| Exam Board | OCR |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2013 |
| Session | January |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Linear transformations |
| Type | Decompose matrix into transformation sequence |
| Difficulty | Standard +0.3 This is a straightforward FP1 matrix transformation question requiring finding a 2×2 matrix from image coordinates, decomposing it into two simple transformations (likely stretch and shear or two stretches), and computing matrix multiplication. All steps are routine applications of standard techniques with no novel insight required, making it slightly easier than average. |
| Spec | 4.03a Matrix language: terminology and notation4.03b Matrix operations: addition, multiplication, scalar4.03d Linear transformations 2D: reflection, rotation, enlargement, shear |
| Answer | Marks | Guidance |
|---|---|---|
| Answer: \(\begin{pmatrix}1 & 2\\0 & 2\end{pmatrix}\) | B1 B1 [2] | Each column correct |
| Answer | Marks | Guidance |
|---|---|---|
| Answer: Either P: \(\begin{pmatrix}1 & 0\\0 & 2\end{pmatrix}\) or \(\begin{pmatrix}1 & 2\\0 & 1\end{pmatrix}\); Q: \(\begin{pmatrix}1 & 1\\0 & 1\end{pmatrix}\) or \(\begin{pmatrix}1 & 0\\0 & 2\end{pmatrix}\) | B1 DB1, B1; B1 DB1, B1 [6] | Either: Stretch, s.f. 2 in y direction or Shear, x-axis invariant e.g. (0,1) → (2,1); Correct matrix; Shear, x axis invariant e.g. (0, 1) → (1, 1) or Stretch, s.f.2 in y direction; Correct matrix. N.B. "in the x/y axis" is incorrect |
| Answer | Marks | Guidance |
|---|---|---|
| Answer: PQ: \(\begin{pmatrix}1 & 1\\0 & 2\end{pmatrix}\) and \(\begin{pmatrix}1 & 4\\0 & 2\end{pmatrix}\) | M1, A1 [2] | Attempt at matrix multiplication of two 2 x 2 matrices from (ii); Obtain correct result cao |
### (i)
Answer: $\begin{pmatrix}1 & 2\\0 & 2\end{pmatrix}$ | B1 B1 [2] | Each column correct
### (ii)
Answer: Either P: $\begin{pmatrix}1 & 0\\0 & 2\end{pmatrix}$ or $\begin{pmatrix}1 & 2\\0 & 1\end{pmatrix}$; Q: $\begin{pmatrix}1 & 1\\0 & 1\end{pmatrix}$ or $\begin{pmatrix}1 & 0\\0 & 2\end{pmatrix}$ | B1 DB1, B1; B1 DB1, B1 [6] | Either: Stretch, s.f. 2 in y direction or Shear, x-axis invariant e.g. (0,1) → (2,1); Correct matrix; Shear, x axis invariant e.g. (0, 1) → (1, 1) or Stretch, s.f.2 in y direction; Correct matrix. N.B. "in the x/y axis" is incorrect
### (iii)
Answer: PQ: $\begin{pmatrix}1 & 1\\0 & 2\end{pmatrix}$ and $\begin{pmatrix}1 & 4\\0 & 2\end{pmatrix}$ | M1, A1 [2] | Attempt at matrix multiplication of two 2 x 2 matrices from (ii); Obtain correct result cao\includegraphics{figure_6}
The diagram shows the unit square $OABC$, and its image $OA'B'C'$ after a transformation. The points have the following coordinates: $A(1, 0)$, $B(1, 1)$, $C(0, 1)$, $B'(3, 2)$ and $C'(2, 2)$.
\begin{enumerate}[label=(\roman*)]
\item Write down the matrix, X, for this transformation. [2]
\item The transformation represented by X is equivalent to a transformation P followed by a transformation Q. Give geometrical descriptions of a pair of possible transformations P and Q and state the matrices that represent them. [6]
\item Find the matrix that represents transformation Q followed by transformation P. [2]
\end{enumerate}
\hfill \mbox{\textit{OCR FP1 2013 Q6 [10]}}