Find the number of combinations of \(n\) letters \(r\) at a time (1) when they are all unlike, (2) when \(r\) of them are alike and the rest unlike. \par Shew that the number of combinations taken \(n\) together of \(3n\) letters, of which \(n\) are \(a\), \(n\) are \(b\), and the rest are unlike, is \(2^{n-1}(n+2)\).
Shew how to sum the series \(a_0+a_1x+\dots+a_nx^n+\dots\), whose coefficients satisfy the relation \(a_n+pa_{n-1}+qa_{n-2}=0\), \(p,q\) being given numbers. \par In the case where \(a_n-8a_{n-1}+7a_{n-2}=0\) and \(a_0=1, a_1=8\), shew that \[ a_n = \tfrac{1}{6}(7^{n+1}-1). \]
If \(p_n\) is the numerator of the \(n\)th convergent of \(a_1+\frac{1}{a_2+}\frac{1}{a_3+}\dots\), shew that \(p_n=a_np_{n-1}+p_{n-2}\). \par Prove that \[ p_nq_{n-4}-q_np_{n-4} = (-1)^{n-1}(a_na_{n-1}a_{n-2}+a_n+a_{n-2}). \]
Shew that if \(n\) can be found so that \(\frac{v_m}{u_m}\) is finite whenever \(m>n\), and the series \(u_1+u_2+\dots\) converges, then the series \(v_1+v_2+\dots\) converges. \par Examine the convergence of the series whose \(n\)th term is \(\frac{x^n}{n^2-x^{2n}}\) for all values of \(x\).
From a variable point \(P\) on a fixed line \(OX\), tangents \(PA, PB\) are drawn to a given circle; prove that \(\tan\angle XPA \cot\angle XPB\) is constant. \par Shew that all the lines for which this constant has the same value with respect to two given circles pass through one or other of two fixed points.
If \(A+B+C=\pi\), prove that
Express the radius \(R\) of the circumcircle of a triangle \(ABC\) in terms of the sides, and prove that if \(K\) is its centre and \(O\) is the orthocentre \[ OK^2 = R^2(1-8\cos A\cos B\cos C). \] Prove that if \(A>B>C\), and \(p,q,r\) are the lengths of the perpendiculars from \(A, B, C\) on \(OK\), then \(p-q+r=0\).
A mound on a level plane has the form of a portion of a sphere. At the bottom its surface has a slope \(\alpha\) and at a point distant \(a\) from the bottom the elevation of the highest visible point is \(\beta\). Shew that the height of the mound is \[ a\sin\beta\sin^2\frac{\alpha}{2}\operatorname{cosec}^2\frac{\alpha-\beta}{2}. \]
Give definitions of \(e^z, \sin z, \cos z\) where \(z\) is a complex number and verify that \[ \sin(z_1+z_2) = \sin z_1 \cos z_2 + \cos z_1 \sin z_2. \] Prove that \[ x+\frac{x^4}{4!}+\frac{x^7}{7!}+\dots = \frac{1}{3}e^x+\frac{2}{3}e^{-\frac{x}{2}}\sin\left(\frac{x\sqrt{3}}{2}-\frac{\pi}{6}\right), \] where \(x\) is real.
The distances of a point from the vertices of an equilateral triangle of unknown size are given. Show how the triangle may be constructed by making first a triangle with the lengths of its sides equal to the three given distances.