| Exam Board | CAIE |
|---|---|
| Module | Further Paper 1 (Further Paper 1) |
| Year | 2023 |
| Session | June |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Invariant lines and eigenvalues and vectors |
| Type | Find invariant lines through origin |
| Difficulty | Standard +0.8 This is a multi-part Further Maths question requiring understanding of matrix transformations, eigenvalues/eigenvectors for invariant lines, composition of transformations, and area scale factors. Part (a) requires proving no rotation by showing the matrix doesn't satisfy rotation properties. Part (b) involves finding eigenvalues and eigenvectors of a general matrix with parameters. Parts (c-d) are more routine applications once the theory is understood. The parametric nature and proof element elevate this above standard exercises, but the individual techniques are well-practiced in Further Maths. |
| Spec | 4.03b Matrix operations: addition, multiplication, scalar4.03d Linear transformations 2D: reflection, rotation, enlargement, shear4.03g Invariant points and lines4.03i Determinant: area scale factor and orientation |
| Answer | Marks | Guidance |
|---|---|---|
| \(\begin{pmatrix}a & b^2 \\ c^2 & a\end{pmatrix} = \begin{pmatrix}\cos\theta & -\sin\theta \\ \sin\theta & \cos\theta\end{pmatrix}\) | M1 | Uses correct matrix for rotation. |
| \(b^2 = -c^2\) which is impossible since \(b\) and \(c\) are real and \(b \neq 0\). | A1 | AG |
| Answer | Marks | Guidance |
|---|---|---|
| \(\begin{pmatrix}a & b^2 \\ c^2 & a\end{pmatrix}\begin{pmatrix}x \\ y\end{pmatrix} = \begin{pmatrix}ax+b^2y \\ c^2x+ay\end{pmatrix}\) | B1 | Transforms \(\begin{pmatrix}x\\y\end{pmatrix}\) to \(\begin{pmatrix}X\\Y\end{pmatrix}\). |
| \(c^2x + amx = m(ax + b^2mx)\) | M1 A1 | Uses \(y=mx\) and \(Y=mX\). |
| \(c^2 + am = ma + b^2m^2 \Rightarrow c^2 = b^2m^2\) | A1 | |
| \(y = \frac{c}{b}x\) and \(y = -\frac{c}{b}x\) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| \(\mathbf{M} = \begin{pmatrix}1 & 5 \\ 0 & 1\end{pmatrix}\begin{pmatrix}5 & 0 \\ 0 & 5\end{pmatrix}\) | M1 A1* | Award M1 if matrices correct but order is wrong. |
| \(\begin{pmatrix}5 & 5^2 \\ 0 & 5\end{pmatrix}\) | DA1 | Dep: previous A1 |
| Answer | Marks | Guidance |
|---|---|---|
| \(12 \times \det\mathbf{M}\) | M1 | Using *their* M. |
| \(300 \text{ cm}^2\) | A1FT |
## Question 4:
**Part 4(a):**
$\begin{pmatrix}a & b^2 \\ c^2 & a\end{pmatrix} = \begin{pmatrix}\cos\theta & -\sin\theta \\ \sin\theta & \cos\theta\end{pmatrix}$ | M1 | Uses correct matrix for rotation.
$b^2 = -c^2$ which is impossible since $b$ and $c$ are real and $b \neq 0$. | A1 | AG
**Part 4(b):**
$\begin{pmatrix}a & b^2 \\ c^2 & a\end{pmatrix}\begin{pmatrix}x \\ y\end{pmatrix} = \begin{pmatrix}ax+b^2y \\ c^2x+ay\end{pmatrix}$ | B1 | Transforms $\begin{pmatrix}x\\y\end{pmatrix}$ to $\begin{pmatrix}X\\Y\end{pmatrix}$.
$c^2x + amx = m(ax + b^2mx)$ | M1 A1 | Uses $y=mx$ and $Y=mX$.
$c^2 + am = ma + b^2m^2 \Rightarrow c^2 = b^2m^2$ | A1 |
$y = \frac{c}{b}x$ and $y = -\frac{c}{b}x$ | A1 |
**Part 4(c):**
$\mathbf{M} = \begin{pmatrix}1 & 5 \\ 0 & 1\end{pmatrix}\begin{pmatrix}5 & 0 \\ 0 & 5\end{pmatrix}$ | M1 A1* | Award M1 if matrices correct but order is wrong.
$\begin{pmatrix}5 & 5^2 \\ 0 & 5\end{pmatrix}$ | DA1 | Dep: previous A1
**Part 4(d):**
$12 \times \det\mathbf{M}$ | M1 | Using *their* **M**.
$300 \text{ cm}^2$ | A1FT |
---4 The matrix $\mathbf { M }$ is given by $\mathbf { M } = \left( \begin{array} { l l } \mathrm { a } & \mathrm { b } ^ { 2 } \\ \mathrm { c } ^ { 2 } & \mathrm { a } \end{array} \right)$, where $a , b , c$ are real constants and $b \neq 0$.
\begin{enumerate}[label=(\alph*)]
\item Show that $\mathbf { M }$ does not represent a rotation about the origin.
\item Find the equations of the invariant lines, through the origin, of the transformation in the $x - y$ plane represented by $\mathbf { M }$.\\
It is given that $\mathbf { M }$ represents the sequence of two transformations in the $x - y$ plane given by an enlargement, centre the origin, scale factor 5 followed by a shear, $x$-axis fixed, with $( 0,1 )$ mapped to $( 5,1 )$.
\item Find $\mathbf { M }$.
\item The triangle $D E F$ in the $x - y$ plane is transformed by $\mathbf { M }$ onto triangle $P Q R$.
Given that the area of triangle $D E F$ is $12 \mathrm {~cm} ^ { 2 }$, find the area of triangle $P Q R$.
\end{enumerate}
\hfill \mbox{\textit{CAIE Further Paper 1 2023 Q4 [12]}}