10 The roots of the equation

$$x ^ { 3 } - 9 x ^ { 2 } + 27 x - 29 = 0$$

are denoted by $\alpha , \beta$ and $\gamma$, where $\alpha$ is real and $\beta$ and $\gamma$ are complex.\\
(i) Write down the value of $\alpha + \beta + \gamma$.\\
(ii) It is given that $\beta = p + \mathrm { i } q$, where $q > 0$. Find the value of $p$, in terms of $\alpha$.\\
(iii) Write down the value of $\alpha \beta \gamma$.\\
(iv) Find the value of $q$, in terms of $\alpha$ only.

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

Further Pure Mathematics 1

\begin{center}
\begin{tabular}{|l|l|l|}
\hline
\includegraphics[max width=\textwidth, alt={}]{a8f7bd80-4733-40b9-be96-9fb0878dbb26-08_43_563_743_388}
 & Morning & 1 hour 30 minutes \\
\hline
Additional materials: 8 page answer booklet Graph paper List of Formulae (MF1) &  &  \\
\hline
\end{tabular}
\end{center}

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.\\
$\mathbf { 1 }$ The matrices $\mathbf { A }$ and $\mathbf { B }$ are given by $\mathbf { A } = \left( \begin{array} { l l } 4 & 1 \\ 0 & 2 \end{array} \right)$ and $\mathbf { B } = \left( \begin{array} { r r } 1 & 1 \\ 0 & - 1 \end{array} \right)$.\\
(i) Find $\mathbf { A } + 3 \mathbf { B }$.\\
(ii) Show that $\mathbf { A } - \mathbf { B } = k \mathbf { I }$, where $\mathbf { I }$ is the identity matrix and $k$ is a constant whose value should be stated.
\end{itemize}

2 The transformation S is a shear parallel to the $x$-axis in which the image of the point ( 1,1 ) is the point $( 0,1 )$.\\
(i) Draw a diagram showing the image of the unit square under S .\\
(ii) Write down the matrix that represents S .

3 One root of the quadratic equation $x ^ { 2 } + p x + q = 0$, where $p$ and $q$ are real, is the complex number 2-3i.\\
(i) Write down the other root.\\
(ii) Find the values of $p$ and $q$.

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

$$\sum _ { r = 1 } ^ { n } \left( r ^ { 3 } + r ^ { 2 } \right) = \frac { 1 } { 12 } n ( n + 1 ) ( n + 2 ) ( 3 n + 1 )$$

5 The complex numbers $3 - 2 \mathrm { i }$ and $2 + \mathrm { i }$ are denoted by $z$ and $w$ respectively. Find, giving your answers in the form $x + \mathrm { i } y$ and showing clearly how you obtain these answers,\\
(i) $2 z - 3 w$,\\
(ii) $( \mathrm { i } z ) ^ { 2 }$,\\
(iii) $\frac { z } { w }$.

6 In an Argand diagram the loci $C _ { 1 }$ and $C _ { 2 }$ are given by

$$| z | = 2 \quad \text { and } \quad \arg z = \frac { 1 } { 3 } \pi$$

respectively.\\
(i) Sketch, on a single Argand diagram, the loci $C _ { 1 }$ and $C _ { 2 }$.\\
(ii) Hence find, in the form $x + \mathrm { i } y$, the complex number representing the point of intersection of $C _ { 1 }$ and $C _ { 2 }$.

7 The matrix $\mathbf { A }$ is given by $\mathbf { A } = \left( \begin{array} { l l } 2 & 0 \\ 0 & 1 \end{array} \right)$.\\
(i) Find $\mathbf { A } ^ { 2 }$ and $\mathbf { A } ^ { 3 }$.\\
(ii) Hence suggest a suitable form for the matrix $\mathbf { A } ^ { n }$.\\
(iii) Use induction to prove that your answer to part (ii) is correct.

8 The matrix $\mathbf { M }$ is given by $\mathbf { M } = \left( \begin{array} { l l l } a & 4 & 2 \\ 1 & a & 0 \\ 1 & 2 & 1 \end{array} \right)$.\\
(i) Find, in terms of $a$, the determinant of $\mathbf { M }$.\\
(ii) Hence find the values of $a$ for which $\mathbf { M }$ is singular.\\
(iii) State, giving a brief reason in each case, whether the simultaneous equations

$$\begin{aligned}
a x + 4 y + 2 z & = 3 a \\
x + a y & = 1 \\
x + 2 y + z & = 3
\end{aligned}$$

have any solutions when\\
(a) $a = 3$,\\
(b) $a = 2$.

9 (i) Use the method of differences to show that

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

(ii) Show that $( r + 1 ) ^ { 3 } - r ^ { 3 } \equiv 3 r ^ { 2 } + 3 r + 1$.\\
(iii) Use the results in parts (i) and (ii) and the standard result for $\sum _ { r = 1 } ^ { n } r$ to show that

$$3 \sum _ { r = 1 } ^ { n } r ^ { 2 } = \frac { 1 } { 2 } n ( n + 1 ) ( 2 n + 1 )$$

10 The cubic equation $x ^ { 3 } - 2 x ^ { 2 } + 3 x + 4 = 0$ has roots $\alpha , \beta$ and $\gamma$.\\
(i) Write down the values of $\alpha + \beta + \gamma , \alpha \beta + \beta \gamma + \gamma \alpha$ and $\alpha \beta \gamma$.

The cubic equation $x ^ { 3 } + p x ^ { 2 } + 10 x + q = 0$, where $p$ and $q$ are constants, has roots $\alpha + 1 , \beta + 1$ and $\gamma + 1$.\\
(ii) Find the value of $p$.\\
(iii) Find the value of $q$.

