| Exam Board | CAIE |
|---|---|
| Module | Further Paper 1 (Further Paper 1) |
| Year | 2022 |
| Session | November |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Invariant lines and eigenvalues and vectors |
| Type | Find line of invariant points |
| Difficulty | Standard +0.3 This is a structured Further Maths question on matrix transformations that guides students through standard procedures: identifying transformations from matrices, finding matrix inverses, and finding invariant points by solving (M-I)x=0. While it requires multiple techniques, each part follows well-established methods with no novel insight needed. The invariant points calculation is algebraically straightforward, and the area relationship uses the determinant property. Slightly easier than average due to the scaffolding and routine nature of all parts. |
| Spec | 4.03a Matrix language: terminology and notation4.03d Linear transformations 2D: reflection, rotation, enlargement, shear4.03g Invariant points and lines4.03o Inverse 3x3 matrix4.03p Inverse properties: (AB)^(-1) = B^(-1)*A^(-1) |
| Answer | Marks | Guidance |
|---|---|---|
| Shear [in the \(y\)-direction] followed by a stretch [parallel to the \(y\)-axis, scale factor \(k\)]. | B2 | Award B1 if given in the wrong order. |
| Answer | Marks | Guidance |
|---|---|---|
| \(\begin{pmatrix}1 & 0\\k & 1\end{pmatrix}^{-1} = \begin{pmatrix}1 & 0\\-k & 1\end{pmatrix},\ \begin{pmatrix}1 & 0\\0 & k\end{pmatrix}^{-1} = \begin{pmatrix}1 & 0\\0 & \frac{1}{k}\end{pmatrix}\) | B1 | |
| \(\mathbf{M}^{-1} = \begin{pmatrix}1 & 0\\-k & 1\end{pmatrix}\begin{pmatrix}1 & 0\\0 & \frac{1}{k}\end{pmatrix}\) | B1 | Correct order. |
| Answer | Marks | Guidance |
|---|---|---|
| \(\mathbf{M} = \begin{pmatrix}1 & 0\\k^2 & k\end{pmatrix}\) | B1 | |
| \(\begin{pmatrix}1 & 0\\k^2 & k\end{pmatrix}\begin{pmatrix}x\\y\end{pmatrix} = \begin{pmatrix}x\\k^2x + ky\end{pmatrix}\) | B1 | Transforms \(\begin{pmatrix}x\\y\end{pmatrix}\) to \(\begin{pmatrix}X\\Y\end{pmatrix}\) |
| \(k^2x + ky = y\) | M1 | Sets \(\begin{pmatrix}X\\Y\end{pmatrix} = \begin{pmatrix}x\\y\end{pmatrix}\) |
| \(y = \frac{k^2}{1-k}x\) | A1 | AG. |
| Answer | Marks | Guidance |
|---|---|---|
| \(1 = | k | \) |
| \(k = -1\) | A1 |
## Question 3(a):
Shear [in the $y$-direction] followed by a stretch [parallel to the $y$-axis, scale factor $k$]. | B2 | Award B1 if given in the wrong order.
**Total: 2**
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## Question 3(b):
$\begin{pmatrix}1 & 0\\k & 1\end{pmatrix}^{-1} = \begin{pmatrix}1 & 0\\-k & 1\end{pmatrix},\ \begin{pmatrix}1 & 0\\0 & k\end{pmatrix}^{-1} = \begin{pmatrix}1 & 0\\0 & \frac{1}{k}\end{pmatrix}$ | B1 |
$\mathbf{M}^{-1} = \begin{pmatrix}1 & 0\\-k & 1\end{pmatrix}\begin{pmatrix}1 & 0\\0 & \frac{1}{k}\end{pmatrix}$ | B1 | Correct order.
**Total: 2**
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## Question 3(c):
$\mathbf{M} = \begin{pmatrix}1 & 0\\k^2 & k\end{pmatrix}$ | B1 |
$\begin{pmatrix}1 & 0\\k^2 & k\end{pmatrix}\begin{pmatrix}x\\y\end{pmatrix} = \begin{pmatrix}x\\k^2x + ky\end{pmatrix}$ | B1 | Transforms $\begin{pmatrix}x\\y\end{pmatrix}$ to $\begin{pmatrix}X\\Y\end{pmatrix}$
$k^2x + ky = y$ | M1 | Sets $\begin{pmatrix}X\\Y\end{pmatrix} = \begin{pmatrix}x\\y\end{pmatrix}$
$y = \frac{k^2}{1-k}x$ | A1 | AG.
**Total: 4**
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## Question 3(d):
$1 = |k|$ | M1 | Uses $|DEF| = |\det\mathbf{M}||ABC|$
$k = -1$ | A1 |
**Total: 2**
---3 The matrix $\mathbf { M }$ is given by $\mathbf { M } = \left( \begin{array} { c c } 1 & 0 \\ 0 & k \end{array} \right) \left( \begin{array} { c c } 1 & 0 \\ k & 1 \end{array} \right)$, where $k$ is a constant and $k \neq 0$ or 1 .
\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 Write $\mathbf { M } ^ { - 1 }$ as the product of two matrices, neither of which is $\mathbf { I }$.
\item Show that the invariant points of the transformation represented by $\mathbf { M }$ lie on the line $\mathrm { y } = \frac { \mathrm { k } ^ { 2 } } { 1 - \mathrm { k } } \mathrm { x }$. [4]
\item The triangle $A B C$ in the $x - y$ plane is transformed by $\mathbf { M }$ onto triangle $D E F$.
Find the value of $k$ for which the area of triangle $D E F$ is equal to the area of triangle $A B C$.
\end{enumerate}
\hfill \mbox{\textit{CAIE Further Paper 1 2022 Q3 [10]}}