OCR FP1 (Further Pure Mathematics 1) 2010 January

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.

1. Find A-4I.
2. 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 \(\gamma\).

1. Use the substitution \(x = u - 1\) to find a cubic equation in \(u\) with integer coefficients.
2. Hence find the value of \(( \alpha + 1 ) ( \beta + 1 ) ( \gamma + 1 )\).

3 The complex number \(z\) satisfies the equation \(z + 2 \mathrm { i } z ^ { * } = 12 + 9 \mathrm { i }\). 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

1. 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 .
2. 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.

1. Find the real root of the cubic equation.
2. Find the values of \(p\) and \(q\).

7

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

8 The complex number \(a\) is such that \(a ^ { 2 } = 5 - 12 \mathrm { i }\).

1. Use an algebraic method to find the two possible values of \(a\).
2. 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 \(a \neq 1\).

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

1. Find \(\mathbf { M } ^ { 2 }\) and \(\mathbf { M } ^ { 3 }\).
2. Hence suggest a suitable form for the matrix \(\mathbf { M } ^ { n }\).
3. Use induction to prove that your answer to part (ii) is correct.
4. Describe fully the single geometrical transformation represented by \(\mathbf { M } ^ { 10 }\).