OCR FP3 (Further Pure Mathematics 3) 2006 June

1. For the infinite group of non-zero complex numbers under multiplication, state the identity element and the inverse of \(1 + 2\mathrm{i}\), giving your answers in the form \(a + ib\). [3]
2. For the group of matrices of the form \(\begin{pmatrix} a & 0 \\ 0 & 0 \end{pmatrix}\) under matrix addition, where \(a \in \mathbb{R}\), state the identity element and the inverse of \(\begin{pmatrix} 3 & 0 \\ 0 & 0 \end{pmatrix}\). [2]

1. Given that \(z_1 = 2e^{\frac{5\pi i}{6}}\) and \(z_2 = 3e^{\frac{2\pi i}{3}}\), express \(z_1z_2\) and \(\frac{z_1}{z_2}\) in the form \(re^{i\theta}\), where \(r > 0\) and \(0 \leq \theta < 2\pi\). [4]
2. Given that \(w = 2(\cos \frac{1}{3}\pi + i \sin \frac{1}{3}\pi)\), express \(w^{-5}\) in the form \(r(\cos \theta + i \sin \theta)\), where \(r > 0\) and \(0 \leq \theta < 2\pi\). [3]

Find the perpendicular distance from the point with position vector \(12\mathbf{i} + 5\mathbf{j} + 3\mathbf{k}\) to the line with equation \(\mathbf{r} = \mathbf{i} + 2\mathbf{j} + 5\mathbf{k} + t(8\mathbf{i} + 3\mathbf{j} - 6\mathbf{k})\). [6]

Find the solution of the differential equation $$\frac{dy}{dx} - \frac{x^2y}{1 + x^3} = x^2$$ for which \(y = 1\) when \(x = 0\), expressing your answer in the form \(y = f(x)\). [8]

A line \(l_1\) has equation \(\frac{x}{2} = \frac{y + 4}{3} = \frac{z + 9}{5}\).

1. Find the cartesian equation of the plane which is parallel to \(l_1\) and which contains the points \((2, 1, 5)\) and \((0, -1, 5)\). [5]
2. Write down the position vector of a point on \(l_1\) with parameter \(t\). [1]
3. Hence, or otherwise, find an equation of the line \(l_2\) which intersects \(l_1\) at right angles and which passes through the point \((-5, 3, 4)\). Give your answer in the form \(\frac{x - a}{p} = \frac{y - b}{q} = \frac{z - c}{r}\). [4]

1. Find the general solution of the differential equation $$\frac{d^2y}{dx^2} + 4y = \sin x.$$ [6]
2. Find the solution of the differential equation for which \(y = 0\) and \(\frac{dy}{dx} = \frac{4}{3}\) when \(x = 0\). [4]

The series \(C\) and \(S\) are defined for \(0 < \theta < \pi\) by \begin{align} C &= 1 + \cos \theta + \cos 2\theta + \cos 3\theta + \cos 4\theta + \cos 5\theta,
S &= \sin \theta + \sin 2\theta + \sin 3\theta + \sin 4\theta + \sin 5\theta. \end{align}

1. Show that \(C + iS = \frac{e^{3i\theta} - e^{-3i\theta}}{e^{i\theta} - e^{-i\theta}} \cdot e^{i\theta}\). [4]
2. Deduce that \(C = \sin 3\theta \cos \frac{5}{2}\theta \operatorname{cosec} \frac{1}{2}\theta\) and write down the corresponding expression for \(S\). [4]
3. Hence find the values of \(\theta\), in the range \(0 < \theta < \pi\), for which \(C = S\). [4]

A group \(D\) of order 10 is generated by the elements \(a\) and \(r\), with the properties \(a^2 = e\), \(r^5 = e\) and \(r^4a = ar\), where \(e\) is the identity. Part of the operation table is shown below. \includegraphics{figure_1}

1. Give a reason why \(D\) is not commutative. [1]
2. Write down the orders of any possible proper subgroups of \(D\). [2]
3. List the elements of a proper subgroup which contains
   1. the element \(a\), [1]
   2. the element \(r\). [1]
4. Determine the order of each of the elements \(r^3\), \(ar\) and \(ar^2\). [4]
5. Copy and complete the section of the table marked E, showing the products of the elements \(ar\), \(ar^2\), \(ar^3\) and \(ar^4\). [5]