OCR FP3 (Further Pure Mathematics 3) 2012 January

1 The variables \(x\) and \(y\) are related by the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 2 x ^ { 2 } + y ^ { 2 } } { x y } .$$

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

2

1. Show that \(\left( z ^ { n } - \mathrm { e } ^ { \mathrm { i } \theta } \right) \left( z ^ { n } - \mathrm { e } ^ { - \mathrm { i } \theta } \right) \equiv z ^ { 2 n } - ( 2 \cos \theta ) z ^ { n } + 1\).
2. Express \(z ^ { 4 } - z ^ { 2 } + 1\) as the product of four factors of the form \(\left( z - e ^ { \mathrm { i } \alpha } \right)\), where \(0 \leqslant \alpha < 2 \pi\).

3 A multiplicative group contains the distinct elements \(e , x\) and \(y\), where \(e\) is the identity.

1. Prove that \(x ^ { - 1 } y ^ { - 1 } = ( y x ) ^ { - 1 }\).
2. Given that \(x ^ { n } y ^ { n } = ( x y ) ^ { n }\) for some integer \(n \geqslant 2\), prove that \(x ^ { n - 1 } y ^ { n - 1 } = ( y x ) ^ { n - 1 }\).
3. If \(x ^ { n - 1 } y ^ { n - 1 } = ( y x ) ^ { n - 1 }\), does it follow that \(x ^ { n } y ^ { n } = ( x y ) ^ { n }\) ? Give a reason for your answer.

4 The line \(l\) has equations \(\frac { x - 1 } { 2 } = \frac { y - 1 } { 3 } = \frac { z + 1 } { 2 }\) and the point \(A\) is ( \(7,3,7\) ). \(M\) is the point where the perpendicular from \(A\) meets \(l\).

1. Find, in either order, the coordinates of \(M\) and the perpendicular distance from \(A\) to \(l\).
2. Find the coordinates of the point \(B\) on \(A M\) such that \(\overrightarrow { A B } = 3 \overrightarrow { B M }\).

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

1. Find the complementary function of the differential equation.
2. Given that there is a particular integral of the form \(y = p x \mathrm { e } ^ { - 2 x }\), find the constant \(p\).
3. Find the solution of the equation for which \(y = 0\) and \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 4\) when \(x = 0\).

6 The plane \(\Pi\) has equation \(\mathbf { r } = \left( \begin{array} { l } 1
6
7 \end{array} \right) + \lambda \left( \begin{array} { r } 2
- 1
- 1 \end{array} \right) + \mu \left( \begin{array} { r } 2
- 3
- 5 \end{array} \right)\) and the line \(l\) has equation \(\mathbf { r } = \left( \begin{array} { l } 7
4
1 \end{array} \right) + t \left( \begin{array} { r } 3
0
- 1 \end{array} \right)\).

1. Express the equation of \(\Pi\) in the form r.n \(= p\).
2. Find the point of intersection of \(l\) and \(\Pi\).
3. The equation of \(\Pi\) may be expressed in the form \(\mathbf { r } = \left( \begin{array} { l } 1
   6
   7 \end{array} \right) + \lambda \left( \begin{array} { r } 2
   - 1
   - 1 \end{array} \right) + \mu \mathbf { c }\), where \(\mathbf { c }\) is perpendicular to \(\left( \begin{array} { r } 2
   - 1
   - 1 \end{array} \right)\). Find \(\mathbf { c }\).

7 The set \(M\) consists of the six matrices \(\left( \begin{array} { l l } 1 & 0
n & 1 \end{array} \right)\), where \(n \in \{ 0,1,2,3,4,5 \}\). It is given that \(M\) forms a group ( \(M , \times\) ) under matrix multiplication, with numerical addition and multiplication both being carried out modulo 6 .

1. Determine whether ( \(M , \times\) ) is a commutative group, justifying your answer.
2. Write down the identity element of the group and find the inverse of \(\left( \begin{array} { l l } 1 & 0
   2 & 1 \end{array} \right)\).
3. State the order of \(\left( \begin{array} { l l } 1 & 0
   3 & 1 \end{array} \right)\) and give a reason why \(( M , \times )\) has no subgroup of order 4.
4. The multiplicative group \(G\) has order 6. All the elements of \(G\), apart from the identity, have order 2 or 3 . Determine whether \(G\) is isomorphic to ( \(M , \times\) ), justifying your answer.

8

1. Use de Moivre's theorem to prove that $$\tan 5 \theta \equiv \frac { 5 \tan \theta - 10 \tan ^ { 3 } \theta + \tan ^ { 5 } \theta } { 1 - 10 \tan ^ { 2 } \theta + 5 \tan ^ { 4 } \theta } .$$
2. Solve the equation \(\tan 5 \theta = 1\), for \(0 \leqslant \theta < \pi\).
3. Show that the roots of the equation $$t ^ { 4 } - 4 t ^ { 3 } - 14 t ^ { 2 } - 4 t + 1 = 0$$ may be expressed in the form \(\tan \alpha\), stating the exact values of \(\alpha\), where \(0 \leqslant \alpha < \pi\). \section*{THERE ARE NO QUESTIONS WRITTEN ON THIS PAGE}