\section*{ADVANCED SUBSIDIARY GCE UNIT MATHEMATICS}
Further Pure Mathematics 1

\section*{THURSDAY 18 JANUARY 2007 }
List of Formulae (MF1)

\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.
\end{itemize}

\begin{itemize}
  \item Read each question carefully and make sure you know what you have to do before starting your answer.
  \item You are reminded of the need for clear presentation in your answers.\\
$\mathbf { 1 }$ The matrices $\mathbf { A }$ and $\mathbf { B }$ are given by $\mathbf { A } = \left( \begin{array} { l l } 2 & 1 \\ 3 & 2 \end{array} \right)$ and $\mathbf { B } = \left( \begin{array} { c c } a & - 1 \\ - 3 & - 2 \end{array} \right)$.\\
(i) Given that $2 \mathbf { A } + \mathbf { B } = \left( \begin{array} { l l } 1 & 1 \\ 3 & 2 \end{array} \right)$, write down the value of $a$.\\
(ii) Given instead that $\mathbf { A B } = \left( \begin{array} { l l } 7 & - 4 \\ 9 & - 7 \end{array} \right)$, find the value of $a$.
\end{itemize}

2 Use an algebraic method to find the square roots of the complex number $15 + 8 \mathrm { i }$.

3 Use the standard results for $\sum _ { r = 1 } ^ { n } r$ and $\sum _ { r = 1 } ^ { n } r ^ { 3 }$ to find

$$\sum _ { r = 1 } ^ { n } r ( r - 1 ) ( r + 1 ) ,$$

expressing your answer in a fully factorised form.

4 (i) Sketch, on an Argand diagram, the locus given by $| z - 1 + \mathrm { i } | = \sqrt { 2 }$.\\
(ii) Shade on your diagram the region given by $1 \leqslant | z - 1 + \mathrm { i } | \leqslant \sqrt { 2 }$.

5 (i) Verify that $z ^ { 3 } - 8 = ( z - 2 ) \left( z ^ { 2 } + 2 z + 4 \right)$.\\
(ii) Solve the quadratic equation $z ^ { 2 } + 2 z + 4 = 0$, giving your answers exactly in the form $x + \mathrm { i } y$. Show clearly how you obtain your answers.\\
(iii) Show on an Argand diagram the roots of the cubic equation $z ^ { 3 } - 8 = 0$.

6 The sequence $u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots$ is defined by $u _ { n } = n ^ { 2 } + 3 n$, for all positive integers $n$.\\
(i) Show that $u _ { n + 1 } - u _ { n } = 2 n + 4$.\\
(ii) Hence prove by induction that each term of the sequence is divisible by 2 .

7 The quadratic equation $x ^ { 2 } + 5 x + 10 = 0$ has roots $\alpha$ and $\beta$.\\
(i) Write down the values of $\alpha + \beta$ and $\alpha \beta$.\\
(ii) Show that $\alpha ^ { 2 } + \beta ^ { 2 } = 5$.\\
(iii) Hence find a quadratic equation which has roots $\frac { \alpha } { \beta }$ and $\frac { \beta } { \alpha }$.

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

$$2 ^ { 2 } \times 1 ! + 3 ^ { 2 } \times 2 ! + 4 ^ { 2 } \times 3 ! + \ldots + ( n + 1 ) ^ { 2 } \times n ! .$$

(iii) State, giving a brief reason, whether the series

$$2 ^ { 2 } \times 1 ! + 3 ^ { 2 } \times 2 ! + 4 ^ { 2 } \times 3 ! + \ldots$$

converges.

9 The matrix $\mathbf { C }$ is given by $\mathbf { C } = \left( \begin{array} { r r } 0 & 3 \\ - 1 & 0 \end{array} \right)$.\\
(i) Draw a diagram showing the unit square and its image under the transformation represented by $\mathbf { C }$.

The transformation represented by $\mathbf { C }$ is equivalent to a rotation, R , followed by another transformation, S.\\
(ii) Describe fully the rotation R and write down the matrix that represents R .\\
(iii) Describe fully the transformation S and write down the matrix that represents S .

10 The matrix $\mathbf { D }$ is given by $\mathbf { D } = \left( \begin{array} { r r r } a & 2 & 0 \\ 3 & 1 & 2 \\ 0 & - 1 & 1 \end{array} \right)$, where $a \neq 2$.\\
(i) Find $\mathbf { D } ^ { - 1 }$.\\
(ii) Hence, or otherwise, solve the equations

$$\begin{aligned}
a x + 2 y & = 3 \\
3 x + y + 2 z & = 4 \\
- y + z & = 1
\end{aligned}$$

\section*{ADVANCED SUBSIDIARY GCE UNIT MATHEMATICS}
Further Pure Mathematics 1

\section*{MONDAY 11 JUNE 2007}
Additional Materials:

Answer Booklet (8 pages) List of Formulae (MF1)

\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.
\end{itemize}

\begin{itemize}
  \item Read each question carefully and make sure you know what you have to do before starting your answer.
  \item You are reminded of the need for clear presentation in your answers.
\end{itemize}

1 The complex number $a + \mathrm { i } b$ is denoted by $z$. Given that $| z | = 4$ and $\arg z = \frac { 1 } { 3 } \pi$, find $a$ and $b$.

2 Prove by induction that, for $n \geqslant 1 , \sum _ { r = 1 } ^ { n } r ^ { 3 } = \frac { 1 } { 4 } n ^ { 2 } ( n + 1 ) ^ { 2 }$.

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

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

4 The matrix $\mathbf { A }$ is given by $\mathbf { A } = \left( \begin{array} { l l } 1 & 1 \\ 3 & 5 \end{array} \right)$.\\
(i) Find $\mathbf { A } ^ { - 1 }$.

The matrix $\mathbf { B } ^ { - 1 }$ is given by $\mathbf { B } ^ { - 1 } = \left( \begin{array} { r r } 1 & 1 \\ 4 & - 1 \end{array} \right)$.\\
(ii) Find $( \mathbf { A B } ) ^ { - 1 }$.

