| Exam Board | CAIE |
|---|---|
| Module | Further Paper 1 (Further Paper 1) |
| Year | 2024 |
| Session | June |
| Marks | 14 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Linear transformations |
| Type | Decompose matrix into transformation sequence |
| Difficulty | Standard +0.3 This is a straightforward Further Maths question testing standard matrix transformation concepts: decomposing a product into component transformations (shear and stretch, both routine to identify), finding invariant lines (standard eigenvalue-related technique), and using determinant for area scaling. All parts follow textbook procedures with no novel problem-solving required, making it slightly easier than average even for Further Maths. |
| Spec | 4.03d Linear transformations 2D: reflection, rotation, enlargement, shear4.03e Successive transformations: matrix products4.03g Invariant points and lines4.03i Determinant: area scale factor and orientation |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| [One-way] stretch, shear | B1 | Both types. |
| Stretch followed by shear | B1 | Correct order. |
| Stretch parallel to the \(x\)-axis, scale factor 7 | B1 | |
| Shear, \(x\)-axis fixed, with \((0,1)\) mapped to \((2,1)\) | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\mathbf{M} = \begin{pmatrix} 7 & 2 \\ 0 & 1 \end{pmatrix}\) | B1 | |
| \(\begin{pmatrix} 7 & 2 \\ 0 & 1 \end{pmatrix}\begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} 7x+2y \\ y \end{pmatrix}\) | B1 FT | Transforms \(\begin{pmatrix} x \\ y \end{pmatrix}\) to \(\begin{pmatrix} X \\ Y \end{pmatrix}\). |
| \(mx = m(7x + 2mx)\) | M1 | Uses \(y = mx\) and \(Y = mX\). |
| \(2m^2 + 6m = 0\) | A1 | |
| \(y = 0\) and \(y = -3x\) | A1 | SCB1 if M0 and both straight lines correct |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Area of \(PQR = | 7 | \times\) Area of \(DEF\) |
| Area of \(DEF = 5 \text{ cm}^2\) | A1 | SCB1 for an answer of 245 (\(35 \times 7\)) |
## Question 3(a):
| Answer | Marks | Guidance |
|--------|-------|----------|
| [One-way] stretch, shear | **B1** | Both types. |
| Stretch followed by shear | **B1** | Correct order. |
| Stretch parallel to the $x$-axis, scale factor 7 | **B1** | |
| Shear, $x$-axis fixed, with $(0,1)$ mapped to $(2,1)$ | **B1** | |
---
## Question 3(b):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\mathbf{M} = \begin{pmatrix} 7 & 2 \\ 0 & 1 \end{pmatrix}$ | **B1** | |
| $\begin{pmatrix} 7 & 2 \\ 0 & 1 \end{pmatrix}\begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} 7x+2y \\ y \end{pmatrix}$ | **B1 FT** | Transforms $\begin{pmatrix} x \\ y \end{pmatrix}$ to $\begin{pmatrix} X \\ Y \end{pmatrix}$. |
| $mx = m(7x + 2mx)$ | **M1** | Uses $y = mx$ and $Y = mX$. |
| $2m^2 + 6m = 0$ | **A1** | |
| $y = 0$ and $y = -3x$ | **A1** | SCB1 if M0 and both straight lines correct |
---
## Question 3(c):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Area of $PQR = |7| \times$ Area of $DEF$ | **M1** | |
| Area of $DEF = 5 \text{ cm}^2$ | **A1** | SCB1 for an answer of 245 ($35 \times 7$) |
---3 The matrix $\mathbf { M }$ is given by $\mathbf { M } = \left( \begin{array} { l l } 1 & 2 \\ 0 & 1 \end{array} \right) \left( \begin{array} { l l } 7 & 0 \\ 0 & 1 \end{array} \right)$.
\begin{enumerate}[label=(\alph*)]
\item The matrix $\mathbf { M }$ represents a sequence of two geometrical transformations in the $x - y$ plane.
Give full details of each transformation, and make clear the order in which they are applied. [4]
\item Find the equations of the invariant lines, through the origin, of the transformation represented by $\mathbf { M }$.\\
The triangle $D E F$ in the $x - y$ plane is transformed by $\mathbf { M }$ onto triangle $P Q R$ .
\item Given that the area of triangle $P Q R$ is $35 \mathrm {~cm} ^ { 2 }$ ,find the area of triangle $D E F$ .
\end{enumerate}
\hfill \mbox{\textit{CAIE Further Paper 1 2024 Q3 [14]}}