The numbers \(a, b, c, d\) have the property that there exist \(x_1, x_2\), not both zero, such that \begin{align} ax_1 + bx_2 &= 0,\\ cx_1 + dx_2 &= 0. \end{align} Show that there exist numbers \(y_1, y_2\), not both zero, such that \begin{align} ay_1 + cy_2 &= 0,\\ by_1 + dy_2 &= 0. \end{align} [If you use any result about determinants, you should prove it.]
(i) Using integration by parts, or otherwise, show that \begin{align} \int_{0}^{\pi/2} \cos 2x \ln (\textrm{cosec} x) \, dx = \frac{\pi}{4}. \end{align} (ii) Evaluate the integral \begin{align} \int_{0}^{\pi/2} \sin 2x \ln (\textrm{cosec} x) \, dx. \end{align} [You may assume that \(u \ln u \to 0\) as \(u \to 0\).]
The variables \(x\) and \(y\) satisfy the differential equations \begin{align} \frac{dx}{dt} &= 2x + y + e^t,\\ \frac{dy}{dt} &= x + 2y. \end{align} Solve these equations subject to the initial conditions \(x(0) = 0, y(0) = 1\). [You may find it helpful to set \(z = x + \lambda y\) and find the two values of \(\lambda\) such that \(z\) satisfies a first order differential equation which does not explicitly involve \(x\) or \(y\).]
Let \(N = p_1^{a_1} \cdots p_r^{a_r}\), where \(p_1, \ldots, p_r\) are distinct primes and \(a_1, \ldots, a_r\) are positive integers. Find an expression for the number of divisors of \(N\) (including 1 and \(N\)) and show that the sum of these divisors is \begin{align} \prod_{i=1}^{r} \frac{(p_i^{a_i+1}-1)}{(p_i-1)}. \end{align}
Prove that the set of matrices of the type $\begin{pmatrix} 1 & 0 & 0 \\ x & 1 & 0 \\ y & z & 1 \end{pmatrix}$ with \(x, y, z\) real numbers, forms a group \(G\) under matrix multiplication. [You may assume that matrix multiplication is associative.] Does the subset consisting of those matrices where \(x, y, z\) are restricted to be integers form a subgroup of \(G\)? Is there an element \(a\) in \(G\), with \(a\) not equal to the identity matrix, such that \(ab = ba\) for all \(b\) belonging to \(G\)? Justify your answers.
Prove that \begin{align} (X + Y + Z)(X + \omega Y + \omega^2 Z)(X + \omega^2 Y + \omega Z) = X^3 + Y^3 + Z^3 - 3XYZ, \end{align} where \(\omega = e^{2\pi i/3}\). Hence, or otherwise, find the roots of the equation \begin{align} x^3 - 6x + 6 = 0. \end{align}
A parabola is given by \(x = at^2 + b, y = ct + d\) where \(a\) and \(c\) are not zero. Find the equation of the tangent at the point \(t\). Show that all the points of intersection of pairs of perpendicular tangents lie on the same straight line.
\(P, A, B, C\) are distinct points in three-dimensional Euclidean space, and \(L, M, N\) are the midpoints of \(BC, CA, AB\) respectively. Prove that the lines through \(L, M, N\) parallel to \(PA, PB, PC\) respectively meet in a point.
Find \(a, b\) such that the function \(f(x) = \frac{(ax + b)}{(x - 1)(x - 4)}\) has a stationary value at \(x = 2\) with \(f(2) = -1\). Show that \(f(x)\) has a maximum at \(x = 2\), and sketch the curve.
Let \(I\) be the integral \begin{align} I = \int_{0}^{\pi/2} \ln (\sin x) \, dx. \end{align} Show, by means of changes of variable, that \begin{align} I &= \int_{0}^{\pi/2} \ln (\cos y) \, dy = \int_{\pi/2}^{\pi} \ln (\sin z) \, dz \end{align} By considering \(\int_{0}^{\pi/2} \ln (\sin 2x) \, dx\), or otherwise, prove that \begin{align} I = -\frac{\pi}{2} \ln 2 \end{align} [You may assume that all these integrals converge.]