AS June 2023 Paper 1 Q3
3.
\[\mathbf{A} = \begin{pmatrix}1 & 0 & 0\\[4pt] 0 & \dfrac{\sqrt{3}}{2} & -\dfrac{1}{2}\\[8pt] 0 & \dfrac{1}{2} & \dfrac{\sqrt{3}}{2}\end{pmatrix}\](a) Describe fully the single geometric transformation \(A\) represented by the matrix \(\mathbf{A}\). (2)
\[\mathbf{B} = \begin{pmatrix}1 & 3 & 0\\ \sqrt{3} & 0 & 5\sqrt{3}\\ 1 & 2 & 0\end{pmatrix}\]The transformation \(B\) is represented by the matrix \(\mathbf{B}\).
The transformation \(A\) followed by the transformation \(B\) is the transformation \(C\), which is represented by the matrix \(\mathbf{C}\).
To determine matrix \(\mathbf{C}\), a student attempts the following matrix multiplication.
\[\begin{pmatrix}1 & 0 & 0\\[4pt] 0 & \dfrac{\sqrt{3}}{2} & -\dfrac{1}{2}\\[8pt] 0 & \dfrac{1}{2} & \dfrac{\sqrt{3}}{2}\end{pmatrix}\begin{pmatrix}1 & 3 & 0\\ \sqrt{3} & 0 & 5\sqrt{3}\\ 1 & 2 & 0\end{pmatrix}\](b) State the error made by the student. (1)
(c) Determine the correct matrix \(\mathbf{C}\). (1)
| Scheme | Marks | AO |
|---|---|---|
| Rotation | B1 | 1.1b |
| 30 degrees or \(\dfrac{\pi}{6}\) about the \(x\)-axis Ignore any reference to direction | B1 | 1.1b |
| (2) |
Notes
B1: Identifies the single transformation as a rotation only
B1: Correct angle and axis. Ignore any reference to direction.
Note \(x\)-plane, \(zy\)-plane and \(x = 0\) are 2nd B0
Any additional incorrect statements is 2nd B0
| Scheme | Marks | AO |
|---|---|---|
| They have found AB when they should find BA Multiplication is the wrong way round It should be BA Matrix B should be on the left instead of the right Student has done transformation B followed by transformation A It should be \(\begin{pmatrix}1 & 3 & 0\\ \sqrt{3} & 0 & 5\sqrt{3}\\ 1 & 2 & 0\end{pmatrix}\begin{pmatrix}1 & 0 & 0\\[4pt] 0 & \dfrac{\sqrt{3}}{2} & -\dfrac{1}{2}\\[8pt] 0 & \dfrac{1}{2} & \dfrac{\sqrt{3}}{2}\end{pmatrix}\) | B1 | 2.3 |
| (1) |
Notes
B1: Explains that they should be multiplied the other way around
| Scheme | Marks | AO |
|---|---|---|
| \[\left\{\begin{pmatrix}1 & 3 & 0\\ \sqrt{3} & 0 & 5\sqrt{3}\\ 1 & 2 & 0\end{pmatrix}\begin{pmatrix}1 & 0 & 0\\[4pt] 0 & \dfrac{\sqrt{3}}{2} & -\dfrac{1}{2}\\[8pt] 0 & \dfrac{1}{2} & \dfrac{\sqrt{3}}{2}\end{pmatrix} =\right\}\begin{pmatrix}1 & \dfrac{3\sqrt{3}}{2} & -\dfrac{3}{2}\\[8pt] \sqrt{3} & \dfrac{5\sqrt{3}}{2} & \dfrac{15}{2}\\[8pt] 1 & \sqrt{3} & -1\end{pmatrix}\]\[\left\{\begin{pmatrix}1 & 3 & 0\\ \sqrt{3} & 0 & 5\sqrt{3}\\ 1 & 2 & 0\end{pmatrix}\begin{pmatrix}1 & 0 & 0\\[4pt] 0 & \dfrac{\sqrt{3}}{2} & -\dfrac{1}{2}\\[8pt] 0 & \dfrac{1}{2} & \dfrac{\sqrt{3}}{2}\end{pmatrix} =\right\}\begin{pmatrix}1 & \dfrac{3\sqrt{3}}{2} & -1.5\\[8pt] \sqrt{3} & \dfrac{5\sqrt{3}}{2} & 7.5\\[8pt] 1 & \sqrt{3} & -1\end{pmatrix}\] | B1 | 1.1b |
| (1) | ||
| (4 marks) |
Notes
B1: Correct exact matrix
Note: \(5\sqrt{3} \times \dfrac{\sqrt{3}}{2}\) must be simplified to \(\dfrac{15}{2}\)
Condone \(\dfrac{2\sqrt{3}}{2}\) not simplified