OCR FP3 (Further Pure Mathematics 3) 2010 June

The line \(l_1\) passes through the points \((0, 0, 10)\) and \((7, 0, 0)\) and the line \(l_2\) passes through the points \((4, 6, 0)\) and \((3, 3, 1)\). Find the shortest distance between \(l_1\) and \(l_2\). [7]

A multiplicative group with identity \(e\) contains distinct elements \(a\) and \(r\), with the properties \(r^6 = e\) and \(ar = r^2a\).

1. Prove that \(rar = a\). [2]
2. Prove, by induction or otherwise, that \(r^n ar^n = a\) for all positive integers \(n\). [4]

In this question, \(w\) denotes the complex number \(\cos \frac{2\pi}{5} + i \sin \frac{2\pi}{5}\).

1. Express \(w^2\), \(w^3\) and \(w^4\) in polar form, with arguments in the interval \(0 \leq \theta < 2\pi\). [4]
2. The points in an Argand diagram which represent the numbers $$1, \quad 1 + w, \quad 1 + w + w^2, \quad 1 + w + w^2 + w^3, \quad 1 + w + w^2 + w^3 + w^4$$ are denoted by \(A\), \(B\), \(C\), \(D\), \(E\) respectively. Sketch the Argand diagram to show these points and join them in the order stated. (Your diagram need not be exactly to scale, but it should show the important features.) [4]
3. Write down a polynomial equation of degree 5 which is satisfied by \(w\). [1]

1. Use the substitution \(y = xz\) to find the general solution of the differential equation $$x \frac{dy}{dx} - y = x \cos \left(\frac{y}{x}\right),$$ giving your answer in a form without logarithms. (You may quote an appropriate result given in the List of Formulae (MF1).) [6]
2. Find the solution of the differential equation for which \(y = \pi\) when \(x = 4\). [2]

Convergent infinite series \(C\) and \(S\) are defined by \begin{align} C &= 1 + \frac{1}{4} \cos \theta + \frac{1}{4} \cos 2\theta + \frac{1}{8} \cos 3\theta + \ldots,
S &= \frac{1}{2} \sin \theta + \frac{1}{4} \sin 2\theta + \frac{1}{8} \sin 3\theta + \ldots. \end{align}

1. Show that \(C + iS = \frac{2}{2 - e^{i\theta}}\). [4]
2. Hence show that \(C = \frac{4 - 2\cos \theta}{5 - 4\cos \theta}\) and find a similar expression for \(S\). [4]

1. Find the general solution of the differential equation $$\frac{d^2y}{dx^2} + 2 \frac{dy}{dx} + 17y = 17x + 36.$$ [7]
2. Show that, when \(x\) is large and positive, the solution approximates to a linear function, and state its equation. [2]

A line \(l\) has equation \(\mathbf{r} = \begin{pmatrix} -7 \\ -3 \\ 0 \end{pmatrix} + \lambda \begin{pmatrix} 2 \\ -2 \\ 3 \end{pmatrix}\). A plane \(\Pi\) passes through the points \((1, 3, 5)\) and \((5, 2, 5)\), and is parallel to \(l\).

1. Find an equation of \(\Pi\), giving your answer in the form \(\mathbf{r} \cdot \mathbf{n} = p\). [4]
2. Find the distance between \(l\) and \(\Pi\). [4]
3. Find an equation of the line which is the reflection of \(l\) in \(\Pi\), giving your answer in the form \(\mathbf{r} = \mathbf{a} + t\mathbf{b}\). [4]

A set of matrices \(M\) is defined by $$A = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}, \quad B = \begin{pmatrix} \omega & 0 \\ 0 & \omega^2 \end{pmatrix}, \quad C = \begin{pmatrix} \omega^2 & 0 \\ 0 & \omega \end{pmatrix}, \quad D = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}, \quad E = \begin{pmatrix} 0 & \omega^2 \\ \omega & 0 \end{pmatrix}, \quad F = \begin{pmatrix} 0 & \omega \\ \omega^2 & 0 \end{pmatrix},$$ where \(\omega\) and \(\omega^2\) are the complex cube roots of 1. It is given that \(M\) is a group under matrix multiplication.

1. Write down the elements of a subgroup of order 2. [1]
2. Explain why there is no element \(X\) of the group, other than \(A\), which satisfies the equation \(X^2 = A\). [2]
3. By finding \(BE\) and \(EB\), verify the closure property for the pair of elements \(B\) and \(E\). [4]
4. Find the inverses of \(B\) and \(E\). [3]
5. Determine whether the group \(M\) is isomorphic to the group \(N\) which is defined as the set of numbers \(\{1, 2, 4, 8, 7, 5\}\) under multiplication modulo 9. Justify your answer clearly. [3]