Edexcel FP1 2012 January — Question 4 11 marks

Exam BoardEdexcel
ModuleFP1 (Further Pure Mathematics 1)
Year2012
SessionJanuary
Marks11
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicLinear transformations
TypeFind image coordinates under transformation
DifficultyModerate -0.8 This is a straightforward multi-part question testing basic matrix transformation mechanics: applying a 2×2 matrix to points, identifying a simple reflection, multiplying two 2×2 matrices, finding a determinant, and using the area scale factor property. All parts are routine FP1 procedures with no problem-solving or insight required, making it easier than average even for Further Maths.
Spec4.03b Matrix operations: addition, multiplication, scalar4.03d Linear transformations 2D: reflection, rotation, enlargement, shear4.03h Determinant 2x2: calculation4.03i Determinant: area scale factor and orientation
4. A right angled triangle \(T\) has vertices \(A ( 1,1 ) , B ( 2,1 )\) and \(C ( 2,4 )\). When \(T\) is transformed by the matrix \(\mathbf { P } = \left( \begin{array} { l l } 0 & 1 \\ 1 & 0 \end{array} \right)\), the image is \(T ^ { \prime }\).
  1. Find the coordinates of the vertices of \(T ^ { \prime }\).
  2. Describe fully the transformation represented by \(\mathbf { P }\). The matrices \(\mathbf { Q } = \left( \begin{array} { c c } 4 & - 2 \\ 3 & - 1 \end{array} \right)\) and \(\mathbf { R } = \left( \begin{array} { l l } 1 & 2 \\ 3 & 4 \end{array} \right)\) represent two transformations. When \(T\) is transformed by the matrix \(\mathbf { Q R }\), the image is \(T ^ { \prime \prime }\).
  3. Find \(\mathbf { Q R }\).
  4. Find the determinant of \(\mathbf { Q R }\).
  5. Using your answer to part (d), find the area of \(T ^ { \prime \prime }\).
Question 4:
Part (a):
AnswerMarks Guidance
Answer/WorkingMark Guidance
\(\begin{pmatrix}0 & 1\\1 & 0\end{pmatrix}\begin{pmatrix}1 & 2 & 2\\1 & 1 & 4\end{pmatrix} = \begin{pmatrix}1 & 1 & 4\\1 & 2 & 2\end{pmatrix}\)M1 Attempt to multiply the right way round with at least 4 correct elements
\(T'\) has coordinates \((1,1)\), \((1,2)\) and \((4,2)\)A1 Correct coordinates or vectors
Part (b):
AnswerMarks Guidance
Answer/WorkingMark Guidance
ReflectionB1
in the line \(y = x\)B1
Part (c):
AnswerMarks Guidance
Answer/WorkingMark Guidance
\(\mathbf{QR} = \begin{pmatrix}4 & -2\\3 & -1\end{pmatrix}\begin{pmatrix}1 & 2\\3 & 4\end{pmatrix} = \begin{pmatrix}-2 & 0\\0 & 2\end{pmatrix}\)M1 2 correct elements
A1Correct matrix
> Note: \(\mathbf{RQ} = \begin{pmatrix}1 & 2\\3 & 4\end{pmatrix}\begin{pmatrix}4 & -2\\3 & -1\end{pmatrix} = \begin{pmatrix}10 & -4\\24 & -10\end{pmatrix}\) scores M0A0 in (c) but allow all marks in (d) and (e)
Part (d):
AnswerMarks Guidance
Answer/WorkingMark Guidance
\(\det(\mathbf{QR}) = -2 \times 2 - 0 = -4\)M1 "\(-2\)"\(\times\)"\(2\)" \(-\) "\(0\)"\(\times\)"\(0\)"
A1\(-4\)
Part (e):
AnswerMarks Guidance
Answer/WorkingMark Guidance
Area of \(T = \frac{1}{2} \times 1 \times 3 = \frac{3}{2}\)B1 Correct area for \(T\)
Area of \(T'' = \frac{3}{2} \times 4 = 6\)M1 Attempt at \(\frac{3}{2} \times \pm 4\)
A1ft\(6\) or follow through their \(\det(\mathbf{QR}) \times\) their triangle area, provided area \(> 0\) 
## Question 4:

### Part (a):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\begin{pmatrix}0 & 1\\1 & 0\end{pmatrix}\begin{pmatrix}1 & 2 & 2\\1 & 1 & 4\end{pmatrix} = \begin{pmatrix}1 & 1 & 4\\1 & 2 & 2\end{pmatrix}$ | M1 | Attempt to multiply the right way round with at least 4 correct elements |
| $T'$ has coordinates $(1,1)$, $(1,2)$ and $(4,2)$ | A1 | Correct coordinates or vectors |

### Part (b):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Reflection | B1 | |
| in the line $y = x$ | B1 | |

### Part (c):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\mathbf{QR} = \begin{pmatrix}4 & -2\\3 & -1\end{pmatrix}\begin{pmatrix}1 & 2\\3 & 4\end{pmatrix} = \begin{pmatrix}-2 & 0\\0 & 2\end{pmatrix}$ | M1 | 2 correct elements |
| | A1 | Correct matrix |

> Note: $\mathbf{RQ} = \begin{pmatrix}1 & 2\\3 & 4\end{pmatrix}\begin{pmatrix}4 & -2\\3 & -1\end{pmatrix} = \begin{pmatrix}10 & -4\\24 & -10\end{pmatrix}$ scores M0A0 in (c) but allow all marks in (d) and (e)

### Part (d):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\det(\mathbf{QR}) = -2 \times 2 - 0 = -4$ | M1 | "$-2$"$\times$"$2$" $-$ "$0$"$\times$"$0$" |
| | A1 | $-4$ |

### Part (e):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Area of $T = \frac{1}{2} \times 1 \times 3 = \frac{3}{2}$ | B1 | Correct area for $T$ |
| Area of $T'' = \frac{3}{2} \times 4 = 6$ | M1 | Attempt at $\frac{3}{2} \times \pm 4$ |
| | A1ft | $6$ or follow through their $\det(\mathbf{QR}) \times$ their triangle area, provided area $> 0$ |

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4. A right angled triangle $T$ has vertices $A ( 1,1 ) , B ( 2,1 )$ and $C ( 2,4 )$. When $T$ is transformed by the matrix $\mathbf { P } = \left( \begin{array} { l l } 0 & 1 \\ 1 & 0 \end{array} \right)$, the image is $T ^ { \prime }$.
\begin{enumerate}[label=(\alph*)]
\item Find the coordinates of the vertices of $T ^ { \prime }$.
\item Describe fully the transformation represented by $\mathbf { P }$.

The matrices $\mathbf { Q } = \left( \begin{array} { c c } 4 & - 2 \\ 3 & - 1 \end{array} \right)$ and $\mathbf { R } = \left( \begin{array} { l l } 1 & 2 \\ 3 & 4 \end{array} \right)$ represent two transformations. When $T$ is transformed by the matrix $\mathbf { Q R }$, the image is $T ^ { \prime \prime }$.
\item Find $\mathbf { Q R }$.
\item Find the determinant of $\mathbf { Q R }$.
\item Using your answer to part (d), find the area of $T ^ { \prime \prime }$.
\end{enumerate}

\hfill \mbox{\textit{Edexcel FP1 2012 Q4 [11]}}
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