| Exam Board | Edexcel |
|---|---|
| Module | CP AS (Core Pure AS) |
| Year | 2021 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Linear transformations |
| Type | Describe rotation from matrix |
| Difficulty | Easy -1.2 This question tests basic recognition of standard 2×2 transformation matrices (rotation and stretch) and simple matrix multiplication. Parts (a) require only recall of standard forms, (b) is straightforward matrix multiplication, and (c) is routine determinant calculation with standard interpretation. No problem-solving or novel insight required—purely procedural application of well-rehearsed techniques. |
| Spec | 4.03c Matrix multiplication: properties (associative, not commutative)4.03d Linear transformations 2D: reflection, rotation, enlargement, shear4.03e Successive transformations: matrix products4.03h Determinant 2x2: calculation |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Rotation | B1 | Identifies the transformation as a rotation |
| 90 degrees anticlockwise about the origin | B1 | Correct angle (allow equivalents in degrees or radians), direction and centre the origin |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Stretch | B1 | Identifies the transformation as a stretch |
| Scale factor 3 parallel to the \(y\)-axis | B1 | Correct scale factor and parallel to/in/along the \(y\)-axis/\(y\) direction |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\mathbf{QP} = \begin{pmatrix} 1 & 0 \\ 0 & 3 \end{pmatrix}\begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix} = \begin{pmatrix} 0 & -1 \\ 3 & 0 \end{pmatrix}\) | B1 | Correct matrix |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \( | \mathbf{R} | = 3\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| The area scale factor of the transformation | B1 | Correct explanation, must include area. Note: "scale factor of the transformation" is B0 |
## Question 1:
**Part (a)(i):**
| Answer | Marks | Guidance |
|--------|-------|----------|
| Rotation | B1 | Identifies the transformation as a rotation |
| 90 degrees anticlockwise about the origin | B1 | Correct angle (allow equivalents in degrees or radians), direction and centre the origin |
**Part (a)(ii):**
| Answer | Marks | Guidance |
|--------|-------|----------|
| Stretch | B1 | Identifies the transformation as a stretch |
| Scale factor 3 parallel to the $y$-axis | B1 | Correct scale factor and parallel to/in/along the $y$-axis/$y$ direction |
**Part (b):**
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\mathbf{QP} = \begin{pmatrix} 1 & 0 \\ 0 & 3 \end{pmatrix}\begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix} = \begin{pmatrix} 0 & -1 \\ 3 & 0 \end{pmatrix}$ | B1 | Correct matrix |
**Part (c)(i):**
| Answer | Marks | Guidance |
|--------|-------|----------|
| $|\mathbf{R}| = 3$ | B1ft | Correct value for the determinant (follow through their $\mathbf{R}$) |
**Part (c)(ii):**
| Answer | Marks | Guidance |
|--------|-------|----------|
| The **area** scale factor of the transformation | B1 | Correct explanation, must include **area**. Note: "scale factor of the transformation" is B0 |1.
$$\mathbf { P } = \left( \begin{array} { r r }
0 & - 1 \\
1 & 0
\end{array} \right) \quad \mathbf { Q } = \left( \begin{array} { l l }
1 & 0 \\
0 & 3
\end{array} \right)$$
\begin{enumerate}[label=(\alph*)]
\item \begin{enumerate}[label=(\roman*)]
\item Describe fully the single geometrical transformation $P$ represented by the matrix $\mathbf { P }$.
\item Describe fully the single geometrical transformation $Q$ represented by the matrix $\mathbf { Q }$.
The transformation $P$ followed by the transformation $Q$ is the transformation $R$, which is represented by the matrix $\mathbf { R }$.
\end{enumerate}\item Determine $\mathbf { R }$.
\item \begin{enumerate}[label=(\roman*)]
\item Evaluate the determinant of $\mathbf { R }$.
\item Explain how the value obtained in (c)(i) relates to the transformation $R$.
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{Edexcel CP AS 2021 Q1 [7]}}