OCR FP3 (Further Pure Mathematics 3) 2011 January

1. Find the general solution of the differential equation $$\frac{dy}{dx} + xy = xe^{\frac{x^2}{2}},$$ giving your answer in the form \(y = f(x)\). [4]
2. Find the particular solution for which \(y = 1\) when \(x = 0\). [2]

Two intersecting lines, lying in a plane \(p\), have equations $$\frac{x-1}{2} = \frac{y-3}{1} = \frac{z-4}{-3} \quad \text{and} \quad \frac{x-1}{-1} = \frac{y-3}{2} = \frac{z-4}{4}.$$

1. Obtain the equation of \(p\) in the form \(2x - y + z = 3\). [3]
2. Plane \(q\) has equation \(2x - y + z = 21\). Find the distance between \(p\) and \(q\). [3]

1. Express \(\sin \theta\) in terms of \(e^{i\theta}\) and \(e^{-i\theta}\) and show that $$\sin^4 \theta \equiv \frac{1}{8}(\cos 4\theta - 4\cos 2\theta + 3).$$ [4]
2. Hence find the exact value of \(\int_0^{\frac{\pi}{4}} \sin^4 \theta \, d\theta\). [4]

The cube roots of 1 are denoted by \(1\), \(\omega\) and \(\omega^2\), where the imaginary part of \(\omega\) is positive.

1. Show that \(1 + \omega + \omega^2 = 0\). [2]

\includegraphics{figure_1} In the diagram, \(ABC\) is an equilateral triangle, labelled anticlockwise. The points \(A\), \(B\) and \(C\) represent the complex numbers \(z_1\), \(z_2\) and \(z_3\) respectively.

1. State the geometrical effect of multiplication by \(\omega\) and hence explain why \(z_1 - z_3 = \omega(z_3 - z_2)\). [4]
2. Hence show that \(z_1 + \omega z_2 + \omega^2 z_3 = 0\). [2]

1. Find the general solution of the differential equation $$3\frac{d^2y}{dx^2} + 5\frac{dy}{dx} - 2y = -2x + 13.$$ [7]
2. Find the particular solution for which \(y = -\frac{7}{2}\) and \(\frac{dy}{dx} = 0\) when \(x = 0\). [5]
3. Write down the function to which \(y\) approximates when \(x\) is large and positive. [1]

\(Q\) is a multiplicative group of order 12.

1. Two elements of \(Q\) are \(a\) and \(r\). It is given that \(r\) has order 6 and that \(a^2 = r^3\). Find the orders of the elements \(a\), \(a^2\), \(a^3\) and \(r^2\). [4]

The table below shows the number of elements of \(Q\) with each possible order. \(G\) and \(H\) are the non-cyclic groups of order 4 and 6 respectively.

1. Construct two tables, similar to the one above, to show the number of elements with each possible order for the groups \(G\) and \(H\). Hence explain why there are no non-cyclic proper subgroups of \(Q\). [5]

Three planes \(\Pi_1\), \(\Pi_2\) and \(\Pi_3\) have equations $$\mathbf{r} \cdot (\mathbf{i} + \mathbf{j} - 2\mathbf{k}) = 5, \quad \mathbf{r} \cdot (\mathbf{i} - \mathbf{j} + 3\mathbf{k}) = 6, \quad \mathbf{r} \cdot (\mathbf{i} + 5\mathbf{j} - 12\mathbf{k}) = 12,$$ respectively. Planes \(\Pi_1\) and \(\Pi_2\) intersect in a line \(l\); planes \(\Pi_2\) and \(\Pi_3\) intersect in a line \(m\).

1. Show that \(l\) and \(m\) are in the same direction. [5]
2. Write down what you can deduce about the line of intersection of planes \(\Pi_1\) and \(\Pi_3\). [1]
3. By considering the cartesian equations of \(\Pi_1\), \(\Pi_2\) and \(\Pi_3\), or otherwise, determine whether or not the three planes have a common line of intersection. [4]

The operation \(*\) is defined on the elements \((x, y)\), where \(x, y \in \mathbb{R}\), by $$(a, b) * (c, d) = (ac, ad + b).$$ It is given that the identity element is \((1, 0)\).

1. Prove that \(*\) is associative. [3]
2. Find all the elements which commute with \((1, 1)\). [3]
3. It is given that the particular element \((m, n)\) has an inverse denoted by \((p, q)\), where $$(m, n) * (p, q) = (p, q) * (m, n) = (1, 0).$$ Find \((p, q)\) in terms of \(m\) and \(n\). [2]
4. Find all self-inverse elements. [3]
5. Give a reason why the elements \((x, y)\), under the operation \(*\), do not form a group. [1]