5 (i) Show that

$$\frac { 1 } { r } - \frac { 1 } { r + 1 } = \frac { 1 } { r ( r + 1 ) }$$

(ii) Hence find an expression, in terms of $n$, for

$$\frac { 1 } { 2 } + \frac { 1 } { 6 } + \frac { 1 } { 12 } + \ldots + \frac { 1 } { n ( n + 1 ) }$$

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

6 The cubic equation $3 x ^ { 3 } - 9 x ^ { 2 } + 6 x + 2 = 0$ has roots $\alpha , \beta$ and $\gamma$.\\
(i) (a) Write down the values of $\alpha + \beta + \gamma$ and $\alpha \beta + \beta \gamma + \gamma \alpha$.\\
(b) Find the value of $\alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 }$.\\
(ii) (a) Use the substitution $x = \frac { 1 } { u }$ to find a cubic equation in $u$ with integer coefficients.\\
(b) Use your answer to part (ii) (a) to find the value of $\frac { 1 } { \alpha } + \frac { 1 } { \beta } + \frac { 1 } { \gamma }$.

7 The matrix $\mathbf { M }$ is given by $\mathbf { M } = \left( \begin{array} { l l l } a & 4 & 0 \\ 0 & a & 4 \\ 2 & 3 & 1 \end{array} \right)$.\\
(i) Find, in terms of $a$, the determinant of $\mathbf { M }$.\\
(ii) In the case when $a = 2$, state whether $\mathbf { M }$ is singular or non-singular, justifying your answer.\\
(iii) In the case when $a = 4$, determine whether the simultaneous equations

$$\begin{aligned}
a x + 4 y \quad = & 6 \\
a y + 4 z & = 8 \\
2 x + 3 y + z & = 1
\end{aligned}$$

have any solutions.

8 The loci $C _ { 1 }$ and $C _ { 2 }$ are given by $| z - 3 | = 3$ and arg $( z - 1 ) = \frac { 1 } { 4 } \pi$ respectively.\\
(i) Sketch, on a single Argand diagram, the loci $C _ { 1 }$ and $C _ { 2 }$.\\
(ii) Indicate, by shading, the region of the Argand diagram for which

$$| z - 3 | \leqslant 3 \text { and } 0 \leqslant \arg ( z - 1 ) \leqslant \frac { 1 } { 4 } \pi$$

9 (i) Write down the matrix, $\mathbf { A }$, that represents an enlargement, centre ( 0,0 ), with scale factor $\sqrt { 2 }$.\\
(ii) The matrix $\mathbf { B }$ is given by $\mathbf { B } = \left( \begin{array} { r r } \frac { 1 } { 2 } \sqrt { 2 } & \frac { 1 } { 2 } \sqrt { 2 } \\ - \frac { 1 } { 2 } \sqrt { 2 } & \frac { 1 } { 2 } \sqrt { 2 } \end{array} \right)$. Describe fully the geometrical transformation represented by $\mathbf { B }$.\\
(iii) Given that $\mathbf { C } = \mathbf { A B }$, show that $\mathbf { C } = \left( \begin{array} { r r } 1 & 1 \\ - 1 & 1 \end{array} \right)$.\\
(iv) Draw a diagram showing the unit square and its image under the transformation represented by $\mathbf { C }$.\\
(v) Write down the determinant of $\mathbf { C }$ and explain briefly how this value relates to the transformation represented by $\mathbf { C }$.

10 (i) Use an algebraic method to find the square roots of the complex number $16 + 30 \mathrm { i }$.\\
(ii) Use your answers to part (i) to solve the equation $z ^ { 2 } - 2 z - ( 15 + 30 \mathrm { i } ) = 0$, giving your answers in the form $x + \mathrm { i } y$.

RECOGNISING ACHIEVEMENT

\section*{ADVANCED SUBSIDIARY GCE MATHEMATICS }
Further Pure Mathematics 1

\section*{FRIDAY 11 JANUARY 2008}
\begin{verbatim}
Additional materials: Answer Booklet (8 pages)
    List of Formulae (MF1)
\end{verbatim}

\begin{itemize}
  \item Write your name, centre number and candidate number in the spaces provided on the answer booklet.
  \item Read each question carefully and make sure you know what you have to do before starting your answer.
  \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 You are reminded of the need for clear presentation in your answers.
\end{itemize}

1 The transformation S is a shear with the $y$-axis invariant (i.e. a shear parallel to the $y$-axis). It is given that the image of the point $( 1,1 )$ is the point $( 1,0 )$.\\
(i) Draw a diagram showing the image of the unit square under the transformation S .\\
(ii) Write down the matrix that represents S .

2 Given that $\sum _ { r = 1 } ^ { n } \left( a r ^ { 2 } + b \right) \equiv n \left( 2 n ^ { 2 } + 3 n - 2 \right)$, find the values of the constants $a$ and $b$.

3 The cubic equation $2 x ^ { 3 } - 3 x ^ { 2 } + 24 x + 7 = 0$ has roots $\alpha , \beta$ and $\gamma$.\\
(i) Use the substitution $x = \frac { 1 } { u }$ to find a cubic equation in $u$ with integer coefficients.\\
(ii) Hence, or otherwise, find the value of $\frac { 1 } { \alpha \beta } + \frac { 1 } { \beta \gamma } + \frac { 1 } { \gamma \alpha }$.

4 The complex number $3 - 4 \mathrm { i }$ is denoted by $z$. Giving your answers in the form $x + \mathrm { i } y$, and showing clearly how you obtain them, find\\
(i) $2 z + 5 z ^ { * }$,\\
(ii) $( z - \mathrm { i } ) ^ { 2 }$,\\
(iii) $\frac { 3 } { z }$.

5 The matrices $\mathbf { A } , \mathbf { B }$ and $\mathbf { C }$ are given by $\mathbf { A } = \left( \begin{array} { l } 3 \\ 1 \\ 2 \end{array} \right) , \mathbf { B } = \left( \begin{array} { l } 4 \\ 0 \\ 3 \end{array} \right)$ and $\mathbf { C } = \left( \begin{array} { l l l } 2 & 4 & - 1 \end{array} \right)$. Find\\
(i) $\mathbf { A } - 4 \mathbf { B }$,\\
(ii) BC ,\\
(iii) CA .

