AS June 2018 Paper 1 Q5
5 A transformation of the \(x\)-\(y\) plane is represented by the matrix \(\begin{pmatrix} \cos\theta & 2\sin\theta \\ 2\sin\theta & -\cos\theta \end{pmatrix}\), where \(\theta\) is a positive acute angle.
(i) Write down the image of the point \((2, 3)\) under this transformation. [2]
(ii) You are given that this image is the point \((a, 0)\). Find the value of \(a\). [5]
| Scheme | Marks | AO |
|---|---|---|
| \((2\cos\theta + 6\sin\theta,\ 4\sin\theta - 3\cos\theta)\) | B1B1 | 1.1,1.1 |
| [2] |
Notes
B1B1: Accept in vector form
| Scheme | Marks | AO |
|---|---|---|
| \(4\sin\theta - 3\cos\theta = 0\) | M1 | 3.1a |
| \(\Rightarrow \tan\theta = \tfrac{3}{4}\) | M1 | 1.1 |
| \(\Rightarrow \theta = 36.9^\circ\) or 0.644 rad | A1 | 1.1 |
| \(a = 2\cos\theta + 6\sin\theta = 5.2\) | M1 | 1.1 |
| A1 | 1.1cao | |
| [5] |
Notes
M1: (1st) their \(4\sin\theta - 3\cos\theta = 0\)
M1: (2nd) \(\tan\theta = \sin\theta / \cos\theta\) used
or \(\sin^2\theta + \cos^2\theta = 1\) used
A1: (1st) \(\theta = 36.9^\circ\) or 0.644 rad or better
or \(\sin\theta = \tfrac{3}{5}, \cos\theta = \tfrac{4}{5}\)
M1: (3rd) substituting their \(\theta\) into their \(2\cos\theta + 6\sin\theta\)
or \(\sin\theta\) and \(\cos\theta\)
A1: (2nd) 5.2