OCR MEI FP1 2008 January — Question 9 13 marks

Exam BoardOCR MEI
ModuleFP1 (Further Pure Mathematics 1)
Year2008
SessionJanuary
Marks13
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicLinear transformations
TypeWrite down transformation matrix
DifficultyModerate -0.8 This is a straightforward Further Pure 1 question that guides students through finding a transformation matrix step-by-step. Parts (i)-(iii) involve simple pattern recognition (P' has same x-coordinate, lies on y=x, so image is (x,x)), leading directly to the matrix. Parts (iv)-(vi) are routine matrix multiplication and interpretation. While it's Further Maths content, the scaffolding and mechanical nature make it easier than average overall.
Spec4.03a Matrix language: terminology and notation4.03d Linear transformations 2D: reflection, rotation, enlargement, shear4.03e Successive transformations: matrix products
9 A transformation T acts on all points in the plane. The image of a general point P is denoted by \(\mathrm { P } ^ { \prime }\). \(\mathrm { P } ^ { \prime }\) always lies on the line \(y = x\) and has the same \(x\)-coordinate as P. This is illustrated in Fig. 9. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{225bff01-f2c4-421f-ac91-c6a0fcb01e6f-4_807_825_402_660} \captionsetup{labelformat=empty} \caption{Fig. 9}
\end{figure}
  1. Write down the image of the point ( \(- 3,7\) ) under transformation T .
  2. Write down the image of the point \(( x , y )\) under transformation T .
  3. Find the \(2 \times 2\) matrix which represents the transformation.
  4. Describe the transformation M represented by the matrix \(\left( \begin{array} { r r } 0 & - 1 \\ 1 & 0 \end{array} \right)\).
  5. Find the matrix representing the composite transformation of T followed by M .
  6. Find the image of the point \(( x , y )\) under this composite transformation. State the equation of the line on which all of these images lie.
Question 9(i):
AnswerMarks Guidance
Answer/WorkingMark Guidance
\((-3,\ -3)\)B1 [1]
Question 9(ii):
AnswerMarks Guidance
Answer/WorkingMark Guidance
\((x,\ x)\)B1, B1 [2]
Question 9(iii):
AnswerMarks Guidance
Answer/WorkingMark Guidance
\(\begin{pmatrix}1 & 0\\1 & 0\end{pmatrix}\)B3 [3] Minus 1 each error to min of 0
Question 9(iv):
AnswerMarks Guidance
Answer/WorkingMark Guidance
Rotation through \(\frac{\pi}{2}\) anticlockwise about the originB1, B1 [2] Rotation and angle (accept \(90°\)); centre and sense
Question 9(v):
AnswerMarks Guidance
Answer/WorkingMark Guidance
\(\begin{pmatrix}0 & -1\\1 & 0\end{pmatrix}\times\begin{pmatrix}1 & 0\\1 & 0\end{pmatrix} = \begin{pmatrix}-1 & 0\\1 & 0\end{pmatrix}\)M1 Attempt to multiply using their T in correct order
A1 [2]c.a.o.
Question 9(vi):
AnswerMarks Guidance
Answer/WorkingMark Guidance
\(\begin{pmatrix}-1 & 0\\1 & 0\end{pmatrix}\begin{pmatrix}x\\y\end{pmatrix} = \begin{pmatrix}-x\\x\end{pmatrix}\)M1, A1(ft) May be implied
So \((-x,\ x)\)
Line \(y = -x\)A1 [3] c.a.o. from correct matrix 
# Question 9(i):

| Answer/Working | Mark | Guidance |
|---|---|---|
| $(-3,\ -3)$ | B1 **[1]** | |

---

# Question 9(ii):

| Answer/Working | Mark | Guidance |
|---|---|---|
| $(x,\ x)$ | B1, B1 **[2]** | |

---

# Question 9(iii):

| Answer/Working | Mark | Guidance |
|---|---|---|
| $\begin{pmatrix}1 & 0\\1 & 0\end{pmatrix}$ | B3 **[3]** | Minus 1 each error to min of 0 |

---

# Question 9(iv):

| Answer/Working | Mark | Guidance |
|---|---|---|
| Rotation through $\frac{\pi}{2}$ anticlockwise about the origin | B1, B1 **[2]** | Rotation and angle (accept $90°$); centre and sense |

---

# Question 9(v):

| Answer/Working | Mark | Guidance |
|---|---|---|
| $\begin{pmatrix}0 & -1\\1 & 0\end{pmatrix}\times\begin{pmatrix}1 & 0\\1 & 0\end{pmatrix} = \begin{pmatrix}-1 & 0\\1 & 0\end{pmatrix}$ | M1 | Attempt to multiply using their **T** in correct order |
| | A1 **[2]** | c.a.o. |

---

# Question 9(vi):

| Answer/Working | Mark | Guidance |
|---|---|---|
| $\begin{pmatrix}-1 & 0\\1 & 0\end{pmatrix}\begin{pmatrix}x\\y\end{pmatrix} = \begin{pmatrix}-x\\x\end{pmatrix}$ | M1, A1(ft) | May be implied |
| So $(-x,\ x)$ | | |
| Line $y = -x$ | A1 **[3]** | c.a.o. from correct matrix |
9 A transformation T acts on all points in the plane. The image of a general point P is denoted by $\mathrm { P } ^ { \prime }$. $\mathrm { P } ^ { \prime }$ always lies on the line $y = x$ and has the same $x$-coordinate as P. This is illustrated in Fig. 9.

\begin{figure}[h]
\begin{center}
  \includegraphics[alt={},max width=\textwidth]{225bff01-f2c4-421f-ac91-c6a0fcb01e6f-4_807_825_402_660}
\captionsetup{labelformat=empty}
\caption{Fig. 9}
\end{center}
\end{figure}

(i) Write down the image of the point ( $- 3,7$ ) under transformation T .\\
(ii) Write down the image of the point $( x , y )$ under transformation T .\\
(iii) Find the $2 \times 2$ matrix which represents the transformation.\\
(iv) Describe the transformation M represented by the matrix $\left( \begin{array} { r r } 0 & - 1 \\ 1 & 0 \end{array} \right)$.\\
(v) Find the matrix representing the composite transformation of T followed by M .\\
(vi) Find the image of the point $( x , y )$ under this composite transformation. State the equation of the line on which all of these images lie.

\hfill \mbox{\textit{OCR MEI FP1 2008 Q9 [13]}}
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