| Exam Board | CAIE |
|---|---|
| Module | Further Paper 1 (Further Paper 1) |
| Year | 2020 |
| Session | November |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Invariant lines and eigenvalues and vectors |
| Type | Find constant from invariant line or area condition |
| Difficulty | Standard +0.3 This is a straightforward Further Maths question combining matrix transformations with invariant lines. Part (a) requires recognizing standard transformation matrices (stretch and shear), part (b) is routine matrix inversion, and part (c) involves applying two standard conditions: determinant equals area scale factor (giving a=2) and the invariant line condition (substituting into Mx=λx to find b). All techniques are direct applications of core Further Maths content with no novel problem-solving required. |
| Spec | 4.03d Linear transformations 2D: reflection, rotation, enlargement, shear4.03e Successive transformations: matrix products4.03g Invariant points and lines4.03o Inverse 3x3 matrix |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| One-way stretch followed by a shear | B2 | Both named correctly. Award B1 if given in the wrong order |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\mathbf{M}^{-1} = \frac{1}{a}\begin{pmatrix} 1 & -b \\ 0 & a \end{pmatrix}\) | M1 A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(a = 2\) | B1 | |
| \(\begin{pmatrix} a & b \\ 0 & 1 \end{pmatrix}\begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} ax+by \\ y \end{pmatrix}\) | B1 | Transforms \(\begin{pmatrix} x \\ y \end{pmatrix}\) to \(\begin{pmatrix} X \\ Y \end{pmatrix}\) |
| \(= \begin{pmatrix} ax - \frac{1}{3}bx \\ -\frac{1}{3}x \end{pmatrix}\) | M1 | Uses \(x + 3y = 0\) |
| \(x = ax - \frac{1}{3}bx \Rightarrow 1 = a - \frac{1}{3}b\) | M1 | Uses that line is invariant (or \(X + 3Y = 0\)) |
| \(b = 3\) | A1 |
## Question 1:
### Part (a):
| Answer | Marks | Guidance |
|--------|-------|----------|
| One-way stretch followed by a shear | **B2** | Both named correctly. Award B1 if given in the wrong order |
---
### Part (b):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\mathbf{M}^{-1} = \frac{1}{a}\begin{pmatrix} 1 & -b \\ 0 & a \end{pmatrix}$ | **M1 A1** | |
---
### Part (c):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $a = 2$ | **B1** | |
| $\begin{pmatrix} a & b \\ 0 & 1 \end{pmatrix}\begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} ax+by \\ y \end{pmatrix}$ | **B1** | Transforms $\begin{pmatrix} x \\ y \end{pmatrix}$ to $\begin{pmatrix} X \\ Y \end{pmatrix}$ |
| $= \begin{pmatrix} ax - \frac{1}{3}bx \\ -\frac{1}{3}x \end{pmatrix}$ | **M1** | Uses $x + 3y = 0$ |
| $x = ax - \frac{1}{3}bx \Rightarrow 1 = a - \frac{1}{3}b$ | **M1** | Uses that line is invariant (or $X + 3Y = 0$) |
| $b = 3$ | **A1** | |1 The matrix $\mathbf { M }$ is given by $\mathbf { M } = \left( \begin{array} { l l } 1 & b \\ 0 & 1 \end{array} \right) \left( \begin{array} { l l } a & 0 \\ 0 & 1 \end{array} \right)$, where $a$ and $b$ are positive constants.
\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 $a$ and $b$, the matrix which transforms parallelogram $O P Q R$ onto the unit square.\\
It is given that the area of $O P Q R$ is $2 \mathrm {~cm} ^ { 2 }$ and that the line $\mathrm { x } + 3 \mathrm { y } = 0$ is invariant under the transformation represented by $\mathbf { M }$.
\item Find the values of $a$ and $b$.
\end{enumerate}
\hfill \mbox{\textit{CAIE Further Paper 1 2020 Q1 [9]}}