OCR FP1 (Further Pure Mathematics 1) 2013 June

1 The complex number \(3 + a \mathrm { i }\), where \(a\) is real, is denoted by \(z\). Given that \(\arg z = \frac { 1 } { 6 } \pi\), find the value of \(a\) and hence find \(| z |\) and \(z ^ { * } - 3\).

2 The matrices \(\mathbf { A } , \mathbf { B }\) and \(\mathbf { C }\) are given by \(\mathbf { A } = \left( \begin{array} { l l } 5 & 1 \end{array} \right) , \mathbf { B } = \left( \begin{array} { l l } 2 & - 5 \end{array} \right)\) and \(\mathbf { C } = \binom { 3 } { 2 }\).

1. Find \(3 \mathbf { A } - 4 \mathbf { B }\).
2. Find CB. Determine whether \(\mathbf { C B }\) is singular or non-singular, giving a reason for your answer.

3 Use an algebraic method to find the square roots of \(11 + ( 12 \sqrt { 5 } ) \mathrm { i }\). Give your answers in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are exact real numbers.

4 The matrix \(\mathbf { M }\) is given by \(\mathbf { M } = \left( \begin{array} { l l } 2 & 2
0 & 1 \end{array} \right)\). Prove by induction that, for \(n \geqslant 1\), $$\mathbf { M } ^ { n } = \left( \begin{array} { c c } 2 ^ { n } & 2 ^ { n + 1 } - 2
0 & 1 \end{array} \right) .$$

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

6
\includegraphics[max width=\textwidth, alt={}, center]{2ba2e0bf-d20a-41ab-a77c-86a08e700b40-2_885_803_1425_630} The Argand diagram above shows a half-line \(l\) and a circle \(C\). The circle has centre 3 i and passes through the origin.

1. Write down, in complex number form, the equations of \(l\) and \(C\).
   [0pt]
2. Write down inequalities that define the region shaded in the diagram. [The shaded region includes the boundaries.]

7

1. Find the matrix that represents a rotation through \(90 ^ { \circ }\) clockwise about the origin.
2. Find the matrix that represents a reflection in the \(x\)-axis.
3. Hence find the matrix that represents a rotation through \(90 ^ { \circ }\) clockwise about the origin, followed by a reflection in the \(x\)-axis.
4. Describe a single transformation that is represented by your answer to part (iii).

8 The cubic equation \(k x ^ { 3 } + 6 x ^ { 2 } + x - 3 = 0\), where \(k\) is a non-zero constant, has roots \(\alpha , \beta\) and \(\gamma\).
Find the value of \(( \alpha + 1 ) ( \beta + 1 ) + ( \beta + 1 ) ( \gamma + 1 ) + ( \gamma + 1 ) ( \alpha + 1 )\) in terms of \(k\).

9

1. Show that \(\frac { 1 } { 3 r - 1 } - \frac { 1 } { 3 r + 2 } \equiv \frac { 3 } { ( 3 r - 1 ) ( 3 r + 2 ) }\).
2. Hence show that \(\sum _ { r = 1 } ^ { 2 n } \frac { 1 } { ( 3 r - 1 ) ( 3 r + 2 ) } = \frac { n } { 2 ( 3 n + 1 ) }\).

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

1. Find the value of \(a\) for which \(\mathbf { A }\) is singular.
2. Given that \(\mathbf { A }\) is non-singular, find \(\mathbf { A } ^ { - 1 }\) and hence solve the equations $$\begin{aligned} a x + 2 y + z & = 1
   x + 3 y + 2 z & = 2
   4 x + y + z & = 3 \end{aligned}$$