OCR FP1 (Further Pure Mathematics 1) 2015 June

1 The complex number \(x + \mathrm { i } y\) is denoted by \(z\). Express \(3 z z ^ { * } - | z | ^ { 2 }\) in terms of \(x\) and \(y\).

2 Find \(\sum _ { r = 1 } ^ { n } \left( 3 r ^ { 2 } - 5 \right)\), expressing your answer in a fully factorised form.

3 The matrix \(\mathbf { A }\) is given by \(\mathbf { A } = \left( \begin{array} { l l } 2 & a
0 & 1 \end{array} \right)\), where \(a\) is a constant.

1. Find \(\mathbf { A } ^ { - 1 }\). The matrix \(\mathbf { B }\) is given by \(\mathbf { B } = \left( \begin{array} { l l } 2 & a
   4 & 1 \end{array} \right)\).
2. Given that \(\mathbf { P A } = \mathbf { B }\), find the matrix \(\mathbf { P }\).

4 Prove by induction that, for \(n \geqslant 1 , \sum _ { r = 1 } ^ { n } r ( 3 r + 1 ) = n ( n + 1 ) ^ { 2 }\).

5 The loci \(C _ { 1 }\) and \(C _ { 2 }\) are given by \(| z + 2 | = 2\) and \(\arg ( z + 2 ) = \frac { 5 } { 6 } \pi\) respectively.

1. Sketch, on a single Argand diagram, the loci \(C _ { 1 }\) and \(C _ { 2 }\).
2. Find the complex number represented by the intersection of \(C _ { 1 }\) and \(C _ { 2 }\).
3. Indicate, by shading, the region of the Argand diagram for which $$| z + 2 | \leqslant 2 \text { and } \frac { 5 } { 6 } \pi \leqslant \arg ( z + 2 ) \leqslant \pi .$$

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

1. The diagram in the Printed Answer Book shows the unit square \(O A B C\). The image of the unit square under the transformation represented by \(\mathbf { M }\) is \(O A ^ { \prime } B ^ { \prime } C ^ { \prime }\). Draw and label \(O A ^ { \prime } B ^ { \prime } C ^ { \prime }\), indicating clearly the coordinates of \(A ^ { \prime } , B ^ { \prime }\) and \(C ^ { \prime }\).
2. The transformation represented by \(\mathbf { M }\) is equivalent to a transformation P followed by a transformation Q. Give geometrical descriptions of a possible pair of transformations P and Q and state the matrices that represent them.

7

1. Use an algebraic method to find the square roots of the complex number \(5 + 12 \mathrm { i }\). You must show sufficient working to justify your answers.
2. Hence solve the quadratic equation \(x ^ { 2 } - 4 x - 1 - 12 \mathrm { i } = 0\).

8

1. Show that \(\frac { 3 } { r - 1 } - \frac { 2 } { r } - \frac { 1 } { r + 1 } \equiv \frac { 4 r + 2 } { r \left( r ^ { 2 } - 1 \right) }\).
2. Hence find an expression, in terms of \(n\), for \(\sum _ { r = 2 } ^ { n } \frac { 4 r + 2 } { r \left( r ^ { 2 } - 1 \right) }\).
3. Hence find the value of \(\sum _ { r = 4 } ^ { \infty } \frac { 4 r + 2 } { r \left( r ^ { 2 } - 1 \right) }\).

9 The matrix \(\mathbf { D }\) is given by \(\mathbf { D } = \left( \begin{array} { l l l } 1 & 3 & 4
2 & a & 3
0 & 1 & a \end{array} \right)\).

1. Find the values of \(a\) for which \(\mathbf { D }\) is singular.
2. Three simultaneous equations are shown below. $$\begin{array} { r } x + 3 y + 4 z = 3
   2 x + a y + 3 z = 2
   y + a z = 0 \end{array}$$ For each of the following values of \(a\), determine whether or not there is a unique solution. If a unique solution does not exist, determine whether the equations are consistent or inconsistent.
   (a) \(a = 3\)
   (b) \(a = 1\)

10 The cubic equation \(x ^ { 3 } + 4 x + 3 = 0\) has roots \(\alpha , \beta\) and \(\gamma\).

1. Use the substitution \(x = \sqrt { u }\) to obtain a cubic equation in \(u\).
2. Find the value of \(\alpha ^ { 4 } + \beta ^ { 4 } + \gamma ^ { 4 } + \alpha \beta \gamma\).