9 (i) Write down the matrix $\mathbf { C }$ which represents a stretch, scale factor 2 , in the $x$-direction.\\
(ii) The matrix $\mathbf { D }$ is given by $\mathbf { D } = \left( \begin{array} { l l } 1 & 3 \\ 0 & 1 \end{array} \right)$. Describe fully the geometrical transformation represented by $\mathbf { D }$.\\
(iii) The matrix $\mathbf { M }$ represents the combined effect of the transformation represented by $\mathbf { C }$ followed by the transformation represented by $\mathbf { D }$. Show that

$$\mathbf { M } = \left( \begin{array} { l l } 
2 & 3 \\
0 & 1
\end{array} \right)$$

(iv) Prove by induction that $\mathbf { M } ^ { n } = \left( \begin{array} { c c } 2 ^ { n } & 3 \left( 2 ^ { n } - 1 \right) \\ 0 & 1 \end{array} \right)$, for all positive integers $n$.

RECOGNISING ACHIEVEMENT

\section*{OXFORD CAMBRIDGE AND RSA EXAMINATIONS}
\section*{Advanced Subsidiary General Certificate of Education Advanced General Certificate of Education }
\section*{MATHEMATICS}

Further Pure Mathematics 1\\
Wednesday 18 JANUARY 2006 Afternoon 1 hour 30 minutes\\
Additional materials:\\
8 page answer booklet\\
Graph paper\\
List of Formulae (MF1)

TIME 1 hour 30 minutes

\begin{itemize}
  \item Write your name, centre number and candidate number in the spaces provided on the answer booklet.
  \item Answer all the questions.
  \item Give non-exact numerical answers correct to 3 significant figures unless a different degree of accuracy is specified in the question or is clearly appropriate.
  \item You are permitted to use a graphical calculator in this paper.
\end{itemize}

\begin{itemize}
  \item The number of marks is given in brackets [ ] at the end of each question or part question.
  \item The total number of marks for this paper is 72.
  \item Questions carrying smaller numbers of marks are printed earlier in the paper, and questions carrying larger numbers of marks later in the paper.
  \item You are reminded of the need for clear presentation in your answers.
\end{itemize}

1 (i) Express $( 1 + 8 i ) ( 2 - i )$ in the form $x + i y$, showing clearly how you obtain your answer.\\
(ii) Hence express $\frac { 1 + 8 i } { 2 + i }$ in the form $x + i y$.

2 Prove by induction that, for $n \geqslant 1 , \sum _ { r = 1 } ^ { n } r ^ { 2 } = \frac { 1 } { 6 } n ( n + 1 ) ( 2 n + 1 )$.

3 The matrix $\mathbf { M }$ is given by $\mathbf { M } = \left( \begin{array} { l l l } 2 & 1 & 3 \\ 1 & 2 & 1 \\ 1 & 1 & 3 \end{array} \right)$.\\
(i) Find the value of the determinant of $\mathbf { M }$.\\
(ii) State, giving a brief reason, whether $\mathbf { M }$ is singular or non-singular.

4 Use the substitution $x = u + 2$ to find the exact value of the real root of the equation

$$x ^ { 3 } - 6 x ^ { 2 } + 12 x - 13 = 0$$

5 Use the standard results for $\sum _ { r = 1 } ^ { n } r , \sum _ { r = 1 } ^ { n } r ^ { 2 }$ and $\sum _ { r = 1 } ^ { n } r ^ { 3 }$ to show that, for all positive integers $n$,

$$\sum _ { r = 1 } ^ { n } \left( 8 r ^ { 3 } - 6 r ^ { 2 } + 2 r \right) = 2 n ^ { 3 } ( n + 1 )$$

6 The matrix $\mathbf { C }$ is given by $\mathbf { C } = \left( \begin{array} { l l } 1 & 2 \\ 3 & 8 \end{array} \right)$.\\
(i) Find $\mathbf { C } ^ { - 1 }$.\\
(ii) Given that $\mathbf { C } = \mathbf { A B }$, where $\mathbf { A } = \left( \begin{array} { l l } 2 & 1 \\ 1 & 3 \end{array} \right)$, find $\mathbf { B } ^ { - 1 }$.

7 (a) The complex number $3 + 2 \mathrm { i }$ is denoted by $w$ and the complex conjugate of $w$ is denoted by $w ^ { * }$. Find\\
(i) the modulus of $w$,\\
(ii) the argument of $w ^ { * }$, giving your answer in radians, correct to 2 decimal places.\\
(b) Find the complex number $u$ given that $u + 2 u ^ { * } = 3 + 2 \mathrm { i }$.\\
(c) Sketch, on an Argand diagram, the locus given by $| z + 1 | = | z |$.

8 The matrix $\mathbf { T }$ is given by $\mathbf { T } = \left( \begin{array} { r r } 2 & 0 \\ 0 & - 2 \end{array} \right)$.\\
(i) Draw a diagram showing the unit square and its image under the transformation represented by $\mathbf { T }$. [3]\\
(ii) The transformation represented by matrix $\mathbf { T }$ is equivalent to a transformation $A$, followed by a transformation B. Give geometrical descriptions of possible transformations A and B, and state the matrices that represent them.

9 (i) Show that $\frac { 1 } { r } - \frac { 1 } { r + 2 } = \frac { 2 } { r ( r + 2 ) }$.\\
(ii) Hence find an expression, in terms of $n$, for

$$\frac { 2 } { 1 \times 3 } + \frac { 2 } { 2 \times 4 } + \ldots + \frac { 2 } { n ( n + 2 ) }$$

(iii) Hence find the value of\\
(a) $\sum _ { r = 1 } ^ { \infty } \frac { 2 } { r ( r + 2 ) }$,\\
(b) $\sum _ { r = n + 1 } ^ { \infty } \frac { 2 } { r ( r + 2 ) }$.

\hfill \mbox{\textit{OCR FP1  Q9 [3]}}