OCR FP3 (Further Pure Mathematics 3) 2016 June

1 In this question, give all non-real numbers in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\) where \(r > 0\) and \(0 < \theta < 2 \pi\).

1. Solve \(z ^ { 5 } = 1\).
2. Hence, or otherwise, solve \(z ^ { 5 } + 32 = 0\). Sketch an Argand diagram showing the roots.

2 Find the shortest distance between the lines \(\mathbf { r } = \left( \begin{array} { l } 2
1
0 \end{array} \right) + \lambda \left( \begin{array} { c } 1
2
- 1 \end{array} \right)\) and \(\mathbf { r } = \left( \begin{array} { c } - 1
1
2 \end{array} \right) + \mu \left( \begin{array} { l } 3
0
1 \end{array} \right)\).

3 The differential equation $$\frac { 2 } { y } - \frac { x } { y ^ { 2 } } \frac { \mathrm {~d} y } { \mathrm {~d} x } = \frac { 1 } { x ^ { 2 } }$$ is to be solved subject to the condition \(y = 1\) when \(x = 1\).

1. Show that \(y = \frac { 1 } { u }\) transforms the differential equation into $$x \frac { \mathrm {~d} u } { \mathrm {~d} x } + 2 u = \frac { 1 } { x ^ { 2 } } .$$
2. Find \(y\) in terms of \(x\).

4 Let \(A\) be the set of non-zero integers.

1. Show that \(A\) does not form a group under multiplication.
2. State the largest subset of \(A\) which forms a group under multiplication. Show that this is a group.

5 Find the general solution of the differential equation $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 2 \frac { \mathrm {~d} y } { \mathrm {~d} x } + 10 y = 85 \cos x .$$

6 The planes \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\) have equations $$\mathbf { r } \cdot \left( \begin{array} { l } 1
2
1 \end{array} \right) = 3 \text { and } \mathbf { r } \cdot \left( \begin{array} { l } 2
1
4 \end{array} \right) = 5$$ respectively. They intersect in the line \(l\).

1. Find cartesian equations of \(l\). The plane \(\Pi _ { 3 }\) has equation \(\mathbf { r } . \left( \begin{array} { c } 1
   5
   - 1 \end{array} \right) = 1\).
2. Show that \(\Pi _ { 3 }\) is parallel to \(l\) but does not contain it.
3. Verify that \(( 2,0,1 )\) lies on planes \(\Pi _ { 1 }\) and \(\Pi _ { 3 }\). Hence write down a vector equation of the line of intersection of these planes.

1. Use de Moivre's theorem to show that $$\sin 6 \theta \equiv \cos \theta \left( 6 \sin \theta - 32 \sin ^ { 3 } \theta + 32 \sin ^ { 5 } \theta \right)$$
2. Hence show that, for \(\sin 2 \theta \neq 0\), $$- 1 \leqslant \frac { \sin 6 \theta } { \sin 2 \theta } < 3$$

8 A non-commutative multiplicative group \(G\) of order eight has the elements $$\left\{ e , a , a ^ { 2 } , a ^ { 3 } , b , a b , a ^ { 2 } b , a ^ { 3 } b \right\}$$ where \(e\) is the identity and \(a ^ { 4 } = b ^ { 2 } = e\).

1. Show that \(b a \neq a ^ { n }\) for any integer \(n\).
2. Prove, by contradiction, that \(b a \neq a ^ { 2 } b\) and also that \(b a \neq a b\). Deduce that \(b a = a ^ { 3 } b\).
3. Prove that \(b a ^ { 2 } = a ^ { 2 } b\).
4. Construct group tables for the three subgroups of \(G\) of order four. \section*{END OF QUESTION PAPER}