If \(l, m, l', m', l''\) and \(m''\) are integers, and if \(\alpha/\beta\) is not rational, and if \[ l\alpha + m\beta = l'\alpha + m'\beta, \] shew that \(l=l'\), and \(m=m'\). Also shew that no two of the numbers \[ (2l+5m)\alpha+l\beta, \quad (2l'+5m'+1)\alpha+l'\beta, \quad (2l''+5m'')\alpha+(l''+1)\beta \] are equal.
Shew that, if \(n\) is a positive integer, the number of solutions of the equation \[ n = 2n_1 + 3n_2, \] where \(n_1\) and \(n_2\) are positive integers or zero, is equal to \(N\) or \(N+1\) according as \(r\) is or is not equal to 1; here \(N\) denotes the quotient and \(r\) the remainder when \(n\) is divided by 6.
Shew that \[ \sum_{m=0}^{N} \frac{\cos m\phi}{\cos^m \theta} = \frac{\cos^2 \theta - \cos\theta\cos\phi}{\cos^2\theta-2\cos\theta\cos\phi+1} - \frac{\cos(N+1)\phi}{\cos^{N-1}\theta} + \frac{\cos N\phi}{\cos^N\theta}. \] Indicate how the value of \(\sum_{m=0}^{N} \frac{m\sin(m-1)\phi}{\cos^{m+1}\phi}\) could be found from the above equation.
(i) If \(I\) be the in-centre and \(O\) the circumcentre of a triangle \(ABC\), shew that \[ OI^2 = R^2 - 2Rr, \] where \(r\) is the radius of the in-circle, and \(R\) is the radius of the circumcircle. (ii) If \(H\) be the orthocentre of the triangle \(ABC\), shew that \[ OH^2 = R^2 - 8R^2\cos A \cos B \cos C. \]
Express \(bc+ca+ab\) and \(abc\) in terms of \(s, p\) and \(q\), where \[ 2s=a+b+c, \quad 2p=a^2+b^2+c^2, \quad 3q=a^3+b^3+c^3. \] In a triangle the sum of the lengths of the sides, \(a, b\) and \(c\), is constant; shew that the small variation, \(\delta R\), in the radius of the circumcircle when the triangle is slightly changed, is given by \[ 24\Delta^2 \delta R = s[3s(3q-2sp)\delta p + (2s^3-3q)\delta q] \] approximately, where \(\Delta\) is the area of the triangle.
(i) If \[ \frac{d^2y}{dx^2} + \frac{1}{x}\frac{dy}{dx} + y = 0, \] shew that \[ x^2\frac{d^3y}{dy^3} - \left(\frac{dx}{dy}\right)^2 - xy\left(\frac{dx}{dy}\right)^3 = 0. \] (ii) If \[ \frac{d^2y}{dx^2} + \left(\frac{l}{x-a}+\frac{m}{x-b}+\frac{n}{x-c}\right)\frac{dy}{dx} = 0, \] where \(l+m+n=2\), shew that \[ \frac{d^2y}{dX^2} + \left(\frac{l}{X-A}+\frac{m}{X-B}+\frac{n}{X-C}\right)\frac{dY}{dX} = 0, \] where \[ x = \frac{\alpha X + \beta}{\gamma X + \delta}, \quad a = \frac{\alpha A + \beta}{\gamma A + \delta} \text{ etc., and } \alpha\delta - \beta\gamma = 1. \]
Assuming that \(\pi[ab-h^2]^{-\frac{1}{2}}\) is the area of the ellipse \(ax^2+2hxy+by^2=1\), shew that the minimum area of an ellipse which passes through the points \((\pm p, 0), (q, r)\) and \((-q, -r)\) is equal to \(\pi pr\).
Shew that if \(m\) and \(n\) are integers \[ \int_0^{\frac{\pi}{2}} \sin^n\theta \cos^m\theta d\theta \] is decreased when either \(m\) or \(n\) is increased. Hence shew that \[ \frac{(n!)^2 2^{2n}}{(2n+1)(2n-1)^2 \dots 3^2} < \frac{\pi}{2} < \frac{(n+1)(n!)^2 2^{2n+1}}{(2n+1)^2 \dots 3^2}. \]
Shew that \[ \int_0^{\frac{\pi}{4}} \sec^3 x dx = \frac{1}{2}\sqrt{2} + \frac{1}{2}\log(1+\sqrt{2}). \] Evaluate \[ \int_0^1 \frac{dx}{(1+x)^2(2x+1)}. \]
Give a rough sketch of the curve \(y^2 = x^5(a-x)(b-x)\), where \(0 < a < b\). Shew that if \(a/b\) is small, the area of the loop is \(\frac{\pi a^4 \sqrt{b}}{2^8} \left(20 - \frac{7a}{b}\right)\) approximately.