| Exam Board | CAIE |
|---|---|
| Module | Further Paper 1 (Further Paper 1) |
| Year | 2021 |
| Session | November |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Invariant lines and eigenvalues and vectors |
| Type | Find invariant lines through origin |
| Difficulty | Challenging +1.2 This question requires understanding of matrix transformations and invariant lines, but follows a standard pattern. Part (a) is straightforward recognition of rotation and stretch matrices. Part (b) requires knowing that one invariant line occurs when eigenvalues are equal (repeated root), leading to a standard trigonometric equation. While it's Further Maths content, the execution is relatively routine once the key concept is recalled. |
| Spec | 4.03d Linear transformations 2D: reflection, rotation, enlargement, shear4.03e Successive transformations: matrix products4.03g Invariant points and lines |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| One-way stretch [scale factor 3 parallel to the \(x\)-axis] followed by a rotation [anticlockwise centred at the origin, through an angle \(\theta\)] | B2 | Award B1 if given in the wrong order |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\mathbf{M} = \begin{pmatrix} 3\cos\theta & -\sin\theta \\ 3\sin\theta & \cos\theta \end{pmatrix}\) | B1 | |
| \(\begin{pmatrix} 3\cos\theta & -\sin\theta \\ 3\sin\theta & \cos\theta \end{pmatrix}\begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} 3x\cos\theta - y\sin\theta \\ 3x\sin\theta + y\cos\theta \end{pmatrix}\) | B1 | Transforms \(\begin{pmatrix} x \\ y \end{pmatrix}\) to \(\begin{pmatrix} X \\ Y \end{pmatrix}\) |
| \(3x\sin\theta + mx\cos\theta = 3mx\cos\theta - m^2 x\sin\theta\) | M1 A1 | Uses \(y = mx\) and \(Y = mX\) |
| \(m^2\sin\theta - 2m\cos\theta + 3\sin\theta = 0\) | A1 | |
| \(4\cos^2\theta - 12\sin^2\theta = 0\) | M1 A1 | Sets discriminant equal to 0 |
| \(\tan\theta = \pm\frac{1}{\sqrt{3}}\) or \(4\cos^2\theta - 12(1-\cos^2\theta) = 0\) leading to \(16\cos^2\theta - 12 = 0\) leading to \(\cos\theta = \pm\frac{\sqrt{3}}{2}\) | M1 | Uses an appropriate trigonometric identity |
| \(\theta = \frac{1}{6}\pi\) and \(\theta = \frac{5}{6}\pi\) | A1 | Allow \(30°\) and \(150°\) |
## Question 4(a):
| Answer | Marks | Guidance |
|--------|-------|----------|
| One-way stretch [scale factor 3 parallel to the $x$-axis] followed by a rotation [anticlockwise centred at the origin, through an angle $\theta$] | B2 | Award B1 if given in the wrong order |
## Question 4(b):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\mathbf{M} = \begin{pmatrix} 3\cos\theta & -\sin\theta \\ 3\sin\theta & \cos\theta \end{pmatrix}$ | B1 | |
| $\begin{pmatrix} 3\cos\theta & -\sin\theta \\ 3\sin\theta & \cos\theta \end{pmatrix}\begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} 3x\cos\theta - y\sin\theta \\ 3x\sin\theta + y\cos\theta \end{pmatrix}$ | B1 | Transforms $\begin{pmatrix} x \\ y \end{pmatrix}$ to $\begin{pmatrix} X \\ Y \end{pmatrix}$ |
| $3x\sin\theta + mx\cos\theta = 3mx\cos\theta - m^2 x\sin\theta$ | M1 A1 | Uses $y = mx$ and $Y = mX$ |
| $m^2\sin\theta - 2m\cos\theta + 3\sin\theta = 0$ | A1 | |
| $4\cos^2\theta - 12\sin^2\theta = 0$ | M1 A1 | Sets discriminant equal to 0 |
| $\tan\theta = \pm\frac{1}{\sqrt{3}}$ or $4\cos^2\theta - 12(1-\cos^2\theta) = 0$ leading to $16\cos^2\theta - 12 = 0$ leading to $\cos\theta = \pm\frac{\sqrt{3}}{2}$ | M1 | Uses an appropriate trigonometric identity |
| $\theta = \frac{1}{6}\pi$ and $\theta = \frac{5}{6}\pi$ | A1 | Allow $30°$ and $150°$ |4 The matrix $\mathbf { M }$ is given by $\mathbf { M } = \left( \begin{array} { c c } \cos \theta & - \sin \theta \\ \sin \theta & \cos \theta \end{array} \right) \left( \begin{array} { l l } 3 & 0 \\ 0 & 1 \end{array} \right)$.
\begin{enumerate}[label=(\alph*)]
\item The matrix $\mathbf { M }$ represents a sequence of two geometrical transformations.
State the type of each transformation, and make clear the order in which they are applied.
\item Find the values of $\theta$, for $0 \leqslant \theta \leqslant \pi$, for which the transformation represented by $\mathbf { M }$ has exactly one invariant line through the origin, giving your answers in terms of $\pi$.
\end{enumerate}
\hfill \mbox{\textit{CAIE Further Paper 1 2021 Q4 [11]}}