OCR FP3 (Further Pure Mathematics 3) 2009 January

1 In this question \(G\) is a group of order \(n\), where \(3 \leqslant n < 8\).

1. In each case, write down the smallest possible value of \(n\) :
   (a) if \(G\) is cyclic,
   (b) if \(G\) has a proper subgroup of order 3,
   (c) if \(G\) has at least two elements of order 2 .
2. Another group has the same order as \(G\), but is not isomorphic to \(G\). Write down the possible value(s) of \(n\).

2

1. Express \(\frac { \sqrt { 3 } + \mathrm { i } } { \sqrt { 3 } - \mathrm { i } }\) in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\), where \(r > 0\) and \(0 \leqslant \theta < 2 \pi\).
2. Hence find the smallest positive value of \(n\) for which \(\left( \frac { \sqrt { 3 } + \mathrm { i } } { \sqrt { 3 } - \mathrm { i } } \right) ^ { n }\) is real and positive.

3 Two skew lines have equations $$\frac { x } { 2 } = \frac { y + 3 } { 1 } = \frac { z - 6 } { 3 } \quad \text { and } \quad \frac { x - 5 } { 3 } = \frac { y + 1 } { 1 } = \frac { z - 7 } { 5 } .$$

1. Find the direction of the common perpendicular to the lines.
2. Find the shortest distance between the lines.

4 Find the general solution of the differential equation $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 4 \frac { \mathrm {~d} y } { \mathrm {~d} x } + 5 y = 65 \sin 2 x$$

5 The variables \(x\) and \(y\) are related by the differential equation $$x ^ { 3 } \frac { \mathrm {~d} y } { \mathrm {~d} x } = x y + x + 1 .$$

1. Use the substitution \(y = u - \frac { 1 } { x }\), where \(u\) is a function of \(x\), to show that the differential equation may be written as $$x ^ { 2 } \frac { \mathrm {~d} u } { \mathrm {~d} x } = u .$$
2. Hence find the general solution of the differential equation (A), giving your answer in the form \(y = \mathrm { f } ( x )\).

6
\includegraphics[max width=\textwidth, alt={}, center]{bc975428-c594-427b-a32e-268412b3cd26-3_554_825_264_660} The cuboid \(O A B C D E F G\) shown in the diagram has \(\overrightarrow { O A } = 4 \mathbf { i } , \overrightarrow { O C } = 2 \mathbf { j } , \overrightarrow { O D } = 3 \mathbf { k }\), and \(M\) is the mid-point of \(G F\).

1. Find the equation of the plane \(A C G E\), giving your answer in the form r.n \(= p\).
2. The plane \(O E F C\) has equation \(\mathbf { r } \cdot ( 3 \mathbf { i } - 4 \mathbf { k } ) = 0\). Find the acute angle between the planes \(O E F C\) and \(A C G E\).
3. The line \(A M\) meets the plane \(O E F C\) at the point \(W\). Find the ratio \(A W : W M\).

7

1. The operation \(*\) is defined by \(x * y = x + y - a\), where \(x\) and \(y\) are real numbers and \(a\) is a real constant.
   (a) Prove that the set of real numbers, together with the operation \(*\), forms a group.
   (b) State, with a reason, whether the group is commutative.
   (c) Prove that there are no elements of order 2.
2. The operation \(\circ\) is defined by \(x \circ y = x + y - 5\), where \(x\) and \(y\) are positive real numbers. By giving a numerical example in each case, show that two of the basic group properties are not necessarily satisfied.

8

1. By expressing \(\sin \theta\) in terms of \(\mathrm { e } ^ { \mathrm { i } \theta }\) and \(\mathrm { e } ^ { - \mathrm { i } \theta }\), show that $$\sin ^ { 6 } \theta \equiv - \frac { 1 } { 32 } ( \cos 6 \theta - 6 \cos 4 \theta + 15 \cos 2 \theta - 10 )$$
2. Replace \(\theta\) by ( \(\frac { 1 } { 2 } \pi - \theta\) ) in the identity in part (i) to obtain a similar identity for \(\cos ^ { 6 } \theta\).
3. Hence find the exact value of \(\int _ { 0 } ^ { \frac { 1 } { 4 } \pi } \left( \sin ^ { 6 } \theta - \cos ^ { 6 } \theta \right) \mathrm { d } \theta\).