Pre-U Pre-U 9795/1 (Pre-U Further Mathematics Paper 1) 2011 June

Given that the matrix \(\mathbf{A} = \begin{pmatrix} 2 & k \\ 1 & -3 \end{pmatrix}\), where \(k\) is real, is such that \(\mathbf{A}^3 = \mathbf{I}\), find the value of \(k\) and the numerical value of \(\det \mathbf{A}\). [4]

The cubic equation \(x^3 + x^2 + 7x - 1 = 0\) has roots \(\alpha\), \(\beta\) and \(\gamma\).

1. Show that \(\alpha^2 + \beta^2 + \gamma^2 = -13\). [3]
2. State what can be deduced about the nature of these roots. [2]

1. Express \(\text{f}(r - 1) - \text{f}(r)\) as a single algebraic fraction, where \(\text{f}(r) = \frac{1}{(2r + 1)^2}\). [1]
2. Hence, using the method of differences, show that $$\sum_{r=1}^{n} \frac{r}{(4r^2 - 1)^2} = \frac{n(n + 1)}{2(2n + 1)^2}$$ for all positive integers \(n\). [4]

1. On a single diagram, sketch the graphs of \(y = \tanh x\) and \(y = \cosh x - 1\), and use your diagram to explain why the equation \(\text{f}(x) = 0\) has exactly two roots, where $$\text{f}(x) = 1 + \tanh x - \cosh x.$$ [3]
2. The non-zero root of \(\text{f}(x) = 0\) is \(\alpha\).
   1. Verify that \(1 < \alpha < 1.5\). [1]
   2. Taking \(x_1 = 1.25\) as an initial approximation to \(\alpha\), use the Newton-Raphson iterative method to find \(x_3\), giving your answer to 5 decimal places. [4]

Find the general solution of the differential equation \(\frac{d^2 y}{dx^2} + y = 8x^2\). [7]

Consider the set \(S\) of all matrices of the form \(\begin{pmatrix} p & p \\ p & p \end{pmatrix}\), where \(p\) is a non-zero rational number.

1. Show that \(S\), under the operation of matrix multiplication, forms a group, \(G\). [5]
2. Find a subgroup of \(G\) of order 2 and show that \(G\) contains no subgroups of order 3. [4]

Sketch the curve with equation \(y = \frac{x^2 + 4x}{2x - 1}\), justifying all significant features. [11]

1. Determine the two values of \(k\) for which the system of equations \begin{align} x + 2y + 3z &= 4
   2x + 3y + kz &= 9
   x + ky + 6z &= 1 \end{align} has no unique solution. [3]
2. Show that the system is consistent for one of these values of \(k\) and inconsistent for the other. [4]

1. The points \(A\), \(B\) and \(C\) have position vectors $$\mathbf{a} = \begin{pmatrix} 19 \\ 3 \\ 10 \end{pmatrix}, \quad \mathbf{b} = \begin{pmatrix} 12 \\ 7 \\ -1 \end{pmatrix} \quad \text{and} \quad \mathbf{c} = \begin{pmatrix} 5 \\ 15 \\ 3 \end{pmatrix}$$ respectively, and \(O\) is the origin. Calculate the volume of the tetrahedron \(OABC\). [3]
2. 1. The plane \(\Pi_1\) has equation \(\mathbf{r} = \begin{pmatrix} 2 \\ 1 \\ 4 \end{pmatrix} + \lambda \begin{pmatrix} 3 \\ 1 \\ -1 \end{pmatrix} + \mu \begin{pmatrix} 6 \\ 2 \\ 5 \end{pmatrix}\). Determine an equation for \(\Pi_1\) in the form \(\mathbf{r} \cdot \mathbf{n} = d\). [4]
   2. A second plane, \(\Pi_2\), has equation \(\mathbf{r} \cdot \begin{pmatrix} 1 \\ 4 \\ 7 \end{pmatrix} = 13\). Find a vector equation for the line of intersection of \(\Pi_1\) and \(\Pi_2\). [4]

1. Use de Moivre's theorem to show that \(\cos 3\theta = 4\cos^3 \theta - 3\cos \theta\). [2]
2. The sequence \(\{u_n\}\) is such that \(u_0 = 1\), \(u_1 = \cos \theta\) and, for \(n \geqslant 1\), $$u_{n+1} = (2\cos \theta)u_n - u_{n-1}.$$
   1. Determine \(u_2\) and \(u_3\) in terms of powers of \(\cos \theta\) only. [2]
   2. Suggest a simple expression for \(u_n\), the \(n\)th term of the sequence, and prove it for all positive integers \(n\) using induction. [6]

1. Let \(I_n = \int_0^{\frac{\pi}{2}} \sec^n t \, dt\) for positive integers \(n\). Prove that, for \(n \geqslant 2\), $$(n - 1)I_n = \frac{2^{n-2}}{(\sqrt{3})^{n-1}} + (n - 2)I_{n-2}.$$ [5]
2. The curve with parametric equations \(x = \tan t\), \(y = \frac{1}{4}\sec^2 t\), for \(0 \leqslant t \leqslant \frac{1}{4}\pi\), is rotated through \(2\pi\) radians about the \(x\)-axis to form a surface of revolution of area \(S\). Show that \(S = \pi I_5\) and evaluate \(S\) exactly. [10]

The complex number \(z_1\) is such that \(z_1 = a + ib\), where \(a\) and \(b\) are positive real numbers.

1. Given that \(z_1^2 = 2 + 2i\), show that \(a = \sqrt{\sqrt{2} + 1}\) and find the exact value of \(b\) in a similar form. [5]

The complex number \(z_2\) is such that \(z_2 = -a + ib\).

1. 1. Determine \(\arg z_2\) as a rational multiple of \(\pi\). [You may use the result \(\tan(\frac{1}{8}\pi) = \sqrt{2} - 1\).] [2]
   2. The point \(P_n\) in an Argand diagram represents the complex number \(z_2^n\), for positive integers \(n\). Find the least value of \(n\) for which \(P_n\) lies on the half-line with equation $$\arg(z) = \frac{1}{4}\pi.$$ [3]

1. 1. Given that \(t = \tan x\), prove that \(\frac{2}{2 - \sin 2x} = \frac{1 + t^2}{1 - t + t^2}\). [2]
   2. Hence determine the value of the constant \(k\) for which $$\frac{d}{dx}\left\{\tan^{-1}\left(\frac{1 - 2\tan x}{\sqrt{3}}\right)\right\} = \frac{k}{2 - \sin 2x}.$$ [4]
2. The curve \(C\) has cartesian equation \(x^2 - xy + y^2 = 72\).
   1. Determine a polar equation for \(C\) in the form \(r^2 = f(\theta)\), and deduce the polar coordinates \((r, \theta)\), where \(0 \leqslant \theta < 2\pi\), of the points on \(C\) which are furthest from the pole \(O\). [7]
   2. Find the exact area of the region of the plane in the first quadrant bounded by \(C\), the \(x\)-axis and the line \(y = x\). Deduce the total area of the region of the plane which lies inside \(C\) and within the first quadrant. [5]