6 The loci $C _ { 1 }$ and $C _ { 2 }$ are given by

$$| z | = | z - 4 \mathbf { i } | \quad \text { and } \quad \arg z = \frac { 1 } { 6 } \pi$$

respectively.\\
(i) Sketch, on a single Argand diagram, the loci $C _ { 1 }$ and $C _ { 2 }$.\\
(ii) Hence find, in the form $x +$ i $y$, the complex number represented by the point of intersection of $C _ { 1 }$ and $C _ { 2 }$.

7 The matrix $\mathbf { A }$ is given by $\mathbf { A } = \left( \begin{array} { c c } a & 3 \\ - 2 & 1 \end{array} \right)$.\\
(i) Given that $\mathbf { A }$ is singular, find $a$.\\
(ii) Given instead that $\mathbf { A }$ is non-singular, find $\mathbf { A } ^ { - 1 }$ and hence solve the simultaneous equations

$$\begin{aligned}
a x + 3 y & = 1 \\
- 2 x + y & = - 1
\end{aligned}$$

8 The sequence $u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots$ is defined by $u _ { 1 } = 1$ and $u _ { n + 1 } = u _ { n } + 2 n + 1$.\\
(i) Show that $u _ { 4 } = 16$.\\
(ii) Hence suggest an expression for $u _ { n }$.\\
(iii) Use induction to prove that your answer to part (ii) is correct.

9 (i) Show that $\alpha ^ { 3 } + \beta ^ { 3 } = ( \alpha + \beta ) ^ { 3 } - 3 \alpha \beta ( \alpha + \beta )$.\\
(ii) The quadratic equation $x ^ { 2 } - 5 x + 7 = 0$ has roots $\alpha$ and $\beta$. Find a quadratic equation with roots $\alpha ^ { 3 }$ and $\beta ^ { 3 }$.\\
(i) Show that $\frac { 2 } { r } - \frac { 1 } { r + 1 } - \frac { 1 } { r + 2 } = \frac { 3 r + 4 } { r ( r + 1 ) ( r + 2 ) }$.\\
(ii) Hence find an expression, in terms of $n$, for

$$\sum _ { r = 1 } ^ { n } \frac { 3 r + 4 } { r ( r + 1 ) ( r + 2 ) }$$

(iii) Hence write down the value of $\sum _ { r = 1 } ^ { \infty } \frac { 3 r + 4 } { r ( r + 1 ) ( r + 2 ) }$.\\
(iv) Given that $\sum _ { r = N + 1 } ^ { \infty } \frac { 3 r + 4 } { r ( r + 1 ) ( r + 2 ) } = \frac { 7 } { 10 }$, find the value of $N$.

RECOGNISING ACHIEVEMENT

ADVANCED SUBSIDIARY GCE

MATHEMATICS\\
Further Pure Mathematics 1\\
MONDAY 2 JUNE 2008\\
Morning\\
Time: 1 hour 30 minutes\\
Additional materials: Answer Booklet (8 pages)\\
List of Formulae (MF1)

\begin{itemize}
  \item Write your name in capital letters, your Centre Number and Candidate Number in the spaces provided on the Answer Booklet.
  \item Read each question carefully and make sure you know what you have to do before starting your answer.
  \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 You are reminded of the need for clear presentation in your answers.
\end{itemize}

1 The matrix $\mathbf { A }$ is given by $\mathbf { A } = \left( \begin{array} { l l } 4 & 1 \\ 5 & 2 \end{array} \right)$ and $\mathbf { I }$ is the $2 \times 2$ identity matrix. Find\\
(i) $\mathbf { A } - 3 \mathbf { I }$,\\
(ii) $\mathrm { A } ^ { - 1 }$.

2 The complex number $3 + 4 \mathrm { i }$ is denoted by $a$.\\
(i) Find $| a |$ and $\arg a$.\\
(ii) Sketch on a single Argand diagram the loci given by\\
(a) $| z - a | = | a |$,\\
(b) $\arg ( z - 3 ) = \arg a$.

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

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

4 The matrix $\mathbf { A }$ is given by $\mathbf { A } = \left( \begin{array} { l l } 3 & 1 \\ 0 & 1 \end{array} \right)$. Prove by induction that, for $n \geqslant 1$,

$$\mathbf { A } ^ { n } = \left( \begin{array} { c c } 
3 ^ { n } & \frac { 1 } { 2 } \left( 3 ^ { n } - 1 \right) \\
0 & 1
\end{array} \right)$$

5 Find $\sum _ { r = 1 } ^ { n } r ^ { 2 } ( r - 1 )$, expressing your answer in a fully factorised form.

6 The cubic equation $x ^ { 3 } + a x ^ { 2 } + b x + c = 0$, where $a , b$ and $c$ are real, has roots ( $3 + \mathrm { i }$ ) and 2 .\\
(i) Write down the other root of the equation.\\
(ii) Find the values of $a , b$ and $c$.

7 Describe fully the geometrical transformation represented by each of the following matrices:\\
(i) $\left( \begin{array} { l l } 6 & 0 \\ 0 & 6 \end{array} \right)$,\\
(ii) $\left( \begin{array} { l l } 0 & 1 \\ 1 & 0 \end{array} \right)$,\\
(iii) $\left( \begin{array} { l l } 1 & 0 \\ 0 & 6 \end{array} \right)$,\\
(iv) $\left( \begin{array} { r r } 0.8 & 0.6 \\ - 0.6 & 0.8 \end{array} \right)$.

8 The quadratic equation $x ^ { 2 } + k x + 2 k = 0$, where $k$ is a non-zero constant, has roots $\alpha$ and $\beta$. Find a quadratic equation with roots $\frac { \alpha } { \beta }$ and $\frac { \beta } { \alpha }$.

9 (i) Use an algebraic method to find the square roots of the complex number $5 + 12 \mathrm { i }$.\\
(ii) Find $( 3 - 2 \mathrm { i } ) ^ { 2 }$.\\
(iii) Hence solve the quartic equation $x ^ { 4 } - 10 x ^ { 2 } + 169 = 0$.

