Edexcel FP1 2010 January — Question 9 12 marks

Exam BoardEdexcel
ModuleFP1 (Further Pure Mathematics 1)
Year2010
SessionJanuary
Marks12
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicLinear transformations
TypeDescribe rotation from matrix
DifficultyStandard +0.3 This is a straightforward FP1 matrix transformation question requiring recognition of a rotation matrix (standard form with cos and sin of 45°), solving a simple matrix equation, calculating distance, matrix multiplication, and applying the result. All steps are routine applications of standard techniques with no novel problem-solving required, making it slightly easier than average.
Spec4.03b Matrix operations: addition, multiplication, scalar4.03d Linear transformations 2D: reflection, rotation, enlargement, shear4.03i Determinant: area scale factor and orientation
9. $$\mathbf { M } = \left( \begin{array} { c c } \frac { 1 } { \sqrt { 2 } } & - \frac { 1 } { \sqrt { 2 } } \\ \frac { 1 } { \sqrt { 2 } } & \frac { 1 } { \sqrt { 2 } } \end{array} \right)$$
  1. Describe fully the geometrical transformation represented by the matrix \(\mathbf { M }\). The transformation represented by \(\mathbf { M }\) maps the point \(A\) with coordinates \(( p , q )\) onto the point \(B\) with coordinates \(( 3 \sqrt { } 2,4 \sqrt { } 2 )\).
  2. Find the value of \(p\) and the value of \(q\).
  3. Find, in its simplest surd form, the length \(O A\), where \(O\) is the origin.
  4. Find \(\mathbf { M } ^ { 2 }\). The point \(B\) is mapped onto the point \(C\) by the transformation represented by \(\mathbf { M } ^ { 2 }\).
  5. Find the coordinates of \(C\).
Question 9:
Part (a):
AnswerMarks Guidance
Answer/WorkingMarks Guidance
\(45°\) or \(\frac{\pi}{4}\) rotation (anticlockwise), about the originB1, B1 More than one transformation \(0/2\)
Part (b):
AnswerMarks Guidance
Answer/WorkingMarks Guidance
\(\begin{pmatrix} \frac{1}{\sqrt{2}} & -\frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} \end{pmatrix}\begin{pmatrix} p \\ q \end{pmatrix} = \begin{pmatrix} 3\sqrt{2} \\ 4\sqrt{2} \end{pmatrix}\)M1
\(p - q = 6\) and \(p + q = 8\)M1 A1 Or equivalent
\(p = 7\) and \(q = 1\)A1 Both correct
Part (c):
AnswerMarks Guidance
Answer/WorkingMarks Guidance
Length of \(OA\) \(= \sqrt{7^2 + 1^2} = \sqrt{50} = 5\sqrt{2}\)M1, A1 Correct use of their \(p\) and \(q\) for M1
Part (d):
AnswerMarks Guidance
Answer/WorkingMarks Guidance
\(M^2 = \begin{pmatrix} \frac{1}{\sqrt{2}} & -\frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} \end{pmatrix}\begin{pmatrix} \frac{1}{\sqrt{2}} & -\frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} \end{pmatrix} = \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}\)M1 A1 Order of matrix multiplication must be correct
Part (e):
AnswerMarks Guidance
Answer/WorkingMarks Guidance
\(\begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}\begin{pmatrix} 3\sqrt{2} \\ 4\sqrt{2} \end{pmatrix}\), coordinates are \((-4\sqrt{2},\ 3\sqrt{2})\)M1 A1 Accept column vector for final A1 
## Question 9:

### Part (a):

| Answer/Working | Marks | Guidance |
|---|---|---|
| $45°$ or $\frac{\pi}{4}$ rotation (anticlockwise), about the origin | B1, B1 | More than one transformation $0/2$ |

### Part (b):

| Answer/Working | Marks | Guidance |
|---|---|---|
| $\begin{pmatrix} \frac{1}{\sqrt{2}} & -\frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} \end{pmatrix}\begin{pmatrix} p \\ q \end{pmatrix} = \begin{pmatrix} 3\sqrt{2} \\ 4\sqrt{2} \end{pmatrix}$ | M1 | |
| $p - q = 6$ and $p + q = 8$ | M1 A1 | Or equivalent |
| $p = 7$ and $q = 1$ | A1 | Both correct |

### Part (c):

| Answer/Working | Marks | Guidance |
|---|---|---|
| Length of $OA$ $= \sqrt{7^2 + 1^2} = \sqrt{50} = 5\sqrt{2}$ | M1, A1 | Correct use of their $p$ and $q$ for M1 |

### Part (d):

| Answer/Working | Marks | Guidance |
|---|---|---|
| $M^2 = \begin{pmatrix} \frac{1}{\sqrt{2}} & -\frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} \end{pmatrix}\begin{pmatrix} \frac{1}{\sqrt{2}} & -\frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} \end{pmatrix} = \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}$ | M1 A1 | Order of matrix multiplication must be correct |

### Part (e):

| Answer/Working | Marks | Guidance |
|---|---|---|
| $\begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}\begin{pmatrix} 3\sqrt{2} \\ 4\sqrt{2} \end{pmatrix}$, coordinates are $(-4\sqrt{2},\ 3\sqrt{2})$ | M1 A1 | Accept column vector for final A1 |
9.

$$\mathbf { M } = \left( \begin{array} { c c } 
\frac { 1 } { \sqrt { 2 } } & - \frac { 1 } { \sqrt { 2 } } \\
\frac { 1 } { \sqrt { 2 } } & \frac { 1 } { \sqrt { 2 } }
\end{array} \right)$$
\begin{enumerate}[label=(\alph*)]
\item Describe fully the geometrical transformation represented by the matrix $\mathbf { M }$.

The transformation represented by $\mathbf { M }$ maps the point $A$ with coordinates $( p , q )$ onto the point $B$ with coordinates $( 3 \sqrt { } 2,4 \sqrt { } 2 )$.
\item Find the value of $p$ and the value of $q$.
\item Find, in its simplest surd form, the length $O A$, where $O$ is the origin.
\item Find $\mathbf { M } ^ { 2 }$.

The point $B$ is mapped onto the point $C$ by the transformation represented by $\mathbf { M } ^ { 2 }$.
\item Find the coordinates of $C$.
\end{enumerate}

\hfill \mbox{\textit{Edexcel FP1 2010 Q9 [12]}}
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