A2 October 2021 Paper 2 Q6
6 In this question you must show detailed reasoning.
The matrix \(\mathbf{A}\) is given by \(\mathbf{A} = \begin{pmatrix} 1 & 2 \\ 0 & 1 \end{pmatrix}\).
(a) Define the transformation represented by \(\mathbf{A}\). [1]
(b) Show that the area of any object shape is invariant under the transformation represented by \(\mathbf{A}\). [1]
The matrix \(\mathbf{B}\) is given by \(\mathbf{B} = \begin{pmatrix} 7 & 2 \\ 21 & 7 \end{pmatrix}\). You are given that \(\mathbf{B}\) represents the transformation which is the result of applying the following three transformations in the given order.
- A shear which leaves the \(y\)-axis invariant and which transforms the point \((1, 1)\) to the point \((1, 4)\).
- The transformation represented by \(\mathbf{A}\).
- A stretch of scale factor \(p\) which leaves the \(x\)-axis invariant.
(c) Determine the value of \(p\). [4]
| Scheme | Marks | AO |
|---|---|---|
| DR A shear which leaves the \(x\)-axis invariant and which transforms the point \((0, 1)\) to the point \((2, 1)\). | B1 | 2.2a |
| [1] |
Notes
B1: Or any useful point transformed to its image
not “scale factor” or sf
| Scheme | Marks | AO |
|---|---|---|
| DR \(\det\mathbf{A} = 1 \times 1 - 0 \times 2 = 1\) and this is the area scale factor | B1 | 2.4 |
| [1] |
Notes
B1: Both
Detailed calculation must be shown
| Scheme | Marks | AO |
|---|---|---|
| DR \(\begin{pmatrix} 1 & 0 \\ 3 & 1 \end{pmatrix}\) seen | B1 | 3.1a |
| \(\begin{pmatrix} 1 & 2 \\ 0 & 1 \end{pmatrix}\begin{pmatrix} 1 & 0 \\ 3 & 1 \end{pmatrix} = \begin{pmatrix} 7 & 2 \\ 3 & 1 \end{pmatrix}\) | B1 | 1.1 |
| \(\begin{pmatrix} 1 & 0 \\ 0 & p \end{pmatrix}\begin{pmatrix} 7 & 2 \\ 3 & 1 \end{pmatrix}\) | M1 | 1.1 |
| \(= \begin{pmatrix} 7 & 2 \\ 3p & p \end{pmatrix} \Rightarrow p = 7\) | A1 | 1.1 |
| [4] |
Notes
B1: (second) BC
M1: Correct form for stretch multiplied into their matrix in either order
A1: Correct multiplication