Show that, if \(p \neq 0\) and \(4p^3 + 27q^2 \neq 0\), the cubic polynomial \(x^3 + px + q\) can be expressed in the form \[ \frac{\alpha(x - \beta)^3 - \beta(x - \alpha)^3}{\alpha - \beta} \] where \(\alpha\) and \(\beta\) are certain constants. Hence, or otherwise, prove that, if \(4p^3 + 27q^2 \neq 0\), the cubic equation \(x^3 + px + q = 0\) has three unequal roots, which can be found by solving a quadratic equation and a cubic equation of the special type \(x^3 = a\).
Three collinear points \(A, B, C\) are given. Give a construction, making use of a ruler only, for the harmonic conjugate of \(C\) with respect to \(A\) and \(B\). Show how to construct, by using a ruler only, a line through a given point \(O\) to cut the sides \(YZ, ZX, XY\), of a given triangle in points \(P, Q, R\) respectively so that the range \((OPQR)\) may be harmonic.
If a rigid body is in equilibrium under the action of two coplanar couples, deduce from the triangle (or parallelogram) of forces that the sum of the moments of the couples is zero. Show that forces \(1, 2, 3, 4, -11, 6\) acting along the sides \(AB, BC, CD, DE, EF, FA\), respectively, of a regular hexagonal lamina have a resultant acting along the line joining the mid-points of the sides \(AB, CD\), and find its magnitude.
Give a short account of the method of generalisation by projection. Obtain the projective generalisations of concentric circles, right angle, rectangular hyperbola. Prove the following theorem. A fixed line through \(A\) meets at \(P\) a variable circle through two fixed points \(A, B\); the tangent at \(P\) to the circle envelops a parabola with focus \(B\). Generalise this theorem by projection. State the dual of the generalised theorem and obtain from it a result for a system of parabolas.
If \[ D_n = \begin{vmatrix} a & b & 0 & 0 & \dots & 0 & 0 \\ c & a & b & 0 & \dots & 0 & 0 \\ 0 & c & a & b & \dots & 0 & 0 \\ 0 & 0 & c & a & \dots & 0 & 0 \\ \vdots & \vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\ 0 & 0 & 0 & 0 & \dots & a & b \\ 0 & 0 & 0 & 0 & \dots & c & a \end{vmatrix}, \] where the determinant is of order \(n\), obtain a relation connecting \(D_n\), \(D_{n-1}\) and \(D_{n-2}\). Hence, or otherwise, prove (i) that, if \(a=2, b=c=1\), then \(D_n = n+1\); and (ii) that if \(a=b=c=1\), then \(D_n=1, 1, 0, -1, -1, 0\), according as \(n\) leaves the remainder \(0, 1, 2, 3, 4, 5\), when divided by 6.
Two points \(P\) and \(Q\) are inverse with respect to a circle \(\Sigma\). The inverses of \(\Sigma, P, Q\) with respect to another circle \(\Gamma\) are \(\Sigma', P', Q'\). Prove that \(P'\) and \(Q'\) are inverse with respect to \(\Sigma'\). Prove that the locus of the inverse points of a given point with respect to a system of coaxal circles is a circle which cuts the system orthogonally.
A segment of height \(\frac{1}{4}a\) is cut off by a plane from a uniform solid sphere of radius \(a\). If this segment can rest with its curved surface in contact with an inclined plane that is rough enough to prevent slipping, show that the angle between the plane and the horizontal cannot exceed \(\sin^{-1}(27/40)\).
A given conic has equation \(S=0\); the tangent at a fixed point \(P\) of the conic has equation \(t=0\). Write down equations representing the most general conics satisfying the following conditions:
By induction, or otherwise, prove the identity \[ \frac{(1-x^{n+1})(1-x^{n+2})(1-x^{n+3})\dots(1-x^{2n})}{(1-x)(1-x^3)(1-x^5)\dots(1-x^{2n-1})} = (1+x)(1+x^2)(1+x^3)\dots(1+x^n), \] where \(n\) is any positive integer. Prove that the expansions of \[ (1+x)(1+x^2)(1+x^3)\dots(1+x^n) \] and \[ (1-x)^{-1}(1-x^3)^{-1}(1-x^5)^{-1}\dots(1-x^{2n-1})^{-1} \] in ascending powers of \(x\) agree as far as the terms in \(x^n\). Hence, or otherwise, prove that the number of ways of expressing a positive integer \(n\) as a sum of one or more unequal positive integers is the same as the number of ways of expressing \(n\) as a sum of one or more odd (not necessarily unequal) positive integers.
A point \(P\) moves along a fixed line and \(O\) is a fixed point not on the line; find the envelope of the line through \(P\) perpendicular to \(OP\). A variable ellipse has a given focus and touches two given lines; prove that the envelope of its minor axis is a parabola.