OCR FP3 (Further Pure Mathematics 3) 2009 June

1 Find the cube roots of \(\frac { 1 } { 2 } \sqrt { 3 } + \frac { 1 } { 2 } \mathrm { i }\), giving your answers in the form \(\cos \theta + \mathrm { i } \sin \theta\), where \(0 \leqslant \theta < 2 \pi\).

2 It is given that the set of complex numbers of the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\) for \(- \pi < \theta \leqslant \pi\) and \(r > 0\), under multiplication, forms a group.

1. Write down the inverse of \(5 \mathrm { e } ^ { \frac { 1 } { 3 } \pi \mathrm { i } }\).
2. Prove the closure property for the group.
3. \(Z\) denotes the element \(\mathrm { e } ^ { \mathrm { i } \gamma }\), where \(\frac { 1 } { 2 } \pi < \gamma < \pi\). Express \(Z ^ { 2 }\) in the form \(\mathrm { e } ^ { \mathrm { i } \theta }\), where \(- \pi < \theta < 0\).

3 A line \(l\) has equation \(\frac { x - 6 } { - 4 } = \frac { y + 7 } { 8 } = \frac { z + 10 } { 7 }\) and a plane \(p\) has equation \(3 x - 4 y - 2 z = 8\).

1. Find the point of intersection of \(l\) and \(p\).
2. Find the equation of the plane which contains \(l\) and is perpendicular to \(p\), giving your answer in the form \(a x + b y + c z = d\).

4 The differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } + \frac { 1 } { 1 - x ^ { 2 } } y = ( 1 - x ) ^ { \frac { 1 } { 2 } } , \quad \text { where } | x | < 1$$ can be solved by the integrating factor method.

1. Use an appropriate result given in the List of Formulae (MF1) to show that the integrating factor can be written as \(\left( \frac { 1 + x } { 1 - x } \right) ^ { \frac { 1 } { 2 } }\).
2. Hence find the solution of the differential equation for which \(y = 2\) when \(x = 0\), giving your answer in the form \(y = \mathrm { f } ( x )\).

5 The variables \(x\) and \(y\) satisfy the differential equation $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } - 6 \frac { \mathrm {~d} y } { \mathrm {~d} x } + 9 y = \mathrm { e } ^ { 3 x }$$

1. Find the complementary function.
2. Explain briefly why there is no particular integral of either of the forms \(y = k \mathrm { e } ^ { 3 x }\) or \(y = k x \mathrm { e } ^ { 3 x }\).
3. Given that there is a particular integral of the form \(y = k x ^ { 2 } \mathrm { e } ^ { 3 x }\), find the value of \(k\).

6 The plane \(\Pi _ { 1 }\) has equation \(\mathbf { r } = \left( \begin{array} { l } 2
2
1 \end{array} \right) + \lambda \left( \begin{array} { l } 1
1
0 \end{array} \right) + \mu \left( \begin{array} { r } 1
- 5
- 2 \end{array} \right)\).

1. Express the equation of \(\Pi _ { 1 }\) in the form r.n \(= p\). The plane \(\Pi _ { 2 }\) has equation \(\mathbf { r } . \left( \begin{array} { r } 7
   17
   - 3 \end{array} \right) = 21\).
2. Find an equation of the line of intersection of \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\), giving your answer in the form \(\mathbf { r } = \mathbf { a } + t \mathbf { b }\).

7

1. Use de Moivre's theorem to prove that $$\tan 3 \theta \equiv \frac { \tan \theta \left( 3 - \tan ^ { 2 } \theta \right) } { 1 - 3 \tan ^ { 2 } \theta } .$$
2. (a) By putting \(\theta = \frac { 1 } { 12 } \pi\) in the identity in part (i), show that \(\tan \frac { 1 } { 12 } \pi\) is a solution of the equation $$t ^ { 3 } - 3 t ^ { 2 } - 3 t + 1 = 0 .$$ (b) Hence show that \(\tan \frac { 1 } { 12 } \pi = 2 - \sqrt { 3 }\).
3. Use the substitution \(t = \tan \theta\) to show that $$\int _ { 0 } ^ { 2 - \sqrt { 3 } } \frac { t \left( 3 - t ^ { 2 } \right) } { \left( 1 - 3 t ^ { 2 } \right) \left( 1 + t ^ { 2 } \right) } \mathrm { d } t = a \ln b$$ where \(a\) and \(b\) are positive constants to be determined.

8 A multiplicative group \(Q\) of order 8 has elements \(\left\{ e , p , p ^ { 2 } , p ^ { 3 } , a , a p , a p ^ { 2 } , a p ^ { 3 } \right\}\), where \(e\) is the identity. The elements have the properties \(p ^ { 4 } = e\) and \(a ^ { 2 } = p ^ { 2 } = ( a p ) ^ { 2 }\).

1. Prove that \(a = p a p\) and that \(p = a p a\).
2. Find the order of each of the elements \(p ^ { 2 } , a , a p , a p ^ { 2 }\).
3. Prove that \(\left\{ e , a , p ^ { 2 } , a p ^ { 2 } \right\}\) is a subgroup of \(Q\).
4. Determine whether \(Q\) is a commutative group.