OCR FP3 (Further Pure Mathematics 3) 2014 June

1

1. Find a vector equation of the line of intersection of the planes \(2 x + y - z = 4\) and \(3 x + 5 y + 2 z = 13\).
2. Find the exact distance of the point \(( 2,5 , - 2 )\) from the plane \(2 x + y - z = 4\).

2 Use the substitution \(u = y ^ { 2 }\) to find the general solution of the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } - 2 y = \frac { \mathrm { e } ^ { x } } { y }$$ for \(y\) in terms of \(x\).

3

1. Solve the equation \(z ^ { 6 } = 1\), giving your answers in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\), and sketch an Argand diagram showing the positions of the roots.
2. Show that \(( 1 + \mathrm { i } ) ^ { 6 } = - 8 \mathrm { i }\).
3. Hence, or otherwise, solve the equation \(z ^ { 6 } + 8 \mathrm { i } = 0\), giving your answers in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\).

4 The group \(G\) consists of the set \(\{ 1,3,7,9,11,13,17,19 \}\) combined under multiplication modulo 20.

1. Find the inverse of each element.
2. Show that \(G\) is not cyclic.
3. Find two isomorphic subgroups of order 4 and state an isomorphism between them.

5 Solve the differential equation $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 5 \frac { \mathrm {~d} y } { \mathrm {~d} x } + 6 y = \mathrm { e } ^ { - x }$$ subject to the conditions \(y = \frac { \mathrm { d } y } { \mathrm {~d} x } = 0\) when \(x = 0\).

6 The line \(l\) has equations \(\frac { x - 1 } { 2 } = \frac { y + 2 } { 3 } = \frac { z - 7 } { 5 }\). The plane \(\Pi\) has equation \(4 x - y - z = 8\).

1. Show that \(l\) is parallel to \(\Pi\) but does not lie in \(\Pi\).
2. The point \(A ( 1 , - 2,7 )\) is on \(l\). Write down a vector equation of the line through \(A\) which is perpendicular to \(\Pi\). Hence find the position vector of the point on \(\Pi\) which is closest to \(A\).
3. Hence write down a vector equation of the line in \(\Pi\) which is parallel to \(l\) and closest to it.

7

1. By expressing \(\sin \theta\) in terms of \(\mathrm { e } ^ { \mathrm { i } \theta }\) and \(\mathrm { e } ^ { - \mathrm { i } \theta }\), show that $$\sin ^ { 5 } \theta \equiv \frac { 1 } { 16 } ( \sin 5 \theta - 5 \sin 3 \theta + 10 \sin \theta ) .$$
2. Hence solve the equation $$\sin 5 \theta + 4 \sin \theta = 5 \sin 3 \theta$$ for \(- \frac { 1 } { 2 } \pi \leqslant \theta \leqslant \frac { 1 } { 2 } \pi\). 8 consists of the set of matrices of the form \(\left( \begin{array} { c c } a & - b
   b & a \end{array} \right)\), where \(a\) and \(b\) are real and \(a ^ { 2 } + b ^ { 2 } \neq 0\), combined under the operation of matrix multiplication.
3. Prove that \(G\) is a group. You may assume that matrix multiplication is associative.
4. Determine whether \(G\) is commutative.
5. Find the order of \(\left( \begin{array} { c c } 0 & - 1
   1 & 0 \end{array} \right)\).