10 The matrix $\mathbf { A }$ is given by $\mathbf { A } = \left( \begin{array} { r r r } a & 8 & 10 \\ 2 & 1 & 2 \\ 4 & 3 & 6 \end{array} \right)$. The matrix $\mathbf { B }$ is such that $\mathbf { A B } = \left( \begin{array} { l l l } a & 6 & 1 \\ 1 & 1 & 0 \\ 1 & 3 & 0 \end{array} \right)$.\\
(i) Show that $\mathbf { A B }$ is non-singular.\\
(ii) Find $( \mathbf { A B } ) ^ { - 1 }$.\\
(iii) Find $\mathbf { B } ^ { - 1 }$.

RECOGNISING ACHIEVEMENT

\section*{ADVANCED SUBSIDIARY GCE MATHEMATICS}
Further Pure Mathematics 1

Candidates answer on the Answer Booklet\\
OCR Supplied Materials:

\begin{itemize}
  \item 8 page Answer Booklet
  \item List of Formulae (MF1)
\end{itemize}

Other Materials Required: None

Thursday 15 January 2009\\
Morning\\
Duration: 1 hour 30 minutes\\
\includegraphics[max width=\textwidth, alt={}, center]{a8f7bd80-4733-40b9-be96-9fb0878dbb26-23_122_382_1018_1430}

\begin{itemize}
  \item Write your name clearly in capital letters, your Centre Number and Candidate Number in the spaces provided on the Answer Booklet.
  \item Use black ink. Pencil may be used for graphs and diagrams only.
  \item Read each question carefully and make sure that you know what you have to do before starting your answer.
  \item Answer all the questions.
  \item Do not write in the bar codes.
  \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 You are reminded of the need for clear presentation in your answers.
  \item The total number of marks for this paper is $\mathbf { 7 2 }$.
  \item This document consists of $\mathbf { 4 }$ pages. Any blank pages are indicated.
\end{itemize}

1 Express $\frac { 2 + 3 \mathrm { i } } { 5 - \mathrm { i } }$ in the form $x + \mathrm { i } y$, showing clearly how you obtain your answer.

2 The matrix $\mathbf { A }$ is given by $\mathbf { A } = \left( \begin{array} { l l } 2 & 0 \\ a & 5 \end{array} \right)$. Find\\
(i) $\mathbf { A } ^ { - 1 }$,\\
(ii) $2 \mathbf { A } - \left( \begin{array} { l l } 1 & 2 \\ 0 & 4 \end{array} \right)$.

3 Find $\sum _ { r = 1 } ^ { n } \left( 4 r ^ { 3 } + 6 r ^ { 2 } + 2 r \right)$, expressing your answer in a fully factorised form.

4 Given that $\mathbf { A }$ and $\mathbf { B }$ are $2 \times 2$ non-singular matrices and $\mathbf { I }$ is the $2 \times 2$ identity matrix, simplify

$$\mathbf { B } ( \mathbf { A B } ) ^ { - 1 } \mathbf { A } - \mathbf { I } .$$

5 By using the determinant of an appropriate matrix, or otherwise, find the value of $k$ for which the simultaneous equations

$$\begin{aligned}
2 x - y + z & = 7 \\
3 y + z & = 4 \\
x + k y + k z & = 5
\end{aligned}$$

do not have a unique solution for $x , y$ and $z$.

6 (i) The transformation P is represented by the matrix $\left( \begin{array} { r r } 1 & 0 \\ 0 & - 1 \end{array} \right)$. Give a geometrical description of transformation P .\\
(ii) The transformation Q is represented by the matrix $\left( \begin{array} { r r } 0 & - 1 \\ - 1 & 0 \end{array} \right)$. Give a geometrical description of transformation Q.\\
(iii) The transformation R is equivalent to transformation P followed by transformation Q . Find the matrix that represents R .\\
(iv) Give a geometrical description of the single transformation that is represented by your answer to part (iii).

7 It is given that $u _ { n } = 13 ^ { n } + 6 ^ { n - 1 }$, where $n$ is a positive integer.\\
(i) Show that $u _ { n } + u _ { n + 1 } = 14 \times 13 ^ { n } + 7 \times 6 ^ { n - 1 }$.\\
(ii) Prove by induction that $u _ { n }$ is a multiple of 7 .

8 (i) Show that $( \alpha - \beta ) ^ { 2 } \equiv ( \alpha + \beta ) ^ { 2 } - 4 \alpha \beta$.

The quadratic equation $x ^ { 2 } - 6 k x + k ^ { 2 } = 0$, where $k$ is a positive constant, has roots $\alpha$ and $\beta$, with $\alpha > \beta$.\\
(ii) Show that $\alpha - \beta = 4 \sqrt { 2 } k$.\\
(iii) Hence find a quadratic equation with roots $\alpha + 1$ and $\beta - 1$.

9 (i) Show that $\frac { 1 } { 2 r - 3 } - \frac { 1 } { 2 r + 1 } = \frac { 4 } { 4 r ^ { 2 } - 4 r - 3 }$.\\
(ii) Hence find an expression, in terms of $n$, for

$$\sum _ { r = 2 } ^ { n } \frac { 4 } { 4 r ^ { 2 } - 4 r - 3 }$$

(iii) Show that $\sum _ { r = 2 } ^ { \infty } \frac { 4 } { 4 r ^ { 2 } - 4 r - 3 } = \frac { 4 } { 3 }$.\\
(i) Use an algebraic method to find the square roots of the complex number $2 + \mathrm { i } \sqrt { 5 }$. Give your answers in the form $x + \mathrm { i } y$, where $x$ and $y$ are exact real numbers.\\
(ii) Hence find, in the form $x + \mathrm { i } y$ where $x$ and $y$ are exact real numbers, the roots of the equation

$$z ^ { 4 } - 4 z ^ { 2 } + 9 = 0$$

(iii) Show, on an Argand diagram, the roots of the equation in part (ii).\\
(iv) Given that $\alpha$ is the root of the equation in part (ii) such that $0 < \arg \alpha < \frac { 1 } { 2 } \pi$, sketch on the same Argand diagram the locus given by $| z - \alpha | = | z |$.

