June 2016 Paper 2 Q23
23 Shape \(A\) maps to shape \(B\) by an enlargement, scale factor 3, centre the origin.
Shape \(B\) maps to shape \(C\) by a rotation through 180°, centre the origin.
Shape \(A\) can be mapped to shape \(C\) by a single transformation.
Use matrices to show that the single transformation is an enlargement, centre the origin.
State the scale factor of the enlargement. [5 marks]
| Answer | Mark | Comments |
|---|---|---|
| Alternative method 1 | ||
| \(\begin{pmatrix} 3 & 0 \\ 0 & 3 \end{pmatrix}\) or \(3\begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}\) | B1 | |
| \(\begin{pmatrix} -1 & 0 \\ 0 & -1 \end{pmatrix}\) | B1 | |
| their \(\begin{pmatrix} -1 & 0 \\ 0 & -1 \end{pmatrix}\ (\times)\) their \(\begin{pmatrix} 3 & 0 \\ 0 & 3 \end{pmatrix}\) | M1 | Either order This mark cannot be implied Must have scored B1 or B2 |
| \(\begin{pmatrix} -3 & 0 \\ 0 & -3 \end{pmatrix}\) or \(-3\begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}\) or \(3\begin{pmatrix} -1 & 0 \\ 0 & -1 \end{pmatrix}\) | M1dep | Correctly multiplies their pair of 2 by 2 matrices |
| \(\begin{pmatrix} -3 & 0 \\ 0 & -3 \end{pmatrix}\) or \(-3\begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}\) or \(3\begin{pmatrix} -1 & 0 \\ 0 & -1 \end{pmatrix}\) and scale factor \(-3\) | A1 | Must gain B1 B1 M1 M1 |
| Alternative method 2 Algebraic method | ||
| \(\begin{pmatrix} 3 & 0 \\ 0 & 3 \end{pmatrix}\) or \(3\begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}\) | B1 | |
| \(\begin{pmatrix} -1 & 0 \\ 0 & -1 \end{pmatrix}\) | B1 | |
| their \(\begin{pmatrix} 3 & 0 \\ 0 & 3 \end{pmatrix}\begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} 3x \\ 3y \end{pmatrix}\) or their \(\begin{pmatrix} -1 & 0 \\ 0 & -1 \end{pmatrix}\ (\times) \begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} -x \\ -y \end{pmatrix}\) | M1 | This mark cannot be implied Must have scored B1 or B2 Multiplications must be correctly worked out |
| their \(\begin{pmatrix} -1 & 0 \\ 0 & -1 \end{pmatrix}\ (\times)\) their \(\begin{pmatrix} 3x \\ 3y \end{pmatrix} = \begin{pmatrix} -3x \\ -3y \end{pmatrix}\) or their \(\begin{pmatrix} 3 & 0 \\ 0 & 3 \end{pmatrix}\ (\times)\) their \(\begin{pmatrix} -x \\ -y \end{pmatrix} = \begin{pmatrix} -3x \\ -3y \end{pmatrix}\) | M1dep | Multiplications must be correctly worked out |
| \(\begin{pmatrix} -3x \\ -3y \end{pmatrix}\) and scale factor \(-3\) | A1 | Must gain B1 B1 M1 M1 |
| Alternative method 3 Unit square method | ||
| \(\begin{pmatrix} 3 & 0 \\ 0 & 3 \end{pmatrix}\) or \(3\begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}\) | B1 | |
| \(\begin{pmatrix} -1 & 0 \\ 0 & -1 \end{pmatrix}\) | B1 | |
| their \(\begin{pmatrix} 3 & 0 \\ 0 & 3 \end{pmatrix}\ (\times) \begin{pmatrix} 1 & 0 & 1 \\ 0 & 1 & 1 \end{pmatrix} = \begin{pmatrix} 3 & 0 & 3 \\ 0 & 3 & 3 \end{pmatrix}\) or their \(\begin{pmatrix} -1 & 0 \\ 0 & -1 \end{pmatrix}\ (\times) \begin{pmatrix} 1 & 0 & 1 \\ 0 & 1 & 1 \end{pmatrix} = \begin{pmatrix} -1 & 0 & -1 \\ 0 & -1 & -1 \end{pmatrix}\) | M1 | This mark cannot be implied Must have scored B1 or B2 Multiplications must be correctly worked out May be seen as three products |
| their \(\begin{pmatrix} -1 & 0 \\ 0 & -1 \end{pmatrix}\ (\times)\) their \(\begin{pmatrix} 3 & 0 & 3 \\ 0 & 3 & 3 \end{pmatrix}\) \(=\begin{pmatrix} -3 & 0 & -3 \\ 0 & -3 & -3 \end{pmatrix}\) or their \(\begin{pmatrix} 3 & 0 \\ 0 & 3 \end{pmatrix}\ (\times)\) their \(\begin{pmatrix} -1 & 0 & -1 \\ 0 & -1 & -1 \end{pmatrix}\) \(=\begin{pmatrix} -3 & 0 & -3 \\ 0 & -3 & -3 \end{pmatrix}\) | M1dep | Multiplications must be correctly worked out May be seen as three products |
| \(\begin{pmatrix} -3 & 0 & -3 \\ 0 & -3 & -3 \end{pmatrix}\) and scale factor \(-3\) | A1 | Must gain B1 B1 M1 M1 May be seen as three 2 by 1 matrices |
Additional guidance
| If both matrices are incorrect | Zero |
| Matrices must be used - ignore diagrams | |
| In matrices, allow missing brackets or inclusion of ‘fraction’ lines | |
| Alt 1 B2 gained then \(\begin{pmatrix} -3 & 0 \\ 0 & -3 \end{pmatrix}\) stated | B2 M0 M0 A0 |
| Allow ‘enlargement \(-3\)’ for ‘scale factor \(-3\)’ Do not allow ‘\(-3\)’ for ‘scale factor \(-3\)’ | |
| Scale factor \(-3\) with no valid working | Zero |
| \(\begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}\begin{pmatrix} 3 & 0 \\ 0 & 3 \end{pmatrix} = \begin{pmatrix} 3 & 0 \\ 0 & 3 \end{pmatrix}\) scores B1 but does not score M1 M1 for the multiplication of two matrices with B1 scored | |
| Alt 3 May also see working for \(\begin{pmatrix} 0 \\ 0 \end{pmatrix}\) |