Pre-U Pre-U 9795/1 (Pre-U Further Mathematics Paper 1) 2013 November

For real values of \(t\), the non-singular matrices \(\mathbf{A}\) and \(\mathbf{B}\) are such that $$\mathbf{A}^{-1} = \begin{pmatrix} t & 5 \\ 2 & 8 \end{pmatrix} \quad \text{and} \quad \mathbf{B}^{-1} = \begin{pmatrix} 2 & -t \\ 3 & -1 \end{pmatrix}.$$

1. Determine the values which \(t\) cannot take. [2]
2. Without finding either \(\mathbf{A}\) or \(\mathbf{B}\), determine \((\mathbf{AB})^{-1}\) in terms of \(t\). [2]

Use de Moivre's theorem to express \(\cos 3\theta\) in terms of powers of \(\cos \theta\) only, and deduce the identity \(\cos 6x \equiv \cos 2x(2\cos 4x - 1)\). [5]

The curve \(C\) has equation \(y = \frac{2x}{x^2 + 1}\).

1. Write down the equation of the asymptote of \(C\) and the coordinates of any points where \(C\) meets the coordinate axes. [2]
2. Show that the curve meets the line \(y = k\) if and only if \(-1 \leqslant k \leqslant 1\). Deduce the coordinates of the turning points of the curve. [5]

[Note: You are NOT required to sketch \(C\).]

Let \(f(n) = 2(5^{n-1} + 1)\) for integers \(n = 1, 2, 3, \ldots\).

1. Prove that, if \(f(n)\) is divisible by 8, then \(f(n + 1)\) is also divisible by 8. [3]
2. Explain why this result does not imply that the statement '\(f(n)\) is divisible by 8 for all positive integers \(n\)' follows by mathematical induction. [1]

The curve \(S\) has polar equation \(r = 1 + \sin \theta + \sin^2 \theta\) for \(0 \leqslant \theta < 2\pi\).

1. Determine the polar coordinates of the points on \(S\) where \(\frac{dr}{d\theta} = 0\). [5]
2. Sketch \(S\). [3]

\(G\) is the set \(\{2, 4, 6, 8\}\), \(H\) is the set \(\{1, 5, 7, 11\}\) and \(\times_n\) denotes the operation of multiplication modulo \(n\).

1. Construct the multiplication tables for \((G, \times_{10})\) and \((H, \times_{12})\). [2]
2. By verifying the four group axioms, show that \(G\) and \(H\) are groups under their respective binary operations, and determine whether \(G\) and \(H\) are isomorphic. [6]

[You may assume that \(\times_n\) is associative.]

Relative to an origin \(O\), the points \(P\), \(Q\) and \(R\) have position vectors $$\mathbf{p} = \mathbf{i} + 2\mathbf{j} - 7\mathbf{k}, \quad \mathbf{q} = -3\mathbf{i} + 4\mathbf{j} + \mathbf{k} \quad \text{and} \quad \mathbf{r} = 6\mathbf{i} + 4\mathbf{j} + \alpha\mathbf{k}$$ respectively.

1. Determine \(\mathbf{p} \times \mathbf{q}\). [2]
2. Deduce the value of \(\alpha\) for which
   1. \(OR\) is normal to the plane \(OPQ\), [1]
   2. the volume of tetrahedron \(OPQR\) is 50, [3]
   3. \(R\) lies in the plane \(OPQ\). [2]

1. Determine \(x\) and \(y\) given that the complex number \(z = x + \text{i}y\) simultaneously satisfies $$|z - 1| = 1 \quad \text{and} \quad \arg(z + 1) = \frac{1}{6}\pi.$$ [4]
2. On an Argand diagram, shade the region whose points satisfy $$1 \leqslant |z - 1| \leqslant 2 \quad \text{and} \quad \frac{1}{6}\pi \leqslant \arg(z + 1) \leqslant \frac{1}{4}\pi.$$ [6]

1. Show that there is exactly one value of \(k\) for which the system of equations \begin{align} kx + 2y + kz &= 4
   3x + 10y + 2z &= m
   (k - 1)x - 4y + z &= k \end{align} does not have a unique solution. [4]
2. Given that the system of equations is consistent for this value of \(k\), find the value of \(m\). [4]
3. Explain the geometrical significance of a non-unique solution to a \(3 \times 3\) system of linear equations. [2]

The roots of the equation \(x^4 - 2x^3 + 2x^2 + x - 3 = 0\) are \(\alpha\), \(\beta\), \(\gamma\) and \(\delta\). Determine the values of

1. \(\alpha^2 + \beta^2 + \gamma^2 + \delta^2\), [2]
2. \(\frac{1}{\alpha} + \frac{1}{\beta} + \frac{1}{\gamma} + \frac{1}{\delta}\), [2]
3. \(\alpha^3 + \beta^3 + \gamma^3 + \delta^3\). [4]

1. Given that \(y = -4\) when \(x = 0\) and that $$\frac{dy}{dx} - y = e^{2x} + 3,$$ find the value of \(x\) for which \(y = 0\). [7]
2. Find the general solution of $$\frac{d^2y}{dx^2} - 4\frac{dy}{dx} + 4y = e^{2x} + 3,$$ given that \(y = cx^2e^{2x} + d\) is a suitable form of particular integral. [7]

1. 1. Use the method of differences to prove that $$\sum_{n=k}^N \frac{1}{n(n+1)} = \frac{1}{k} - \frac{1}{N+1}.$$ [4]
   2. Deduce the value of \(\sum_{n=k}^{\infty} \frac{1}{n(n+1)}\) and show that \(\sum_{n=k}^{\infty} \frac{1}{(n+1)^2} < \frac{1}{k}\). [3]
2. Let \(S = \sum_{n=1}^{\infty} \frac{1}{n^2}\). Show that \(\frac{205}{144} < S < \frac{241}{144}\). [3]

1. Let \(I_n = \int_0^{\alpha} \cosh^n x \, dx\) for integers \(n \geqslant 0\), where \(\alpha = \ln 2\).
   1. Prove that, for \(n \geqslant 2\), \(nI_n = \frac{3 \times 5^{n-1}}{4^n} + (n-1)I_{n-2}\). [5]
   2. A curve has parametric equations \(x = 12 \sinh t + 4 \sinh^3 t\), \(y = 3 \cosh^4 t\), \(0 \leqslant t \leqslant \ln 2\). Find the length of the arc of this curve, giving your answer in the form \(a + b \ln 2\) for rational numbers \(a\) and \(b\). [8]
2. The circle with equation \(x^2 + (y - R)^2 = r^2\), where \(r < R\), is rotated through one revolution about the \(x\)-axis to form a solid of revolution called a torus. By using suitable parametric equations for the circle, determine, in terms of \(\pi\), \(R\) and \(r\), the surface area of this torus. [11]