RECOGNISING ACHIEVEMENT

\section*{ADVANCED SUBSIDIARY GCE MATHEMATICS}
Further Pure Mathematics 1

Candidates answer on the Answer Booklet\\
OCR Supplied Materials:

\begin{itemize}
  \item 8 page Answer Booklet
  \item List of Formulae (MF1)
\end{itemize}

Other Materials Required: None

\section*{Friday 5 June 2009 Afternoon}
Duration: 1 hour 30 minutes\\
\includegraphics[max width=\textwidth, alt={}, center]{a8f7bd80-4733-40b9-be96-9fb0878dbb26-26_122_377_1018_1430}

\begin{itemize}
  \item Write your name clearly in capital letters, your Centre Number and Candidate Number in the spaces provided on the Answer Booklet.
  \item Use black ink. Pencil may be used for graphs and diagrams only.
  \item Read each question carefully and make sure that you know what you have to do before starting your answer.
  \item Answer all the questions.
  \item Do not write in the bar codes.
  \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 You are reminded of the need for clear presentation in your answers.
  \item The total number of marks for this paper is $\mathbf { 7 2 }$.
  \item This document consists of $\mathbf { 4 }$ pages. Any blank pages are indicated.
\end{itemize}

1 Evaluate $\sum _ { r = 101 } ^ { 250 } r ^ { 3 }$.

2 The matrices $\mathbf { A }$ and $\mathbf { B }$ are given by $\mathbf { A } = \left( \begin{array} { l l } 3 & 0 \\ 0 & 1 \end{array} \right)$ and $\mathbf { B } = \left( \begin{array} { l l } 5 & 0 \\ 0 & 2 \end{array} \right)$ and $\mathbf { I }$ is the $2 \times 2$ identity matrix. Find the values of the constants $a$ and $b$ for which $a \mathbf { A } + b \mathbf { B } = \mathbf { I }$.

3 The complex numbers $z$ and $w$ are given by $z = 5 - 2 \mathrm { i }$ and $w = 3 + 7 \mathrm { i }$. Giving your answers in the form $x + \mathrm { i } y$ and showing clearly how you obtain them, find\\
(i) $4 z - 3 w$,\\
(ii) $z ^ { * } w$.

4 The roots of the quadratic equation $x ^ { 2 } + x - 8 = 0$ are $p$ and $q$. Find the value of $p + q + \frac { 1 } { p } + \frac { 1 } { q }$.

5 The cubic equation $x ^ { 3 } + 5 x ^ { 2 } + 7 = 0$ has roots $\alpha , \beta$ and $\gamma$.\\
(i) Use the substitution $x = \sqrt { u }$ to find a cubic equation in $u$ with integer coefficients.\\
(ii) Hence find the value of $\alpha ^ { 2 } \beta ^ { 2 } + \beta ^ { 2 } \gamma ^ { 2 } + \gamma ^ { 2 } \alpha ^ { 2 }$.

6 The complex number $3 - 3 \mathrm { i }$ is denoted by $a$.\\
(i) Find $| a |$ and $\arg a$.\\
(ii) Sketch on a single Argand diagram the loci given by\\
(a) $| z - a | = 3 \sqrt { 2 }$,\\
(b) $\quad \arg ( z - a ) = \frac { 1 } { 4 } \pi$.\\
(iii) Indicate, by shading, the region of the Argand diagram for which

$$| z - a | \geqslant 3 \sqrt { 2 } \quad \text { and } \quad 0 \leqslant \arg ( z - a ) \leqslant \frac { 1 } { 4 } \pi$$

7 (i) Use the method of differences to show that

$$\sum _ { r = 1 } ^ { n } \left\{ ( r + 1 ) ^ { 4 } - r ^ { 4 } \right\} = ( n + 1 ) ^ { 4 } - 1$$

(ii) Show that $( r + 1 ) ^ { 4 } - r ^ { 4 } \equiv 4 r ^ { 3 } + 6 r ^ { 2 } + 4 r + 1$.\\
(iii) Hence show that

$$4 \sum _ { r = 1 } ^ { n } r ^ { 3 } = n ^ { 2 } ( n + 1 ) ^ { 2 }$$

8 The matrix $\mathbf { C }$ is given by $\mathbf { C } = \left( \begin{array} { l l } 3 & 2 \\ 1 & 1 \end{array} \right)$.\\
(i) Draw a diagram showing the image of the unit square under the transformation represented by $\mathbf { C }$.

The transformation represented by $\mathbf { C }$ is equivalent to a transformation S followed by another transformation T.\\
(ii) Given that S is a shear with the $y$-axis invariant in which the image of the point ( 1,1 ) is ( 1,2 ), write down the matrix that represents $S$.\\
(iii) Find the matrix that represents transformation T and describe fully the transformation T .

9 The matrix $\mathbf { A }$ is given by $\mathbf { A } = \left( \begin{array} { l l l } a & 1 & 1 \\ 1 & a & 1 \\ 1 & 1 & 2 \end{array} \right)$.\\
(i) Find, in terms of $a$, the determinant of $\mathbf { A }$.\\
(ii) Hence find the values of $a$ for which $\mathbf { A }$ is singular.\\
(iii) State, giving a brief reason in each case, whether the simultaneous equations

$$\begin{aligned}
a x + y + z & = 2 a \\
x + a y + z & = - 1 \\
x + y + 2 z & = - 1
\end{aligned}$$

have any solutions when\\
(a) $a = 0$,\\
(b) $a = 1$.

10 The sequence $u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots$ is defined by $u _ { 1 } = 3$ and $u _ { n + 1 } = 3 u _ { n } - 2$.\\
(i) Find $u _ { 2 }$ and $u _ { 3 }$ and verify that $\frac { 1 } { 2 } \left( u _ { 4 } - 1 \right) = 27$.\\
(ii) Hence suggest an expression for $u _ { n }$.\\
(iii) Use induction to prove that your answer to part (ii) is correct.

