OCR MEI FP1 2006 January — Question 9 12 marks

Exam BoardOCR MEI
ModuleFP1 (Further Pure Mathematics 1)
Year2006
SessionJanuary
Marks12
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicLinear transformations
TypeFind image coordinates under transformation
DifficultyStandard +0.3 This is a structured multi-part question on linear transformations that guides students through understanding a projection. Parts (i)-(v) involve geometric reasoning with clear constraints (same y-coordinate, lies on y=2x), while (vi)-(vii) require finding the matrix and showing singularity. The question is methodical rather than requiring novel insight, making it slightly easier than average for Further Maths FP1.
Spec4.03a Matrix language: terminology and notation4.03d Linear transformations 2D: reflection, rotation, enlargement, shear4.03h Determinant 2x2: calculation4.03i Determinant: area scale factor and orientation4.03l Singular/non-singular matrices4.03n Inverse 2x2 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 \(y = 2 x\) and has the same \(y\)-coordinate as P. This is illustrated in Fig. 9. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{4048c232-6a4e-4baa-9262-93428f375203-4_821_837_475_612} \captionsetup{labelformat=empty} \caption{Fig. 9}
\end{figure}
  1. Write down the image of the point \(( 10,50 )\) under transformation T .
  2. P has coordinates \(( x , y )\). State the coordinates of \(\mathrm { P } ^ { \prime }\).
  3. All points on a particular line \(l\) are mapped onto the point \(( 3,6 )\). Write down the equation of the line \(l\).
  4. In part (iii), the whole of the line \(l\) was mapped by T onto a single point. There are an infinite number of lines which have this property under T. Describe these lines.
  5. For a different set of lines, the transformation T has the same effect as translation parallel to the \(x\)-axis. Describe this set of lines.
  6. Find the \(2 \times 2\) matrix which represents the transformation.
  7. Show that this matrix is singular. Relate this result to the transformation.
Question 9:
(i)
AnswerMarks Guidance
\((25, 50)\)B1 (\(y\)-coord same, \(x = 50/2 = 25\))
(ii)
AnswerMarks
P\('\) has coordinates \(\left(\frac{y}{2}, y\right)\)B1 B1
(iii)
AnswerMarks
\(y = 6\), so line \(l\) is \(y = 6\)B1
(iv)
AnswerMarks
Lines parallel to the \(x\)-axis (horizontal lines)B1
(v)
AnswerMarks
Lines parallel to the \(y\)-axis (vertical lines, i.e. \(x = c\))B1
(vi)
AnswerMarks Guidance
\(\begin{pmatrix} x' \\ y' \end{pmatrix} = \mathbf{M}\begin{pmatrix} x \\ y \end{pmatrix}\), need \(x' = \frac{y}{2}\), \(y' = y\)M1
\(\mathbf{M} = \begin{pmatrix} 0 & \frac{1}{2} \\ 0 & 1 \end{pmatrix}\)A2 A1 each row
(vii)
AnswerMarks
\(\det(\mathbf{M}) = (0)(1) - \left(\frac{1}{2}\right)(0) = 0\)M1 A1
Matrix is singular; this relates to the fact that the transformation maps the whole plane onto the line \(y = 2x\), so it is not invertible / many points map to same imageA1 
# Question 9:

**(i)**

$(25, 50)$ | B1 | ($y$-coord same, $x = 50/2 = 25$)

**(ii)**

P$'$ has coordinates $\left(\frac{y}{2}, y\right)$ | B1 B1 |

**(iii)**

$y = 6$, so line $l$ is $y = 6$ | B1 |

**(iv)**

Lines parallel to the $x$-axis (horizontal lines) | B1 |

**(v)**

Lines parallel to the $y$-axis (vertical lines, i.e. $x = c$) | B1 |

**(vi)**

$\begin{pmatrix} x' \\ y' \end{pmatrix} = \mathbf{M}\begin{pmatrix} x \\ y \end{pmatrix}$, need $x' = \frac{y}{2}$, $y' = y$ | M1 |

$\mathbf{M} = \begin{pmatrix} 0 & \frac{1}{2} \\ 0 & 1 \end{pmatrix}$ | A2 | A1 each row

**(vii)**

$\det(\mathbf{M}) = (0)(1) - \left(\frac{1}{2}\right)(0) = 0$ | M1 A1 |

Matrix is singular; this relates to the fact that the transformation maps the whole plane onto the line $y = 2x$, so it is not invertible / many points map to same image | A1 |
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 $y = 2 x$ and has the same $y$-coordinate as P. This is illustrated in Fig. 9.

\begin{figure}[h]
\begin{center}
  \includegraphics[alt={},max width=\textwidth]{4048c232-6a4e-4baa-9262-93428f375203-4_821_837_475_612}
\captionsetup{labelformat=empty}
\caption{Fig. 9}
\end{center}
\end{figure}

(i) Write down the image of the point $( 10,50 )$ under transformation T .\\
(ii) P has coordinates $( x , y )$. State the coordinates of $\mathrm { P } ^ { \prime }$.\\
(iii) All points on a particular line $l$ are mapped onto the point $( 3,6 )$. Write down the equation of the line $l$.\\
(iv) In part (iii), the whole of the line $l$ was mapped by T onto a single point. There are an infinite number of lines which have this property under T. Describe these lines.\\
(v) For a different set of lines, the transformation T has the same effect as translation parallel to the $x$-axis. Describe this set of lines.\\
(vi) Find the $2 \times 2$ matrix which represents the transformation.\\
(vii) Show that this matrix is singular. Relate this result to the transformation.

\hfill \mbox{\textit{OCR MEI FP1 2006 Q9 [12]}}
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