Question 6 - A-Level Maths

AQA FP1 2008 January — Question 6 10 marks

Exam Board	AQA
Module	FP1 (Further Pure Mathematics 1)
Year	2008
Session	January
Marks	10
Paper	Download PDF ↗
Mark scheme	Download PDF ↗
Topic	Linear transformations
Type	Extract enlargement and rotation parameters
Difficulty	Standard +0.3 This is a structured Further Maths question on matrix transformations with clear scaffolding. Part (a) involves routine matrix multiplication and factorization using standard trigonometric values. Part (b) requires recognizing the geometric interpretation from the matrix form (scale factor from q, mirror line from the reflection matrix pattern). Part (c) uses the result from (a)(i) to deduce M^4 = p^2I. While this is Further Maths content, the question guides students through each step methodically, making it slightly easier than average for A-level overall but typical for FP1.
Spec	4.03a Matrix language: terminology and notation 4.03b Matrix operations: addition, multiplication, scalar 4.03d Linear transformations 2D: reflection, rotation, enlargement, shear 4.03e Successive transformations: matrix products 4.03h Determinant 2x2: calculation 4.03n Inverse 2x2 matrix

6 The matrix $\mathbf { M }$ is defined by $$\mathbf { M } = \left[ \begin{array} { c c } \sqrt { 3 } & 3 \\ 3 & - \sqrt { 3 } \end{array} \right]$$

1. Show that $$\mathbf { M } ^ { 2 } = p \mathbf { I }$$ where $p$ is an integer and $\mathbf { I }$ is the $2 \times 2$ identity matrix.
2. Show that the matrix $\mathbf { M }$ can be written in the form $$q \left[ \begin{array} { c c } \cos 60 ^ { \circ } & \sin 60 ^ { \circ } \\ \sin 60 ^ { \circ } & - \cos 60 ^ { \circ } \end{array} \right]$$ where $q$ is a real number. Give the value of $q$ in surd form.
The matrix $\mathbf { M }$ represents a combination of an enlargement and a reflection. Find:
1. the scale factor of the enlargement;
2. the equation of the mirror line of the reflection.
Describe fully the geometrical transformation represented by $\mathbf { M } ^ { 4 }$.

Show mark scheme Show mark scheme source

Question 6(a)(i):

Answer	Marks	Guidance
Working/Answer	Marks	Guidance
$\mathbf{M}^2 = \begin{bmatrix} 12 & 0 \\ 0 & 12 \end{bmatrix}$	M1A1	M1 if zeroes appear in right places
$= 12\mathbf{I}$	A1F	ft provided of right form
Subtotal	3

Question 6(a)(ii):

Answer	Marks	Guidance
Working/Answer	Marks	Guidance
$q\cos 60° = \frac{1}{2}q = \sqrt{3} \Rightarrow q = 2\sqrt{3}$	M1A1	OE SC $q = 2\sqrt{3}$ NMS 1/3
Other entries verified	E1	surd for $\sin 60°$ needed
Subtotal	3

Question 6(b)(i):

Answer	Marks	Guidance
Working/Answer	Marks	Guidance
$SF = q = 2\sqrt{3}$	B1F	ft wrong value for $q$
Subtotal	1

Question 6(b)(ii):

Answer	Marks	Guidance
Working/Answer	Marks	Guidance
Equation is $y = x\tan 30°$	B1
Subtotal	1

Question 6(c):

Answer	Marks	Guidance
Working/Answer	Marks	Guidance
$\mathbf{M}^4 = 144\mathbf{I}$	B1F	PI; ft wrong value in (a)(i)
$\mathbf{M}^4$ gives enlargement SF 144	B1F	ft if $\mathbf{M}^4 = k\mathbf{I}$
Subtotal	2
Total	10

## Question 6(a)(i):

| Working/Answer | Marks | Guidance |
|---|---|---|
| $\mathbf{M}^2 = \begin{bmatrix} 12 & 0 \\ 0 & 12 \end{bmatrix}$ | M1A1 | M1 if zeroes appear in right places |
| $= 12\mathbf{I}$ | A1F | ft provided of right form |
| **Subtotal** | **3** | |

## Question 6(a)(ii):

| Working/Answer | Marks | Guidance |
|---|---|---|
| $q\cos 60° = \frac{1}{2}q = \sqrt{3} \Rightarrow q = 2\sqrt{3}$ | M1A1 | OE SC $q = 2\sqrt{3}$ NMS 1/3 |
| Other entries verified | E1 | surd for $\sin 60°$ needed |
| **Subtotal** | **3** | |

## Question 6(b)(i):

| Working/Answer | Marks | Guidance |
|---|---|---|
| $SF = q = 2\sqrt{3}$ | B1F | ft wrong value for $q$ |
| **Subtotal** | **1** | |

## Question 6(b)(ii):

| Working/Answer | Marks | Guidance |
|---|---|---|
| Equation is $y = x\tan 30°$ | B1 | |
| **Subtotal** | **1** | |

## Question 6(c):

| Working/Answer | Marks | Guidance |
|---|---|---|
| $\mathbf{M}^4 = 144\mathbf{I}$ | B1F | PI; ft wrong value in (a)(i) |
| $\mathbf{M}^4$ gives enlargement SF 144 | B1F | ft if $\mathbf{M}^4 = k\mathbf{I}$ |
| **Subtotal** | **2** | |
| **Total** | **10** | |

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Show LaTeX source

6 The matrix $\mathbf { M }$ is defined by

$$\mathbf { M } = \left[ \begin{array} { c c } 
\sqrt { 3 } & 3 \\
3 & - \sqrt { 3 }
\end{array} \right]$$
\begin{enumerate}[label=(\alph*)]
\item \begin{enumerate}[label=(\roman*)]
\item Show that

$$\mathbf { M } ^ { 2 } = p \mathbf { I }$$

where $p$ is an integer and $\mathbf { I }$ is the $2 \times 2$ identity matrix.
\item Show that the matrix $\mathbf { M }$ can be written in the form

$$q \left[ \begin{array} { c c } 
\cos 60 ^ { \circ } & \sin 60 ^ { \circ } \\
\sin 60 ^ { \circ } & - \cos 60 ^ { \circ }
\end{array} \right]$$

where $q$ is a real number. Give the value of $q$ in surd form.
\end{enumerate}\item The matrix $\mathbf { M }$ represents a combination of an enlargement and a reflection.

Find:
\begin{enumerate}[label=(\roman*)]
\item the scale factor of the enlargement;
\item the equation of the mirror line of the reflection.
\end{enumerate}\item Describe fully the geometrical transformation represented by $\mathbf { M } ^ { 4 }$.
\end{enumerate}

\hfill \mbox{\textit{AQA FP1 2008 Q6 [10]}}

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