OCR FP1 (Further Pure Mathematics 1) 2009 January

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

1. \(\mathbf { A } ^ { - 1 }\),
2. \(2 \mathbf { A } - \left( \begin{array} { l l } 1 & 2
   0 & 4 \end{array} \right)\).

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

1. 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 .
2. 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.
3. The transformation R is equivalent to transformation P followed by transformation Q . Find the matrix that represents R .
4. 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.

1. Show that \(u _ { n } + u _ { n + 1 } = 14 \times 13 ^ { n } + 7 \times 6 ^ { n - 1 }\).
2. Prove by induction that \(u _ { n }\) is a multiple of 7 .

8

1. 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 \(\alpha > \beta\).
2. Show that \(\alpha - \beta = 4 \sqrt { 2 } k\).
3. Hence find a quadratic equation with roots \(\alpha + 1\) and \(\beta - 1\).

9

1. Show that \(\frac { 1 } { 2 r - 3 } - \frac { 1 } { 2 r + 1 } = \frac { 4 } { 4 r ^ { 2 } - 4 r - 3 }\).
2. Hence find an expression, in terms of \(n\), for $$\sum _ { r = 2 } ^ { n } \frac { 4 } { 4 r ^ { 2 } - 4 r - 3 }$$
3. Show that \(\sum _ { r = 2 } ^ { \infty } \frac { 4 } { 4 r ^ { 2 } - 4 r - 3 } = \frac { 4 } { 3 }\).
4. 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.
5. Hence find, in the form \(x + \mathrm { i } y\) where \(x\) and \(y\) are exact real numbers, the roots of the equation $$z ^ { 4 } - 4 z ^ { 2 } + 9 = 0$$
6. Show, on an Argand diagram, the roots of the equation in part (ii).
7. 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 \(| z - \alpha | = | z |\).