OCR FP1 (Further Pure Mathematics 1) 2012 June

1 The complex numbers \(z\) and \(w\) are given by \(z = 6 - \mathrm { i }\) and \(w = 5 + 4 \mathrm { i }\). Giving your answers in the form \(x + \mathrm { i } y\) and showing clearly how you obtain them, find

1. \(z + 3 w\),
2. \(\frac { Z } { W }\).

2 The matrices \(\mathbf { A }\) and \(\mathbf { B }\) are given by \(\mathbf { A } = \left( \begin{array} { l l } 2 & 1 \\ 4 & 3 \end{array} \right)\) and \(\mathbf { B } = \left( \begin{array} { l l } 1 & 0 \\ 3 & 2 \end{array} \right)\). Find

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

3 One root of the quadratic equation \(x ^ { 2 } + a x + b = 0\), where \(a\) and \(b\) are real, is the complex number \(4 - 3 \mathrm { i }\). Find the values of \(a\) and \(b\).

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

5 Prove by induction that, for \(n \geqslant 1 , \sum _ { r = 1 } ^ { n } 4 \times 3 ^ { r } = 6 \left( 3 ^ { n } - 1 \right)\).

6 The quadratic equation \(2 x ^ { 2 } + x + 5 = 0\) has roots \(\alpha\) and \(\beta\).

1. Use the substitution \(x = \frac { 1 } { u + 1 }\) to obtain a quadratic equation in \(u\) with integer coefficients.
2. Hence, or otherwise, find the value of \(\left( \frac { 1 } { \alpha } - 1 \right) \left( \frac { 1 } { \beta } - 1 \right)\).

7 The loci \(C _ { 1 }\) and \(C _ { 2 }\) are given by \(| z - 3 - 4 \mathrm { i } | = 4\) and \(| z | = | z - 8 \mathrm { i } |\) respectively.

1. Sketch, on a single Argand diagram, the loci \(C _ { 1 }\) and \(C _ { 2 }\).
2. Hence find the complex numbers represented by the points of intersection of \(C _ { 1 }\) and \(C _ { 2 }\).
3. Indicate, by shading, the region of the Argand diagram for which $$| z - 3 - 4 i | \leqslant 4 \text { and } | z | \geqslant | z - 8 i | .$$

8

1. Show that \(\frac { 1 } { r } - \frac { 1 } { r + 2 } \equiv \frac { 2 } { r ( r + 2 ) }\).
2. Hence find an expression, in terms of \(n\), for \(\sum _ { r = 1 } ^ { n } \frac { 2 } { r ( r + 2 ) }\).
3. Given that \(\sum _ { r = N + 1 } ^ { \infty } \frac { 2 } { r ( r + 2 ) } = \frac { 11 } { 30 }\), find the value of \(N\).

9

1. The matrix \(\mathbf { X }\) is given by \(\mathbf { X } = \left( \begin{array} { l l } 1 & 2 \\ 0 & 1 \end{array} \right)\). Describe fully the geometrical transformation represented by \(\mathbf { X }\).
2. The matrix \(\mathbf { Z }\) is given by \(\mathbf { Z } = \left( \begin{array} { c c } \frac { 1 } { 2 } & \frac { 1 } { 2 } ( 2 + \sqrt { 3 } ) \\ - \frac { 1 } { 2 } \sqrt { 3 } & \frac { 1 } { 2 } ( 1 - 2 \sqrt { 3 } ) \end{array} \right)\). The transformation represented by \(\mathbf { Z }\) is equivalent to the transformation represented by \(\mathbf { X }\), followed by another transformation represented by the matrix \(\mathbf { Y }\). Find \(\mathbf { Y }\).
3. Describe fully the geometrical transformation represented by \(\mathbf { Y }\).

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

1. Find the determinant of \(\mathbf { D }\) in terms of \(a\).
2. Three simultaneous equations are shown below. $$\begin{array} { r } a x + 2 y - z = 0 \\ 2 x + a y + z = a \\ x + y + a z = a \end{array}$$ For each of the following values of \(a\), determine whether or not there is a unique solution. If the solution is not unique, determine whether the equations are consistent or inconsistent.
   1. \(\quad a = 3\)
   2. \(a = 2\)
   3. \(\quad a = 0\) \section*{THERE ARE NO QUESTIONS WRITTEN ON THIS PAGE}