RECOGNISING ACHIEVEMENT

\section*{ADVANCED SUBSIDIARY GCE MATHEMATICS}
Further Pure Mathematics 1

Candidates answer on the Answer Booklet\\
OCR Supplied Materials:

\begin{itemize}
  \item 8 page Answer Booklet
  \item List of Formulae (MF1)
\end{itemize}

Other Materials Required:\\
None

Wednesday 20 January 2010\\
Afternoon\\
Duration: 1 hour 30 minutes\\
\includegraphics[max width=\textwidth, alt={}, center]{a8f7bd80-4733-40b9-be96-9fb0878dbb26-29_124_385_1016_1427}

\begin{itemize}
  \item Write your name clearly in capital letters, your Centre Number and Candidate Number in the spaces provided on the Answer Booklet.
  \item Use black ink. Pencil may be used for graphs and diagrams only.
  \item Read each question carefully and make sure that you know what you have to do before starting your answer.
  \item Answer all the questions.
  \item Do not write in the bar codes.
  \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 You are reminded of the need for clear presentation in your answers.
  \item The total number of marks for this paper is 72.
  \item This document consists of $\mathbf { 4 }$ pages. Any blank pages are indicated.
\end{itemize}

1 The matrix $\mathbf { A }$ is given by $\mathbf { A } = \left( \begin{array} { l l } a & 2 \\ 3 & 4 \end{array} \right)$ and $\mathbf { I }$ is the $2 \times 2$ identity matrix.\\
(i) Find A-4I.\\
(ii) Given that $\mathbf { A }$ is singular, find the value of $a$.

2 The cubic equation $2 x ^ { 3 } + 3 x - 3 = 0$ has roots $\alpha , \beta$ and $\gamma$.\\
(i) Use the substitution $x = u - 1$ to find a cubic equation in $u$ with integer coefficients.\\
(ii) Hence find the value of $( \alpha + 1 ) ( \beta + 1 ) ( \gamma + 1 )$.

3 The complex number $z$ satisfies the equation $z + 2 \mathbf { i } z ^ { * } = 12 + 9 \mathbf { i }$. Find $z$, giving your answer in the form $x + \mathrm { i } y$.

4 Find $\sum _ { r = 1 } ^ { n } r ( r + 1 ) ( r - 2 )$, expressing your answer in a fully factorised form.

5 (i) The transformation T is represented by the matrix $\left( \begin{array} { r r } 0 & - 1 \\ 1 & 0 \end{array} \right)$. Give a geometrical description of T .\\
(ii) The transformation T is equivalent to a reflection in the line $y = - x$ followed by another transformation S . Give a geometrical description of S and find the matrix that represents S .

6 One root of the cubic equation $x ^ { 3 } + p x ^ { 2 } + 6 x + q = 0$, where $p$ and $q$ are real, is the complex number 5-i.\\
(i) Find the real root of the cubic equation.\\
(ii) Find the values of $p$ and $q$.

7 (i) Show that $\frac { 1 } { r ^ { 2 } } - \frac { 1 } { ( r + 1 ) ^ { 2 } } \equiv \frac { 2 r + 1 } { r ^ { 2 } ( r + 1 ) ^ { 2 } }$.\\
(ii) Hence find an expression, in terms of $n$, for $\sum _ { r = 1 } ^ { n } \frac { 2 r + 1 } { r ^ { 2 } ( r + 1 ) ^ { 2 } }$.\\
(iii) Find $\sum _ { r = 2 } ^ { \infty } \frac { 2 r + 1 } { r ^ { 2 } ( r + 1 ) ^ { 2 } }$.

8 The complex number $a$ is such that $a ^ { 2 } = 5 - 12 \mathrm { i }$.\\
(i) Use an algebraic method to find the two possible values of $a$.\\
(ii) Sketch on a single Argand diagram the two possible loci given by $| z - a | = | a |$.

9 The matrix $\mathbf { A }$ is given by $\mathbf { A } = \left( \begin{array} { r r r } 2 & - 1 & 1 \\ 0 & 3 & 1 \\ 1 & 1 & a \end{array} \right)$, where $a \neq 1$.\\
(i) Find $\mathbf { A } ^ { - 1 }$.\\
(ii) Hence, or otherwise, solve the equations

$$\begin{array} { r } 
2 x - y + z = 1 \\
3 y + z = 2 \\
x + y + a z = 2
\end{array}$$

10 The matrix $\mathbf { M }$ is given by $\mathbf { M } = \left( \begin{array} { l l } 1 & 2 \\ 0 & 1 \end{array} \right)$.\\
(i) Find $\mathbf { M } ^ { 2 }$ and $\mathbf { M } ^ { 3 }$.\\
(ii) Hence suggest a suitable form for the matrix $\mathbf { M } ^ { n }$.\\
(iii) Use induction to prove that your answer to part (ii) is correct.\\
(iv) Describe fully the single geometrical transformation represented by $\mathbf { M } ^ { 10 }$.

\section*{ADVANCED SUBSIDIARY GCE MATHEMATICS}
Further Pure Mathematics 1

\section*{Candidates answer on the Answer Booklet}
\section*{OCR Supplied Materials:}
\begin{itemize}
  \item 8 page Answer Booklet
  \item List of Formulae (MF1)
\end{itemize}

\section*{Other Materials Required:}
\begin{itemize}
  \item Scientific or graphical calculator
\end{itemize}

\section*{Friday 11 June 2010 \\
 Morning}
Duration: 1 hour 30 minutes\\
\includegraphics[max width=\textwidth, alt={}, center]{a8f7bd80-4733-40b9-be96-9fb0878dbb26-32_118_383_1032_1420}

\begin{itemize}
  \item Write your name clearly in capital letters, your Centre Number and Candidate Number in the spaces provided on the Answer Booklet.
  \item Use black ink. Pencil may be used for graphs and diagrams only.
  \item Read each question carefully and make sure that you know what you have to do before starting your answer.
  \item Answer all the questions.
  \item Do not write in the bar codes.
  \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 You are reminded of the need for clear presentation in your answers.
  \item The total number of marks for this paper is $\mathbf { 7 2 }$.
  \item This document consists of $\mathbf { 4 }$ pages. Any blank pages are indicated.
\end{itemize}

1 Prove by induction that, for $n \geqslant 1 , \sum _ { r = 1 } ^ { n } r ( r + 1 ) = \frac { 1 } { 3 } n ( n + 1 ) ( n + 2 )$.

