OCR FP3 (Further Pure Mathematics 3) 2010 January

1 Determine whether the lines $$\frac { x - 1 } { 1 } = \frac { y + 2 } { - 1 } = \frac { z + 4 } { 2 } \quad \text { and } \quad \frac { x + 3 } { 2 } = \frac { y - 1 } { 3 } = \frac { z - 5 } { 4 }$$ intersect or are skew.
\(2 \quad H\) denotes the set of numbers of the form \(a + b \sqrt { 5 }\), where \(a\) and \(b\) are rational. The numbers are combined under multiplication.

1. Show that the product of any two members of \(H\) is a member of \(H\). It is now given that, for \(a\) and \(b\) not both zero, \(H\) forms a group under multiplication.
2. State the identity element of the group.
3. Find the inverse of \(a + b \sqrt { 5 }\).
4. With reference to your answer to part (iii), state a property of the number 5 which ensures that every number in the group has an inverse.

3 Use the integrating factor method to find the solution of the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } + 2 y = \mathrm { e } ^ { - 3 x }$$ for which \(y = 1\) when \(x = 0\). Express your answer in the form \(y = \mathrm { f } ( x )\).

4

1. Write down, in cartesian form, the roots of the equation \(z ^ { 4 } = 16\).
2. Hence solve the equation \(w ^ { 4 } = 16 ( 1 - w ) ^ { 4 }\), giving your answers in cartesian form.

5 A regular tetrahedron has vertices at the points $$A \left( 0,0 , \frac { 2 } { 3 } \sqrt { 6 } \right) , \quad B \left( \frac { 2 } { 3 } \sqrt { 3 } , 0,0 \right) , \quad C \left( - \frac { 1 } { 3 } \sqrt { 3 } , 1,0 \right) , \quad D \left( - \frac { 1 } { 3 } \sqrt { 3 } , - 1,0 \right) .$$

1. Obtain the equation of the face \(A B C\) in the form $$x + \sqrt { 3 } y + \left( \frac { 1 } { 2 } \sqrt { 2 } \right) z = \frac { 2 } { 3 } \sqrt { 3 }$$ (Answers which only verify the given equation will not receive full credit.)
2. Give a geometrical reason why the equation of the face \(A B D\) can be expressed as $$x - \sqrt { 3 } y + \left( \frac { 1 } { 2 } \sqrt { 2 } \right) z = \frac { 2 } { 3 } \sqrt { 3 }$$
3. Hence find the cosine of the angle between two faces of the tetrahedron.

6 The variables \(x\) and \(y\) satisfy the differential equation $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 16 y = 8 \cos 4 x$$

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

7

1. Solve the equation \(\cos 6 \theta = 0\), for \(0 < \theta < \pi\).
2. By using de Moivre's theorem, show that $$\cos 6 \theta \equiv \left( 2 \cos ^ { 2 } \theta - 1 \right) \left( 16 \cos ^ { 4 } \theta - 16 \cos ^ { 2 } \theta + 1 \right)$$
3. Hence find the exact value of $$\cos \left( \frac { 1 } { 12 } \pi \right) \cos \left( \frac { 5 } { 12 } \pi \right) \cos \left( \frac { 7 } { 12 } \pi \right) \cos \left( \frac { 11 } { 12 } \pi \right)$$ justifying your answer.

8 The function f is defined by \(\mathrm { f } : x \mapsto \frac { 1 } { 2 - 2 x }\) for \(x \in \mathbb { R } , x \neq 0 , x \neq \frac { 1 } { 2 } , x \neq 1\). The function g is defined by \(\mathrm { g } ( x ) = \mathrm { ff } ( x )\).

1. Show that \(\mathrm { g } ( x ) = \frac { 1 - x } { 1 - 2 x }\) and that \(\operatorname { gg } ( x ) = x\). It is given that f and g are elements of a group \(K\) under the operation of composition of functions. The element e is the identity, where e : \(x \mapsto x\) for \(x \in \mathbb { R } , x \neq 0 , x \neq \frac { 1 } { 2 } , x \neq 1\).
2. State the orders of the elements f and g .
3. The inverse of the element f is denoted by h . Find \(\mathrm { h } ( x )\).
4. Construct the operation table for the elements e, f, g, h of the group \(K\).