OCR FP3 (Further Pure Mathematics 3) 2013 January

1 Two planes have equations $$x + 2 y + 5 z = 12 \text { and } 2 x - y + 3 z = 5$$

1. Find the acute angle between the planes.
2. Find a vector equation of the line of intersection of the planes.

2 The elements of a group \(G\) are the complex numbers \(a + b \mathrm { i }\) where \(a , b \in \{ 0,1,2,3,4 \}\). These elements are combined under the operation of addition modulo 5 .

1. State the identity element and the order of \(G\).
2. Write down the inverse of \(2 + 4 \mathrm { i }\).
3. Show that every non-zero element of \(G\) has order 5 .

3 Solve the differential equation \(x \frac { \mathrm {~d} y } { \mathrm {~d} x } - 3 y = x ^ { 4 } \mathrm { e } ^ { 2 x }\) for \(y\) in terms of \(x\), given that \(y = 0\) when \(x = 1\).

4 The lines \(l _ { 1 }\) and \(l _ { 2 }\) have equations $$\mathbf { r } = \left( \begin{array} { l } 1
2
1 \end{array} \right) + \lambda \left( \begin{array} { r } 2
3
- 1 \end{array} \right) \text { and } \mathbf { r } = \left( \begin{array} { l } 3
0
1 \end{array} \right) + \mu \left( \begin{array} { r } 4
- 1
- 1 \end{array} \right)$$ respectively.

1. Find the shortest distance between the lines.
2. Find a cartesian equation of the plane which contains \(l _ { 1 }\) and which is parallel to \(l _ { 2 }\).

5

1. Solve the equation \(z ^ { 5 } = 1\), giving your answers in polar form.
2. Hence, by considering the equation \(( z + 1 ) ^ { 5 } = z ^ { 5 }\), show that the roots of $$5 z ^ { 4 } + 10 z ^ { 3 } + 10 z ^ { 2 } + 5 z + 1 = 0$$ can be expressed in the form \(\frac { 1 } { \mathrm { e } ^ { \mathrm { i } \theta } - 1 }\), stating the values of \(\theta\).

6 The differential equation \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 4 y = \sin k x\) is to be solved, where \(k\) is a constant.

1. In the case \(k = 2\), by using a particular integral of the form \(a x \cos 2 x + b x \sin 2 x\), find the general solution.
2. Describe briefly the behaviour of \(y\) when \(x \rightarrow \infty\).
3. In the case \(k \neq 2\), explain whether \(y\) would exhibit the same behaviour as in part (ii) when \(x \rightarrow \infty\).

7 Let \(S = \mathrm { e } ^ { \mathrm { i } \theta } + \mathrm { e } ^ { 2 \mathrm { i } \theta } + \mathrm { e } ^ { 3 \mathrm { i } \theta } + \ldots + \mathrm { e } ^ { 10 \mathrm { i } \theta }\).

1. (a) Show that, for \(\theta \neq 2 n \pi\), where \(n\) is an integer, $$S = \frac { \mathrm { e } ^ { \frac { 1 } { 2 } \mathrm { i } \theta } \left( \mathrm { e } ^ { 10 \mathrm { i } \theta } - 1 \right) } { 2 \mathrm { i } \sin \left( \frac { 1 } { 2 } \theta \right) }$$ (b) State the value of \(S\) for \(\theta = 2 n \pi\), where \(n\) is an integer.
2. Hence show that, for \(\theta \neq 2 n \pi\), where \(n\) is an integer, $$\cos \theta + \cos 2 \theta + \cos 3 \theta + \ldots + \cos 10 \theta = \frac { \sin \left( \frac { 21 } { 2 } \theta \right) } { 2 \sin \left( \frac { 1 } { 2 } \theta \right) } - \frac { 1 } { 2 }$$
3. Hence show that \(\theta = \frac { 1 } { 11 } \pi\) is a root of \(\cos \theta + \cos 2 \theta + \cos 3 \theta + \ldots + \cos 10 \theta = 0\) and find another root in the interval \(0 < \theta < \frac { 1 } { 4 } \pi\).

8 A multiplicative group \(H\) has the elements \(\left\{ e , a , a ^ { 2 } , a ^ { 3 } , w , a w , a ^ { 2 } w , a ^ { 3 } w \right\}\) where \(e\) is the identity, elements \(a\) and \(w\) have orders 4 and 2 respectively and \(w a = a ^ { 3 } w\).

1. Show that \(w a ^ { 2 } = a ^ { 2 } w\) and also that \(w a ^ { 3 } = a w\).
2. Hence show that each of \(a w , a ^ { 2 } w\) and \(a ^ { 3 } w\) has order 2 .
3. Find two non-cyclic subgroups of \(H\) of order 4, and show that they are not cyclic.