2 The matrices $\mathbf { A } , \mathbf { B }$ and $\mathbf { C }$ are given by $\mathbf { A } = \left( \begin{array} { l l } 1 & - 4 \end{array} \right) , \mathbf { B } = \binom { 5 } { 3 }$ and $\mathbf { C } = \left( \begin{array} { r r } 3 & 0 \\ - 2 & 2 \end{array} \right)$. Find\\
(i) $\mathbf { A B }$,\\
(ii) $\mathbf { B A } - 4 \mathbf { C }$.

3 Find $\sum _ { r = 1 } ^ { n } ( 2 r - 1 ) ^ { 2 }$, expressing your answer in a fully factorised form.

4 The complex numbers $a$ and $b$ are given by $a = 7 + 6 \mathrm { i }$ and $b = 1 - 3 \mathrm { i }$. Showing clearly how you obtain your answers, find\\
(i) $| a - 2 b |$ and $\arg ( a - 2 b )$,\\
(ii) $\frac { b } { a }$, giving your answer in the form $x + \mathrm { i } y$.

5 (a) Write down the matrix that represents a reflection in the line $y = x$.\\
(b) Describe fully the geometrical transformation represented by each of the following matrices:

$$\begin{aligned}
& \text { (i) } \left( \begin{array} { c c } 
5 & 0 \\
0 & 1
\end{array} \right) \text {, } \\
& \text { (ii) } \left( \begin{array} { c c } 
\frac { 1 } { 2 } & \frac { 1 } { 2 } \sqrt { 3 } \\
- \frac { 1 } { 2 } \sqrt { 3 } & \frac { 1 } { 2 }
\end{array} \right) \text {. }
\end{aligned}$$

6 (i) Sketch on a single Argand diagram the loci given by\\
(a) $| z - 3 + 4 \mathrm { i } | = 5$,\\
(b) $| z | = | z - 6 |$.\\
(ii) Indicate, by shading, the region of the Argand diagram for which

$$| z - 3 + 4 i | \leqslant 5 \quad \text { and } \quad | z | \geqslant | z - 6 | .$$

7 The quadratic equation $x ^ { 2 } + 2 k x + k = 0$, where $k$ is a non-zero constant, has roots $\alpha$ and $\beta$. Find a quadratic equation with roots $\frac { \alpha + \beta } { \alpha }$ and $\frac { \alpha + \beta } { \beta }$.\\
(i) Show that $\frac { 1 } { \sqrt { r + 2 } + \sqrt { r } } \equiv \frac { \sqrt { r + 2 } - \sqrt { r } } { 2 }$.\\
(ii) Hence find an expression, in terms of $n$, for

$$\sum _ { r = 1 } ^ { n } \frac { 1 } { \sqrt { r + 2 } + \sqrt { r } }$$

(iii) State, giving a brief reason, whether the series $\sum _ { r = 1 } ^ { \infty } \frac { 1 } { \sqrt { r + 2 } + \sqrt { r } }$ converges.

9 The matrix $\mathbf { A }$ is given by $\mathbf { A } = \left( \begin{array} { r r r } a & a & - 1 \\ 0 & a & 2 \\ 1 & 2 & 1 \end{array} \right)$.\\
(i) Find, in terms of $a$, the determinant of $\mathbf { A }$.\\
(ii) Three simultaneous equations are shown below.

$$\begin{aligned}
a x + a y - z & = - 1 \\
a y + 2 z & = 2 a \\
x + 2 y + z & = 1
\end{aligned}$$

For each of the following values of $a$, determine whether the equations are consistent or inconsistent. If the equations are consistent, determine whether or not there is a unique solution.\\
(a) $a = 0$\\
(b) $a = 1$\\
(c) $a = 2$

10 The complex number $z$, where $0 < \arg z < \frac { 1 } { 2 } \pi$, is such that $z ^ { 2 } = 3 + 4 \mathrm { i }$.\\
(i) Use an algebraic method to find $z$.\\
(ii) Show that $z ^ { 3 } = 2 + 11 \mathrm { i }$.

The complex number $w$ is the root of the equation

$$w ^ { 6 } - 4 w ^ { 3 } + 125 = 0$$

for which $- \frac { 1 } { 2 } \pi < \arg w < 0$.\\
(iii) Find $w$.

\section*{ADVANCED SUBSIDIARY GCE MATHEMATICS}
Further Pure Mathematics 1

\section*{QUESTION PAPER}
Candidates answer on the printed answer book.\\
OCR supplied materials:

\begin{itemize}
  \item Printed answer book 4725
  \item List of Formulae (MF1)
\end{itemize}

Other materials required:

\begin{itemize}
  \item Scientific or graphical calculator
\end{itemize}

Wednesday 19 January 2011 Afternoon

Duration: 1 hour 30 minutes

These instructions are the same on the printed answer book and the question paper.

\begin{itemize}
  \item The question paper will be found in the centre of the printed answer book.
  \item Write your name, centre number and candidate number in the spaces provided on the printed answer book. Please write clearly and in capital letters.
  \item Write your answer to each question in the space provided in the printed answer book. Additional paper may be used if necessary but you must clearly show your candidate number, centre number and question number(s).
  \item Use black ink. Pencil may be used for graphs and diagrams only.
  \item Read each question carefully. Make sure you know what you have to do before starting your answer.
  \item Answer all the questions.
  \item Do not write in the bar codes.
  \item You are permitted to use a scientific or graphical calculator in this paper.
  \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.
\end{itemize}

This information is the same on the printed answer book and the question paper.

\begin{itemize}
  \item The number of marks is given in brackets [ ] at the end of each question or part question on the question paper.
  \item You are reminded of the need for clear presentation in your answers.
  \item The total number of marks for this paper is 72.
  \item The printed answer book consists of $\mathbf { 1 2 }$ pages. The question paper consists of $\mathbf { 4 }$ pages. Any blank pages are indicated.
\end{itemize}

\hfill \mbox{\textit{OCR FP1  Q10}}