Solve the simultaneous equations: \begin{align*} x(y+z-x) &= a^2, \\ y(z+x-y) &= b^2, \\ z(x+y-z) &= c^2. \end{align*}
If one root of the equation \(x^3+ax+b=0\) is twice the difference of the other two, prove that one root is \(\frac{13b}{3a}\).
If \(p > q-2\), find the number of ways in which \(p\) positive signs and \(q\) negative signs can be placed in a row so that no two negative signs shall be together.
(i) Sum to \(n\) terms the series \[ \frac{1}{1.3.5} + \frac{2}{3.5.7} + \frac{3}{5.7.9} + \dots. \] (ii) Prove that \[ \frac{2^3}{2!} + \frac{3^3}{3!} + \frac{4^3}{4!} + \dots \text{ to infinity} = 5e-1. \]
If \(p\) is small, so that \(p^3\) is negligible, prove that an approximation to a solution of the equation \(x^{2+p}=a^2\) is \[ x = a - \frac{1}{2}ap\log_e a + \frac{1}{8}ap^2(2+\log_e a)\log_e a. \]
If \(\theta\) and \(\phi\) are unequal and less than \(180^{\circ}\), and if \[ (x-a)\cos 2\theta + y\sin 2\theta = (x-a)\cos 2\phi + y\sin 2\phi = a, \] and \[ \tan\theta - \tan\phi = 2e, \] prove that \[ y^2=2ax-(1-e^2)x^2. \]
\(APQB\) is a straight line, and the lengths of \(AQ, PB\) and \(AB\) are \(2a, 2b\) and \(2c\) respectively. Circles are described on diameters \(AQ, PB, AB\). Prove that the radius of a circle that touches these three circles is \[ \frac{c(c-a)(c-b)}{c^2-ab}. \]
A regular polygon of \(n\) sides is inscribed in a circle of radius \(a\), and from any point \(P\) on the circumference chords are drawn to the angular points; if these chords are of lengths \(c_1, c_2, c_3, \dots, c_n\), beginning with the chord drawn to the nearest angular point and taking the rest in order, find the value of \[ c_1c_2+c_2c_3+c_3c_4+\dots+c_{n-1}c_n+c_nc_1, \] and prove that it is the same for every point \(P\) on the circumference.
Sum to \(n\) terms the series
Prove that \[ \tan^{-1}\frac{\tan 2\alpha+\tanh 2\beta}{\tan 2\alpha-\tanh 2\beta} + \tan^{-1}\frac{\tan\alpha-\tanh\beta}{\tan\alpha+\tanh\beta} = \tan^{-1}(\cot\alpha\coth\beta). \]