June 2025 Paper 2 Q12
12
(a)
Circle the matrix that represents an enlargement, scale factor 3, centre \(O\). [1 mark]
- \(\begin{pmatrix} 3 & 0 \\ 0 & 3 \end{pmatrix}\)
- \(\begin{pmatrix} 3 & 3 \\ 0 & 0 \end{pmatrix}\)
- \(\begin{pmatrix} 0 & 3 \\ 3 & 0 \end{pmatrix}\)
- \(\begin{pmatrix} 3 & 0 \\ 3 & 0 \end{pmatrix}\)
(b)
\(OABC\) is the unit square.
\(O\ (0, 0) \qquad A\ (1, 0) \qquad B\ (1, 1) \qquad C\ (0, 1)\)
K is the transformation represented by \(\begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}\)
L is the transformation represented by \(\begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}\)
\(OABC\) is mapped to \(OA^{\prime}B^{\prime}C^{\prime}\) under a combined transformation of K followed by L.
Work out the coordinates of \(A^{\prime}\), \(B^{\prime}\) and \(C^{\prime}\). [3 marks]
| Answer | Mark | Comments |
|---|---|---|
| \(\begin{pmatrix} 3 & 0 \\ 0 & 3 \end{pmatrix}\) | B1 |
| Answer | Mark | Comments |
|---|---|---|
| \(A'\,(0, -1) \quad B'\,(-1, -1) \quad C'\,(-1, 0)\) | B3 | B2 \(\begin{pmatrix} 0 & -1 \\ -1 & 0 \end{pmatrix}\) or any two of \(A'\,(0, -1) \quad B'\,(-1, -1) \quad C'\,(-1, 0)\) or \(\begin{pmatrix} 0 \\ -1 \end{pmatrix}\) and \(\begin{pmatrix} -1 \\ -1 \end{pmatrix}\) and \(\begin{pmatrix} -1 \\ 0 \end{pmatrix}\) B1 \(\begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}\begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}\) or \(\begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}\begin{pmatrix} 1 \\ 0 \end{pmatrix}\) and \(\begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}\begin{pmatrix} 1 \\ -1 \end{pmatrix}\) and \(\begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}\begin{pmatrix} 0 \\ -1 \end{pmatrix}\) or any one of \(A'\,(0, -1) \quad B'\,(-1, -1) \quad C'\,(-1, 0)\) |
Additional guidance
| If answer line can only be awarded B1 or B0 check the working space to see if more marks can be awarded | |
| No misreads | |
| B1 awarded for products must have matrices in correct order but can be recovered for B2 or B3 eg \(\begin{pmatrix} 1 \\ 0 \end{pmatrix}\begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}\) and \(\begin{pmatrix} 1 \\ -1 \end{pmatrix}\begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}\) and \(\begin{pmatrix} 0 \\ -1 \end{pmatrix}\begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}\) is B0 but if recovered to \(\begin{pmatrix} 0 \\ -1 \end{pmatrix}\) and \(\begin{pmatrix} -1 \\ -1 \end{pmatrix}\) and \(\begin{pmatrix} -1 \\ 0 \end{pmatrix}\) award B2 | |
| \(\begin{pmatrix} 0 \\ -1 \end{pmatrix}\) and \(\begin{pmatrix} -1 \\ -1 \end{pmatrix}\) and \(\begin{pmatrix} -1 \\ 0 \end{pmatrix}\) may be seen as a single matrix eg \(\begin{pmatrix} 0 & -1 & -1 \\ -1 & -1 & 0 \end{pmatrix}\) or \(\begin{pmatrix} -1 & -1 & 0 \\ -1 & 0 & -1 \end{pmatrix}\) | B2 |
| \(\begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}\begin{pmatrix} 1 \\ 0 \end{pmatrix}\) and \(\begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}\begin{pmatrix} 1 \\ -1 \end{pmatrix}\) and \(\begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}\begin{pmatrix} 0 \\ -1 \end{pmatrix}\) may be seen as a single matrix product eg \(\begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}\begin{pmatrix} 1 & 1 & 0 \\ 0 & -1 & -1 \end{pmatrix}\) or \(\begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}\begin{pmatrix} 0 & 1 & 1 \\ -1 & -1 & 0 \end{pmatrix}\) | B1 |
| \(\begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}\begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}\) may be seen embedded eg \(\begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}\begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}\begin{pmatrix} 1 \\ 0 \end{pmatrix}\) | B1 |