OCR FP1 (Further Pure Mathematics 1) 2010 June

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

1. \(\mathbf { A B }\),
2. \(\mathbf { B A } - 4 \mathbf { C }\).

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

1. \(| a - 2 b |\) and \(\arg ( a - 2 b )\),
2. \(\frac { b } { a }\), giving your answer in the form \(x + \mathrm { i } y\).

5

1. Write down the matrix that represents a reflection in the line \(y = x\).
2. 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

1. Sketch on a single Argand diagram the loci given by
   (a) \(| z - 3 + 4 \mathrm { i } | = 5\),
   (b) \(| z | = | z - 6 |\).
2. 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 \(\frac { \alpha + \beta } { \beta }\).

1. Show that \(\frac { 1 } { \sqrt { r + 2 } + \sqrt { r } } \equiv \frac { \sqrt { r + 2 } - \sqrt { r } } { 2 }\).
2. Hence find an expression, in terms of \(n\), for $$\sum _ { r = 1 } ^ { n } \frac { 1 } { \sqrt { r + 2 } + \sqrt { r } }$$
3. 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 \(\mathbf { A } = \left( \begin{array} { r r r } a & a & - 1
0 & a & 2
1 & 2 & 1 \end{array} \right)\).

1. Find, in terms of \(a\), the determinant of \(\mathbf { A }\).
2. 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 \(z ^ { 2 } = 3 + 4 \mathrm { i }\).

1. Use an algebraic method to find \(z\).
2. 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 \(- \frac { 1 } { 2 } \pi < \arg w < 0\).
3. Find \(w\).