| Exam Board | OCR |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2007 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Linear transformations |
| Type | Area scale factor from determinant |
| Difficulty | Moderate -0.3 This is a standard Further Pure 1 matrix transformations question with routine parts: writing down an enlargement matrix, recognizing a rotation with enlargement from matrix entries, matrix multiplication verification, and interpreting determinant as area scale factor. While it requires multiple techniques, each step follows textbook procedures with no novel problem-solving required. |
| Spec | 4.03d Linear transformations 2D: reflection, rotation, enlargement, shear4.03e Successive transformations: matrix products4.03h Determinant 2x2: calculation4.03i Determinant: area scale factor and orientation |
| Answer | Marks | Guidance |
|---|---|---|
| (i) \(\begin{pmatrix} \sqrt{2} & 0 \\ 0 & \sqrt{2} \end{pmatrix}\) | B1 | Correct matrix. |
| (ii) Rotation (centre O), 45°, clockwise. | B1B1B1 | Sensible alternatives OK, must be a single transformation. |
| (iii) | B1 | Matrix multiplication or combination of transformations. |
| (iv) \(\begin{pmatrix} 0 \\ 0 \end{pmatrix} \begin{pmatrix} 1 \\ 1 \end{pmatrix} \begin{pmatrix} 1 \\ -1 \end{pmatrix} \begin{pmatrix} 2 \\ 0 \end{pmatrix}\) | M1 A1 | For at least two correct images. For correct diagram. |
| (v) \(\det C = 2\) | B1 | State correct value. |
| area of square has been doubled | B1 | State correct relation a.e.f. |
(i) $\begin{pmatrix} \sqrt{2} & 0 \\ 0 & \sqrt{2} \end{pmatrix}$ | B1 | Correct matrix. |
(ii) Rotation (centre O), 45°, clockwise. | B1B1B1 | Sensible alternatives OK, must be a single transformation. |
(iii) | B1 | Matrix multiplication or combination of transformations. |
(iv) $\begin{pmatrix} 0 \\ 0 \end{pmatrix} \begin{pmatrix} 1 \\ 1 \end{pmatrix} \begin{pmatrix} 1 \\ -1 \end{pmatrix} \begin{pmatrix} 2 \\ 0 \end{pmatrix}$ | M1 A1 | For at least two correct images. For correct diagram. |
(v) $\det C = 2$ | B1 | State correct value. |
area of square has been doubled | B1 | State correct relation a.e.f. |
**Total: 1 + 3 + 1 + 2 + 1 + 1 = 9 marks**9 (i) Write down the matrix, $\mathbf { A }$, that represents an enlargement, centre ( 0,0 ), with scale factor $\sqrt { 2 }$.\\
(ii) The matrix $\mathbf { B }$ is given by $\mathbf { B } = \left( \begin{array} { r r } \frac { 1 } { 2 } \sqrt { 2 } & \frac { 1 } { 2 } \sqrt { 2 } \\ - \frac { 1 } { 2 } \sqrt { 2 } & \frac { 1 } { 2 } \sqrt { 2 } \end{array} \right)$. Describe fully the geometrical transformation represented by $\mathbf { B }$.\\
(iii) Given that $\mathbf { C } = \mathbf { A B }$, show that $\mathbf { C } = \left( \begin{array} { r r } 1 & 1 \\ - 1 & 1 \end{array} \right)$.\\
(iv) Draw a diagram showing the unit square and its image under the transformation represented by $\mathbf { C }$.\\
(v) Write down the determinant of $\mathbf { C }$ and explain briefly how this value relates to the transformation represented by $\mathbf { C }$.
\hfill \mbox{\textit{OCR FP1 2007 Q9 [9]}}