OCR FP1 (Further Pure Mathematics 1) 2008 June

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

1. \(\mathbf { A } - 3 \mathbf { I }\),
2. \(\mathrm { A } ^ { - 1 }\).

2 The complex number \(3 + 4 \mathrm { i }\) is denoted by \(a\).

1. Find \(| a |\) and \(\arg a\).
2. Sketch on a single Argand diagram the loci given by
   (a) \(| z - a | = | a |\),
   (b) \(\arg ( z - 3 ) = \arg a\).

3

1. Show that \(\frac { 1 } { r ! } - \frac { 1 } { ( r + 1 ) ! } = \frac { r } { ( r + 1 ) ! }\).
2. 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 .

1. Write down the other root of the equation.
2. Find the values of \(a , b\) and \(c\).

7 Describe fully the geometrical transformation represented by each of the following matrices:

1. \(\left( \begin{array} { l l } 6 & 0
   0 & 6 \end{array} \right)\),
2. \(\left( \begin{array} { l l } 0 & 1
   1 & 0 \end{array} \right)\),
3. \(\left( \begin{array} { l l } 1 & 0
   0 & 6 \end{array} \right)\),
4. \(\left( \begin{array} { r r } 0.8 & 0.6
   - 0.6 & 0.8 \end{array} \right)\).

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

1. Use an algebraic method to find the square roots of the complex number \(5 + 12 \mathrm { i }\).
2. Find \(( 3 - 2 \mathrm { i } ) ^ { 2 }\).
3. 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 \(\mathbf { A B } = \left( \begin{array} { l l l } a & 6 & 1
1 & 1 & 0
1 & 3 & 0 \end{array} \right)\).

1. Show that \(\mathbf { A B }\) is non-singular.
2. Find \(( \mathbf { A B } ) ^ { - 1 }\).
3. Find \(\mathbf { B } ^ { - 1 }\).