| Exam Board | OCR |
|---|---|
| Module | Further Pure Core 2 (Further Pure Core 2) |
| Year | 2020 |
| Session | November |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Invariant lines and eigenvalues and vectors |
| Type | Describe geometric transformation from matrix |
| Difficulty | Challenging +1.2 This is a structured Further Maths question requiring understanding of determinants and geometric transformations. Part (a) is routine calculation. Parts (b)-(d) require knowing that det(stretch)×det(rotation)=det(A) and that rotations preserve one axis while stretches don't, but the question heavily scaffolds the approach. The calculations are straightforward once the concepts are understood, making this moderately above average difficulty. |
| Spec | 4.03d Linear transformations 2D: reflection, rotation, enlargement, shear4.03h Determinant 2x2: calculation4.03i Determinant: area scale factor and orientation |
| Answer | Marks | Guidance |
|---|---|---|
| 7 | (a) | detA (= 0.6×1.8 – –0.8×2.4) = 3 |
| [1] | 1.1 | |
| (b) | Determinant of rotation = 1 |
| Answer | Marks |
|---|---|
| stretch = 1×sf = 3 => sf = 3 | B1 |
| Answer | Marks |
|---|---|
| [2] | 1.1 |
| Answer | Marks |
|---|---|
| (c) | Since the second column of A contains |
| Answer | Marks | Guidance |
|---|---|---|
| stretch must be parallel to the y-axis. | B1 | |
| [1] | 2.4 | Or any correct, complete |
| explanation. | May see |
| Answer | Marks |
|---|---|
| (d) | sinθ = –0.8 and cosθ = 0.6 oe |
| awrt –53° (or –0.93 rads) | M1 |
| Answer | Marks |
|---|---|
| [2] | 2.2a |
| 1.1 | Condone if only one equation |
Question 7:
7 | (a) | detA (= 0.6×1.8 – –0.8×2.4) = 3 | B1
[1] | 1.1
(b) | Determinant of rotation = 1
Determinant of rotation × determinant of
stretch = 1×sf = 3 => sf = 3 | B1
B1
[2] | 1.1
2.2a
(c) | Since the second column of A contains
entries bigger than 1 (in magnitude) the
stretch must be parallel to the y-axis. | B1
[1] | 2.4 | Or any correct, complete
explanation. | May see
cosθ −sinθ1 0
sinθ cosθ0 3
cosθ −3sinθ
=
sinθ 3cosθ
or similar
(d) | sinθ = –0.8 and cosθ = 0.6 oe
awrt –53° (or –0.93 rads) | M1
A1
[2] | 2.2a
1.1 | Condone if only one equation
or 53° (0.93 rads) clockwise or
307° (5.36 rads) (anticlockwise).7 The matrix $\mathbf { A }$ is given by $\mathbf { A } = \left( \begin{array} { r r } 0.6 & 2.4 \\ - 0.8 & 1.8 \end{array} \right)$.
\begin{enumerate}[label=(\alph*)]
\item Find $\operatorname { det } \mathbf { A }$.
The matrix A represents a stretch parallel to one of the coordinate axes followed by a rotation about the origin.
\item By considering the determinants of these transformations, determine the scale factor of the stretch.
\item Explain whether the stretch is parallel to the $x$-axis or the $y$-axis, justifying your answer.
\item Find the angle of rotation.
\end{enumerate}
\hfill \mbox{\textit{OCR Further Pure Core 2 2020 Q7 [6]}}