OCR FP3 (Further Pure Mathematics 3) 2010 January

Determine whether the lines $$\frac{x-1}{-1} = \frac{y+2}{2} = \frac{z+4}{2} \quad \text{and} \quad \frac{x+3}{2} = \frac{y-1}{3} = \frac{z-5}{4}$$ intersect or are skew. [5]

\(H\) 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 \(H\). [2] 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. [1]
3. Find the inverse of \(a + b\sqrt{5}\). [2]
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. [1]

Use the integrating factor method to find the solution of the differential equation $$\frac{\text{d}y}{\text{d}x} + 2y = \text{e}^{-3x}$$ for which \(y = 1\) when \(x = 0\). Express your answer in the form \(y = \text{f}(x)\). [6]

1. Write down, in cartesian form, the roots of the equation \(z^4 = 16\). [2]
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 $$A\left(0, 0, \frac{2}{\sqrt{3}}\sqrt{6}\right), \quad B\left(\frac{2}{\sqrt{3}}\sqrt{3}, 0, 0\right), \quad C\left(-\frac{1}{3}\sqrt{3}, 1, 0\right), \quad D\left(-\frac{1}{3}\sqrt{3}, -1, 0\right).$$

1. Obtain the equation of the face \(ABC\) in the form $$x + \sqrt{3}y + \left(\frac{1}{2}\sqrt{2}\right)z = \frac{2}{3}\sqrt{3}.$$ [5] (Answers which only verify the given equation will not receive full credit.)
2. Give a geometrical reason why the equation of the face \(ABD\) can be expressed as $$x - \sqrt{3}y + \left(\frac{1}{2}\sqrt{2}\right)z = \frac{2}{3}\sqrt{3}.$$ [2]
3. Hence find the cosine of the angle between two faces of the tetrahedron. [4]

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

1. Find the complementary function of the differential equation. [2]
2. Given that there is a particular integral of the form \(y = px\sin 4x\), where \(p\) is a constant, find the general solution of the equation. [6]
3. Find the solution of the equation for which \(y = 2\) and \(\frac{\text{d}y}{\text{d}x} = 0\) when \(x = 0\). [4]

1. Solve the equation \(\cos 6\theta = 0\), for \(0 < \theta < \pi\). [3]
2. By using de Moivre's theorem, show that $$\cos 6\theta \equiv (2\cos^2\theta - 1)(16\cos^4\theta - 16\cos^2\theta + 1).$$ [5]
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. [5]

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

1. Show that \(\text{g}(x) = \frac{1-x}{1-2x}\) and that \(\text{gg}(x) = x\). [4]

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 \(\text{e} : x \mapsto x\) for \(x \in \mathbb{R}, x \neq 0, x \neq \frac{1}{2}, x \neq 1\).

1. State the orders of the elements f and g. [2]
2. The inverse of the element f is denoted by h. Find \(\text{h}(x)\). [2]
3. Construct the operation table for the elements e, f, g, h of the group \(K\). [4]