AS June 2025 Paper 1 Q11
11 The \(2 \times 2\) matrix \(\mathbf{A}\) represents an anticlockwise rotation of \(150^\circ\) about the origin.
(a) Find matrix \(\mathbf{A}\)
Write each element in its simplest exact form. [2 marks]
(b) Find matrix \(\mathbf{A}^2\)
Write each element in its simplest exact form. [2 marks]
(c) Fully describe the transformation represented by the matrix \(\mathbf{A}^5\) [2 marks]
(d) Find the least positive integer \(n\) which satisfies the equation\[\mathbf{A}^n = \mathbf{I}\] [3 marks]
| Scheme | Marks | AO |
|---|---|---|
| Obtains at least three correct elements. May be unsimplified. | M1 | 1.1a |
| Obtains \(\begin{bmatrix} -\dfrac{\sqrt{3}}{2} & -\dfrac{1}{2} \\[4pt] \dfrac{1}{2} & -\dfrac{\sqrt{3}}{2} \end{bmatrix}\) Accept \(\dfrac{1}{2}\begin{bmatrix} -\sqrt{3} & -1 \\ 1 & -\sqrt{3} \end{bmatrix}\) or \(-\dfrac{1}{2}\begin{bmatrix} \sqrt{3} & 1 \\ -1 & \sqrt{3} \end{bmatrix}\) | A1 | 1.1b |
| (2) |
Typical solution
\[\begin{bmatrix} \cos 150^\circ & -\sin 150^\circ \\ \sin 150^\circ & \cos 150^\circ \end{bmatrix} = \begin{bmatrix} -\dfrac{\sqrt{3}}{2} & -\dfrac{1}{2} \\[4pt] \dfrac{1}{2} & -\dfrac{\sqrt{3}}{2} \end{bmatrix}\]| Scheme | Marks | AO |
|---|---|---|
| Obtains at least three correct elements. May be unsimplified, eg \(\begin{bmatrix} \dfrac{3}{4} - \dfrac{1}{4} & \dfrac{\sqrt{3}}{4} + \dfrac{\sqrt{3}}{4} \\[4pt] -\dfrac{\sqrt{3}}{4} - \dfrac{\sqrt{3}}{4} & -\dfrac{1}{4} + \dfrac{3}{4} \end{bmatrix}\) | M1 | 1.1a |
| Obtains \(\begin{bmatrix} \dfrac{1}{2} & \dfrac{\sqrt{3}}{2} \\[4pt] -\dfrac{\sqrt{3}}{2} & \dfrac{1}{2} \end{bmatrix}\) Accept \(\dfrac{1}{2}\begin{bmatrix} 1 & \sqrt{3} \\ -\sqrt{3} & 1 \end{bmatrix}\) | A1 | 1.1b |
| (2) |
Typical solution
\[\mathbf{A}^2 = \begin{bmatrix} \cos 300^\circ & -\sin 300^\circ \\ \sin 300^\circ & \cos 300^\circ \end{bmatrix} = \begin{bmatrix} \dfrac{1}{2} & \dfrac{\sqrt{3}}{2} \\[4pt] -\dfrac{\sqrt{3}}{2} & \dfrac{1}{2} \end{bmatrix}\]| Scheme | Marks | AO |
|---|---|---|
| Attempts a correct calculation for the rotation angle eg \(150 \times 5\) oe PI by 750 or 30 or \(\begin{bmatrix} \dfrac{\sqrt{3}}{2} & -\dfrac{1}{2} \\[4pt] \dfrac{1}{2} & \dfrac{\sqrt{3}}{2} \end{bmatrix}\) | M1 | 3.1a |
| Obtains the correct transformation. Accept anticlockwise rotation of \(750^\circ\) about the origin oe Accept omission of ‘anticlockwise’. | A1 | 1.1b |
| (2) |
Typical solution
\[750^\circ - 720^\circ = 30^\circ\]Anticlockwise rotation of \(30^\circ\) about the origin
| Scheme | Marks | AO |
|---|---|---|
| Considers multiples of 150 and/or 360 eg list of three multiples eg \(150n\) eg \(360m\) | M1 | 3.1a |
| Finds a common multiple of 150 and 360 eg 3600 or Writes an expression for \(n\) in terms of another integer, eg \(n = \dfrac{360m}{150}\) where \(m \in \mathbf{N}\) PI by an \(n\)-value which is a multiple of 12 | A1 | 1.1b |
| Obtains 12 | A1 | 2.2a |
| (3) | ||
| (9 marks) |
Typical solution
LCM of 150 and 360
\[\begin{aligned} &= 2^3 \times 3^2 \times 5^2 \\ &= 1800\end{aligned}\]\[n = 1800 \div 150 = 12\]