OCR FP3 (Further Pure Mathematics 3) 2015 June

1 Find the general solution of the differential equation $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 4 \frac { \mathrm {~d} y } { \mathrm {~d} x } + 13 y = \sin x$$

2 The elements of a group \(G\) are polynomials of the form \(a + b x + c x ^ { 2 }\), where \(a , b , c \in \{ 0,1,2,3,4 \}\). The group operation is addition, where the coefficients are added modulo 5 .

1. State the identity element.
2. State the inverse of \(3 + 2 x + x ^ { 2 }\).
3. State the order of \(G\). The proper subgroup \(H\) contains \(2 + x\) and \(1 + x\).
4. Find the order of \(H\), justifying your answer.

3 The plane \(\Pi\) passes through the points \(( 1,2,1 ) , ( 2,3,6 )\) and \(( 4 , - 1,2 )\).

1. Find a cartesian equation of the plane \(\Pi\). The line \(l\) has equation \(\mathbf { r } = \left( \begin{array} { r } - 1
   - 2
   6 \end{array} \right) + \lambda \left( \begin{array} { r } 4
   3
   - 2 \end{array} \right)\).
2. Find the coordinates of the point of intersection of \(\Pi\) and \(l\).
3. Find the acute angle between \(\Pi\) and \(l\).

4 In an Argand diagram, the complex numbers \(0 , z\) and \(z \mathrm { e } ^ { \frac { 1 } { 6 } \mathrm { i } \pi }\) are represented by the points \(O , A\) and \(B\) respectively.

1. Sketch a possible Argand diagram showing the triangle \(O A B\). Show that the triangle is isosceles and state the size of angle \(A O B\). The complex numbers \(1 + \mathrm { i }\) and \(5 + 2 \mathrm { i }\) are represented by the points \(C\) and \(D\) respectively. The complex number \(w\) is represented by the point \(E\), such that \(C D = C E\) and angle \(D C E = \frac { 1 } { 6 } \pi\).
2. Calculate the possible values of \(w\), giving your answers exactly in the form \(a + b \mathrm { i }\).

5 Find the particular solution of the differential equation $$x \frac { \mathrm {~d} y } { \mathrm {~d} x } + 3 y = x ^ { 2 } + x$$ for which \(y = 1\) when \(x = 1\), giving \(y\) in terms of \(x\).

6 Find the shortest distance between the lines with equations $$\frac { x - 1 } { 2 } = \frac { y + 2 } { 3 } = \frac { z - 5 } { - 1 } \quad \text { and } \quad \frac { x - 3 } { 4 } = \frac { y - 1 } { - 2 } = \frac { z + 1 } { 3 } .$$

7

1. Use de Moivre's theorem to show that \(\tan 4 \theta \equiv \frac { 4 \tan \theta - 4 \tan ^ { 3 } \theta } { 1 - 6 \tan ^ { 2 } \theta + \tan ^ { 4 } \theta }\).
2. Hence find the exact roots of \(t ^ { 4 } + 4 \sqrt { 3 } t ^ { 3 } - 6 t ^ { 2 } - 4 \sqrt { 3 } t + 1 = 0\).

8 Let \(G\) be any multiplicative group. \(H\) is a subset of \(G\). \(H\) consists of all elements \(h\) such that \(h g = g h\) for every element \(g\) in \(G\).

1. Prove that \(H\) is a subgroup of \(G\). Now consider the case where \(G\) is given by the following table:

   	\(e\)	\(p\)	\(q\)	\(r\)	\(s\)	\(t\)
   \(e\)	\(e\)	\(p\)	\(q\)	\(r\)	\(s\)	\(t\)
   \(p\)	\(p\)	\(q\)	\(e\)	\(s\)	\(t\)	\(r\)
   \(q\)	\(q\)	\(e\)	\(p\)	\(t\)	\(r\)	\(s\)
   \(r\)	\(r\)	\(t\)	\(s\)	\(e\)	\(q\)	\(p\)
   \(s\)	\(s\)	\(r\)	\(t\)	\(p\)	\(e\)	\(q\)
   \(t\)	\(t\)	\(s\)	\(r\)	\(q\)	\(p\)	\(e\)

2. Show that \(H\) consists of just the identity element.