Prove that, if \(\alpha\) is small, one root of the equation \[ \alpha x^3 = x^2 - 1 \] is approximately \[ 1 + \tfrac{1}{2}\alpha + \tfrac{5}{8}\alpha^2, \] and find approximations to the other two roots.
Prove that the angle of intersection of two curves is unaltered by inversion. \(P\) is one of the points of intersection of two circles \(C_1, C_2\). Prove that the inverse of \(P\) with respect to any circle touching \(C_1\) and \(C_2\) lies on one or other of two fixed orthogonal circles.
If \(P\) and \(Q\) are two non-parallel coplanar forces and \(R\) is their resultant, show that
Prove that the reciprocal of a system of confocal conics with respect to one of the common foci \(S\) is a system of coaxal circles with \(S\) as a limiting point. What is the reciprocal of the second focus \(S'\)? Reciprocate the following theorem (i) with respect to the focus \(S\), (ii) with respect to the focus \(S'\). The tangents to a confocal system at the points where a fixed line through the common focus \(S\) meets them envelop a parabola touching the axis of the system which bisects \(SS'\). Give a direct proof of the original theorem or one of the two reciprocals.
Factorise the determinant \[ \begin{vmatrix} w^3 & w^2 & w & 1 \\ x^3 & x^2 & x & 1 \\ y^3 & y^2 & y & 1 \\ z^3 & z^2 & z & 1 \end{vmatrix}. \] Evaluate in factorised form the four determinants of the third order obtained from this by deleting the first row and each column in turn.
Three collinear points \(A, B, C\) are given. Give a construction, using a straight edge only, for the harmonic conjugate of \(C\) with respect to \(A\) and \(B\). Justify your construction. A circle is given and a point outside it. Give a construction, using a straight edge only, for the tangents from the point to the circle.
Two light struts \(OA, OB\), each 2 ft. long, are smoothly hinged together at \(O\), and their ends \(A, B\) are fixed by smooth hinges at points 2 ft. apart. A force of fixed magnitude \(F\) is applied at \(O\); if its line of action is in the plane \(OAB\) but otherwise variable, find the greatest thrust in either strut.
Tangents are drawn to the conic \[ S \equiv ax^2 + by^2 + cz^2 + 2fyz + 2gzx + 2hxy = 0 \] at the points where it is cut by the line \(lx+my+nz=0\). Prove that their equation is \[ (Al^2 + Bm^2 + Cn^2 + 2Fmn + 2Gnl + 2Hlm)S - \Delta(lx+my+nz)^2 = 0, \] where \(\Delta\) is the discriminant of \(S\), and \(A, B, C, F, G, H\) are the co-factors of the corresponding small letters in \(\Delta\). A line \(L\) meets a conic \(S\) at \(P, Q\) and another conic \(S'\) at \(P', Q'\). Prove that the four points at which the tangents to \(S\) at \(P\) and \(Q\) meet the tangents to \(S'\) at \(P'\) and \(Q'\) lie on a conic \(S''\) through the common points of \(S\) and \(S'\). Shew further that all lines \(L\) which touch a given conic touching the four common tangents of \(S\) and \(S'\) give the same conic \(S''\).
Find the sum of the first \(n\) terms of the series \[ \frac{1}{(1-x)(1-x^2)} + \frac{x}{(1-x^2)(1-x^3)} + \frac{x^2}{(1-x^3)(1-x^4)} + \dots. \] Deduce the sum to infinity in the cases \(|x| < 1\) and \(|x| > 1\). Hence, or otherwise, obtain the sum of the infinite series \[ \frac{1}{\sinh u \sinh 2u} + \frac{1}{\sinh 2u \sinh 3u} + \frac{1}{\sinh 3u \sinh 4u} + \dots, \] where \(u\) is real and different from zero.
A tetrahedron \(ABCD\) is such that \(AB=CD\), \(AC=BD\), \(AD=BC\). Prove that (i) the lengths of the perpendiculars from the vertices to the opposite faces are equal, (ii) if \(P\) is a point on \(AB\), the sum of the lengths of the perpendiculars from \(P\) to the edges \(AC, BD\) is the same for all positions of \(P\) between \(A\) and \(B\).