OCR FP3 (Further Pure Mathematics 3) 2013 June

1 The plane \(\Pi\) passes through the points with coordinates \(( 1,6,2 ) , ( 5,2,1 )\) and \(( 1,0 , - 2 )\).

1. Find a vector equation of \(\Pi\) in the form \(\mathbf { r } = \mathbf { a } + \lambda \mathbf { b } + \mu \mathbf { c }\).
2. Find a cartesian equation of \(\Pi\).
   \(2 G\) consists of the set \(\{ 1,3,5,7 \}\) with the operation of multiplication modulo 8 .

1. Write down the operation table and, assuming associativity, show that \(G\) is a group.
2. State the order of each element.
3. Find all the proper subgroups of \(G\). The group \(H\) consists of the set \(\{ 1,3,7,9 \}\) with the operation of multiplication modulo 10 .
4. Explaining your reasoning, determine whether \(H\) is isomorphic to \(G\).

3 The differential equation $$3 x y ^ { 2 } \frac { \mathrm {~d} y } { \mathrm {~d} x } + 2 y ^ { 3 } = \frac { \cos x } { x }$$ is to be solved for \(x > 0\). Use the substitution \(u = y ^ { 3 }\) to find the general solution for \(y\) in terms of \(x\).

4 The complex numbers 0,3 and \(3 \mathrm { e } ^ { \frac { 1 } { 3 } \pi \mathrm { i } }\) are represented in an Argand diagram by the points \(O , A\) and \(B\) respectively.

1. Sketch the triangle \(O A B\) and show that it is equilateral.
2. Hence express \(3 - 3 e ^ { \frac { 1 } { 3 } \pi i }\) in polar form.
3. Hence find \(\left( 3 - 3 \mathrm { e } ^ { \frac { 1 } { 3 } \pi \mathrm { i } } \right) ^ { 5 }\), giving your answer in the form \(a + b \sqrt { 3 } \mathrm { i }\) where \(a\) and \(b\) are rational numbers.

5 Find the solution of the differential equation \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 2 \frac { \mathrm {~d} y } { \mathrm {~d} x } + 5 y = \mathrm { e } ^ { - x }\) for which \(y = \frac { \mathrm { d } y } { \mathrm {~d} x } = 0\) when \(x = 0\).

6 The plane \(\Pi\) has equation \(x + 2 y - 2 z = 5\). The line \(l\) has equation \(\frac { x - 1 } { 2 } = \frac { y + 1 } { 5 } = \frac { z - 2 } { 1 }\).

1. Find the coordinates of the point of intersection of \(l\) with the plane \(\Pi\).
2. Calculate the acute angle between \(l\) and \(\Pi\).
3. Find the coordinates of the two points on the line \(l\) such that the distance of each point from the plane \(\Pi\) is 2 .

7 A commutative group \(G\) has order 18. The elements \(a , b\) and \(c\) have orders 2, 3 and 9 respectively.

1. Prove that \(a b\) has order 6 .
2. Show that \(G\) is cyclic.

8

1. Use de Moivre's theorem to show that \(\cos 5 \theta \equiv 16 \cos ^ { 5 } \theta - 20 \cos ^ { 3 } \theta + 5 \cos \theta\).
2. Hence find the roots of \(16 x ^ { 4 } - 20 x ^ { 2 } + 5 = 0\) in the form \(\cos \alpha\) where \(0 \leqslant \alpha \leqslant \pi\).
3. Hence find the exact value of \(\cos \frac { 1 } { 10 } \pi\).