| Exam Board | OCR |
|---|---|
| Module | Further Pure Core 2 (Further Pure Core 2) |
| Year | 2021 |
| Session | November |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Linear transformations |
| Type | Decompose matrix into transformation sequence |
| Difficulty | Standard +0.8 This is a Further Maths question requiring students to work with matrix transformations in both directions: identifying a transformation from a matrix (part a), proving area invariance via determinant (part b), and decomposing a composite transformation into constituent matrices to find an unknown parameter (part c). Part (c) requires constructing three transformation matrices, multiplying them in the correct order, equating to B, and solving for p. While systematic, this demands solid understanding of transformation matrices, matrix multiplication, and careful algebraic manipulation—significantly above standard A-level but routine for Further Maths students. |
| Spec | 4.03c Matrix multiplication: properties (associative, not commutative)4.03d Linear transformations 2D: reflection, rotation, enlargement, shear4.03h Determinant 2x2: calculation4.03i Determinant: area scale factor and orientation |
| Answer | Marks | Guidance |
|---|---|---|
| 6 | (a) | DR |
| Answer | Marks | Guidance |
|---|---|---|
| (2, 1). | B1 | |
| [1] | 2.2a | Or any useful point transformed |
| to its image | not “scale factor” or sf | |
| (b) | DR |
| Answer | Marks | Guidance |
|---|---|---|
| scale factor | B1 | |
| [1] | 2.4 | Both |
| Answer | Marks |
|---|---|
| (c) | DR |
| Answer | Marks |
|---|---|
| 3p p | B1 |
| Answer | Marks |
|---|---|
| [4] | 3.1a |
| Answer | Marks |
|---|---|
| 1.1 | BC |
Question 6:
6 | (a) | DR
A shear which leaves the x-axis invariant and
which transforms the point (0, 1) to the point
(2, 1). | B1
[1] | 2.2a | Or any useful point transformed
to its image | not “scale factor” or sf
(b) | DR
detA = 1×1 – 0×2 = 1 and this is the area
scale factor | B1
[1] | 2.4 | Both | Detailed calculation must be
shown
(c) | DR
1 0
seen
3 1
1 21 0 7 2
=
0 13 1 3 1
1 07 2
0 p3 1
7 2
= ⇒ p =7
3p p | B1
B1
M1
A1
[4] | 3.1a
1.1
1.1
1.1 | BC
Correct form for stretch
multiplied into their matrix in
either order
Correct multiplication6 In this question you must show detailed reasoning.\\
The matrix $\mathbf { A }$ is given by $\mathbf { A } = \left( \begin{array} { l l } 1 & 2 \\ 0 & 1 \end{array} \right)$.
\begin{enumerate}[label=(\alph*)]
\item Define the transformation represented by $\mathbf { A }$.
\item Show that the area of any object shape is invariant under the transformation represented by $\mathbf { A }$.
The matrix $\mathbf { B }$ is given by $\mathbf { B } = \left( \begin{array} { r l } 7 & 2 \\ 21 & 7 \end{array} \right)$. You are given that $\mathbf { B }$ represents the transformation which is the result of applying the following three transformations in the given order.
\begin{itemize}
\item A shear which leaves the $y$-axis invariant and which transforms the point $( 1,1 )$ to the point (1, 4).
\item The transformation represented by $\mathbf { A }$.
\item A stretch of scale factor $p$ which leaves the $x$-axis invariant.
\item Determine the value of $p$.
\end{itemize}
\end{enumerate}
\hfill \mbox{\textit{OCR Further Pure Core 2 2021 Q6 [6]}}