix multiplication.
$$\left( \begin{array} { c c c }
1 & 0 & 0 \\
0 & \frac { \sqrt { 3 } } { 2 } & - \frac { 1 } { 2 } \\
0 & \frac { 1 } { 2 } & \frac { \sqrt { 3 } } { 2 }
\end{array} \right) \left( \begin{array} { c c c }
1 & 3 & 0 \\
\sqrt { 3 } & 0 & 5 \sqrt { 3 } \\
1 & 2 & 0
\end{array} \right)$$
\item State the error made by the student.
\item Determine the correct matrix $\mathbf { C }$.
\end{enumerate}
\hfill \mbox{\textit{Edexcel CP AS 2023 Q3 [4]}}