| Exam Board | CAIE |
|---|---|
| Module | Further Paper 1 (Further Paper 1) |
| Year | 2023 |
| Session | November |
| Marks | 8 |
| 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 on matrix transformations requiring identification of basic transformations (stretch and shear), calculation of area via determinant, finding inverse matrix, and verifying an invariant line by showing Mv = λv. All techniques are standard with no novel insight required, making it slightly easier than average even for Further Maths. |
| Spec | 4.03a Matrix language: terminology and notation4.03b Matrix operations: addition, multiplication, scalar4.03d Linear transformations 2D: reflection, rotation, enlargement, shear4.03e Successive transformations: matrix products4.03g Invariant points and lines4.03h Determinant 2x2: calculation4.03n Inverse 2x2 matrix |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Shear followed by a stretch | B2 | Award B1 if given in the wrong order |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \( | OPQR | = |
| \(\mathbf{M}^{-1} = \frac{1}{k}\begin{pmatrix}1 & 0\\-1 & k\end{pmatrix}\) | M1 A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\begin{pmatrix}k & 0\\1 & 1\end{pmatrix}\begin{pmatrix}x\\\frac{1}{k-1}x\end{pmatrix}\) | B1 | Sets \(y = \frac{1}{k-1}x\) |
| \(\begin{pmatrix}k & 0\\1 & 1\end{pmatrix}\begin{pmatrix}x\\\frac{1}{k-1}x\end{pmatrix} = \begin{pmatrix}kx\\x+\frac{1}{k-1}x\end{pmatrix} = \begin{pmatrix}kx\\\frac{k}{k-1}x\end{pmatrix}\) | M1 | Shows that \(Y = \frac{1}{k-1}X\) |
| \(k\begin{pmatrix}x\\\frac{1}{k-1}x\end{pmatrix}\) | A1 | |
| Alternative: \(\begin{pmatrix}k & 0\\1 & 1\end{pmatrix}\begin{pmatrix}x\\y\end{pmatrix} = \begin{pmatrix}kx\\x+y\end{pmatrix}\) | B1 | Transforms \(\begin{pmatrix}x\\y\end{pmatrix}\) to \(\begin{pmatrix}X\\Y\end{pmatrix}\) |
| \(X = kx\) and \(mX = x+y\), \(mkx = x+mx\) | M1 | Uses \(y=mx\) and \(Y=mX\) |
| \(m = \frac{1}{k-1}\), \(y = \frac{1}{k-1}x\) | A1 | AG |
## Question 3(a):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Shear followed by a stretch | B2 | Award B1 if given in the wrong order |
---
## Question 3(b):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $|OPQR| = |\det\mathbf{M}| = |k|$ | B1 | |
| $\mathbf{M}^{-1} = \frac{1}{k}\begin{pmatrix}1 & 0\\-1 & k\end{pmatrix}$ | M1 A1 | |
---
## Question 3(c):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\begin{pmatrix}k & 0\\1 & 1\end{pmatrix}\begin{pmatrix}x\\\frac{1}{k-1}x\end{pmatrix}$ | B1 | Sets $y = \frac{1}{k-1}x$ |
| $\begin{pmatrix}k & 0\\1 & 1\end{pmatrix}\begin{pmatrix}x\\\frac{1}{k-1}x\end{pmatrix} = \begin{pmatrix}kx\\x+\frac{1}{k-1}x\end{pmatrix} = \begin{pmatrix}kx\\\frac{k}{k-1}x\end{pmatrix}$ | M1 | Shows that $Y = \frac{1}{k-1}X$ |
| $k\begin{pmatrix}x\\\frac{1}{k-1}x\end{pmatrix}$ | A1 | |
| **Alternative:** $\begin{pmatrix}k & 0\\1 & 1\end{pmatrix}\begin{pmatrix}x\\y\end{pmatrix} = \begin{pmatrix}kx\\x+y\end{pmatrix}$ | B1 | Transforms $\begin{pmatrix}x\\y\end{pmatrix}$ to $\begin{pmatrix}X\\Y\end{pmatrix}$ |
| $X = kx$ and $mX = x+y$, $mkx = x+mx$ | M1 | Uses $y=mx$ and $Y=mX$ |
| $m = \frac{1}{k-1}$, $y = \frac{1}{k-1}x$ | A1 | AG |
---3 The matrix $\mathbf { M }$ is given by $\mathbf { M } = \left( \begin{array} { l l } k & 0 \\ 0 & 1 \end{array} \right) \left( \begin{array} { l l } 1 & 0 \\ 1 & 1 \end{array} \right)$, where $k$ is a constant and $k \neq 0$ and $k \neq 1$.
\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.\\
The unit square in the $x - y$ plane is transformed by $\mathbf { M }$ onto parallelogram $O P Q R$.
\item Find, in terms of $k$, the area of parallelogram $O P Q R$ and the matrix which transforms $O P Q R$ onto the unit square.
\item Show that the line through the origin with gradient $\frac { 1 } { k - 1 }$ is invariant under the transformation in the $x - y$ plane represented by $\mathbf { M }$.
\end{enumerate}
\hfill \mbox{\textit{CAIE Further Paper 1 2023 Q3 [8]}}