State (without proof) Descartes' rule of signs connecting the number of positive roots of an algebraic equation with the signs of the coefficients, and deduce a similar rule for the number of negative roots. \par Find the numbers of positive and negative roots of the equations \begin{align*} x^4 - x^3 + x^2 - 1 &= 0, \\ x^8 - x^3 - x^2 + 1 &= 0. \end{align*}
The points \(D, E, F\) lie on the sides \(BC, CA, AB\) respectively of a triangle \(ABC\). Prove that a necessary and sufficient condition for \(D, E, F\) to be collinear is that \[ \frac{BD}{DC} \frac{CE}{EA} \frac{AF}{FB} = -1. \] If \[ \frac{BD}{DC} = -\frac{m}{n}, \quad \frac{CE}{EA} = -\frac{n}{l}, \quad \frac{AF}{FB} = -\frac{l}{m}, \] prove that \[ \frac{EF}{l(m-n)} = \frac{FD}{m(n-l)} = \frac{DE}{n(l-m)}. \]
A uniform string of weight \(w\) per unit length hangs freely under gravity, with its two ends fastened to fixed supports. Show that the difference in the tension at any two points is \(w\) times the corresponding difference in the heights of the points. \par Prove that the same result is true for a string in contact with a smooth cylinder of any form of section; the string lies in a vertical plane perpendicular to the generators, which are horizontal.
Show how to perform any three of the following constructions, using a ruler only. Justify your constructions.
Express \[ y = \frac{4}{(1-x)^2(1-x^2)} \] in partial fractions. Show that, when \(x=0\), the value of \[ \frac{1}{n!} \frac{d^n y}{dx^n} \] is equal to \((n+2)^2\) when \(n\) is even, and \((n+1)(n+3)\) when \(n\) is odd.
Two coplanar circles \(S, T\) have radii 9 and 2 units and their centres are 5 units apart. By inverting \(S\) and \(T\) into concentric circles or otherwise, prove that it is possible to draw six circles in the area between \(S\) and \(T\), each of the six circles touching \(S\) and \(T\) and its two neighbours.
A rough plane is inclined at an angle \(\alpha\) to the horizontal. One end of a light rod is pivoted to a point of the plane, and to the other end of the rod is fastened a mass \(M\) which rests on the plane; the coefficient of friction between \(M\) and the plane is \(\mu\), and the friction between the rod and the plane is negligible. If the rod makes an acute angle \(\beta\) with a line of greatest slope of the plane (both directions being measured up the plane), show that the least horizontal force, acting parallel to the plane, that must be applied to \(M\) in order to prevent slipping is \[ Mg (\sin \alpha \tan \beta - \mu \cos \alpha \sec \beta). \] Find also the thrust in the rod.
Obtain the tangential equation of a conic in the form \[ Al^2 + Bm^2 + Cn^2 + 2Fmn + 2Gnl + 2Hlm = 0, \] and find equations giving the coordinates of the foci. Explain how to find the condition that the conic shall be a parabola. \par Given an ellipse and two points \(P, Q\) on it, prove that there is a parabola which touches the axes of the ellipse, the normals at \(P\) and \(Q\), and the chord \(PQ\).
Obtain the quadratic equation whose roots \(\eta\) and \(\bar{\eta}\) are given by \[ \eta = \omega + \omega^3 + \omega^4 + \omega^5 + \omega^9 \quad \text{and} \quad \bar{\eta} = \omega^{-1} + \omega^{-3} + \omega^{-4} + \omega^{-5} + \omega^{-9}, \] where \(\omega = \cos \frac{2\pi}{11} + i \sin \frac{2\pi}{11}\). Deduce that the absolute value of \(\eta\) is \(\sqrt{3}\), and that \[ \sin \frac{2\pi}{11} + \sin \frac{6\pi}{11} + \sin \frac{8\pi}{11} + \sin \frac{10\pi}{11} + \sin \frac{18\pi}{11} = \frac{\sqrt{11}}{2}. \]
Two fixed points \(A, B\) lie on a given tangent to a conic \(S\). \(P\) is the pole with regard to \(S\) of a variable line \(p\) through \(A\). Prove that the locus of the point of intersection of \(p\) and \(BP\) is a straight line.