OCR FP1 (Further Pure Mathematics 1) 2012 January

1 The complex number \(a + 5 \mathrm { i }\), where \(a\) is positive, is denoted by \(z\). Given that \(| z | = 13\), find the value of \(a\) and hence find \(\arg z\).

2 The matrices \(\mathbf { A }\) and \(\mathbf { B }\) are given by \(\mathbf { A } = \left( \begin{array} { r r } 3 & 4
2 & - 3 \end{array} \right)\) and \(\mathbf { B } = \left( \begin{array} { r r } 4 & 6
3 & - 5 \end{array} \right)\), and \(\mathbf { I }\) is the \(2 \times 2\) identity matrix. Given that \(p \mathbf { A } + q \mathbf { B } = \mathbf { I }\), find the values of the constants \(p\) and \(q\).

3 Use an algebraic method to find the square roots of \(3 + ( 6 \sqrt { 2 } )\) i. Give your answers in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are exact real numbers.

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

5

1. Find the matrix that represents a reflection in the line \(y = - x\).
2. The matrix \(\mathbf { C }\) is given by \(\mathbf { C } = \left( \begin{array} { l l } 1 & 0
   0 & 4 \end{array} \right)\).
   1. Describe fully the geometrical transformation represented by \(\mathbf { C }\).
   2. State the value of the determinant of \(\mathbf { C }\) and describe briefly how this value relates to the transformation represented by \(\mathbf { C }\).

6 Sketch, on a single Argand diagram, the loci given by \(| z - \sqrt { 3 } - \mathrm { i } | = 2\) and \(\arg z = \frac { 1 } { 6 } \pi\).

7 The matrix \(\mathbf { M }\) is given by \(\mathbf { M } = \left( \begin{array} { l l } 3 & 0
2 & 1 \end{array} \right)\).

1. Show that \(\mathbf { M } ^ { 4 } = \left( \begin{array} { l l } 81 & 0
   80 & 1 \end{array} \right)\).
2. Hence suggest a suitable form for the matrix \(\mathbf { M } ^ { n }\), where \(n\) is a positive integer.
3. Use induction to prove that your answer to part (ii) is correct.

8

1. Show that \(\frac { r } { r + 1 } - \frac { r - 1 } { r } \equiv \frac { 1 } { r ( r + 1 ) }\).
2. Hence find an expression, in terms of \(n\), for $$\frac { 1 } { 2 } + \frac { 1 } { 6 } + \frac { 1 } { 12 } + \ldots + \frac { 1 } { n ( n + 1 ) }$$
3. Hence find \(\sum _ { r = n + 1 } ^ { \infty } \frac { 1 } { r ( r + 1 ) }\).

\(\mathbf { 9 }\) The matrix \(\mathbf { X }\) is given by \(\mathbf { X } = \left( \begin{array} { r r r } a & 2 & 9
2 & a & 3
1 & 0 & - 1 \end{array} \right)\).

1. Find the determinant of \(\mathbf { X }\) in terms of \(a\).
2. Hence find the values of \(a\) for which \(\mathbf { X }\) is singular.
3. Given that \(\mathbf { X }\) is non-singular, find \(\mathbf { X } ^ { - 1 }\) in terms of \(a\).

10 The cubic equation \(3 x ^ { 3 } - 9 x ^ { 2 } + 6 x + 2 = 0\) has roots \(\alpha , \beta\) and \(\gamma\).

1. Write down the values of \(\alpha + \beta + \gamma , \alpha \beta + \beta \gamma + \gamma \alpha\) and \(\alpha \beta \gamma\). The cubic equation \(x ^ { 3 } + a x ^ { 2 } + b x + c = 0\) has roots \(\alpha ^ { 2 } , \beta ^ { 2 }\) and \(\gamma ^ { 2 }\).
2. Show that \(c = - \frac { 4 } { 9 }\) and find the values of \(a\) and \(b\). \section*{THERE ARE NO QUESTIONS WRITTEN ON THIS PAGE}