A number of the form \(p/q\), where \(p\) and \(q\) are integers (\(q\neq 0\)), is said to be rational. Prove that a rational root of the equation \[ x^n+a_1x^{n-1}+a_2x^{n-2}+\dots+a_n=0, \] where \(a_1, a_2, \dots, a_n\) are integers, is necessarily an integer. Prove further that, if \(|a_n|\) is a prime number, the equation cannot have more than three distinct rational roots. Find the rational roots of the equation \[ 8x^5 - 4x^4 - 2x^3 - 3x^2 + 1 = 0, \] and hence solve the equation completely.
Prove that the locus of a point which moves so that the lines joining it to two fixed points \(H\) and \(K\) are conjugate with respect to a fixed conic \(S\) is a conic \(S'\) through \(H\) and \(K\). \(P\) is a general point of \(S'\) and the tangents to \(S\) from \(P\) meet \(S'\) again in \(L\) and \(M\). The other tangents to \(S\) through \(L\) and \(M\) meet in \(Q\). Prove that \(Q\) lies on \(S'\) and that, as \(P\) varies on \(S'\), the line \(PQ\) passes through the pole of \(HK\) with respect to \(S\).
Four pennies and four half-crowns are placed at random in a row on a table. Find the chance that (i) no two pennies are adjacent; (ii) no two heads are adjacent (and showing).
A circle of radius \(b\) rolls round a fixed circle of larger radius \(a\). Find parametric equations for the curve traced out by a point fixed to the circumference of the moving circle. Show that the length of the arc between consecutive cusps is \(8(a+b)b/a\) or \(8(a-b)b/a\) according as the moving circle is outside or inside the fixed circle. Draw a rough diagram of the curve in each case when \(b=a/3\).
State and prove Leibniz's formula for \(\dfrac{d^n(uv)}{dx^n}\), where \(u\) and \(v\) are functions of \(x\). If \(y_n(x) = \dfrac{d^n}{dx^n}\{(x^2-1)^n\}\), prove the relations
A particle is projected from a point \(P\) on an imperfectly elastic plane which is inclined at an angle of \(30^\circ\) to the horizontal. The direction of projection is in the vertical plane containing the line of greatest slope through \(P\) and makes an angle of \(30^\circ\) both with the vertical and with the plane. Show that, if the coefficient of restitution is \(\frac{2}{3}\), the velocity of the particle immediately after the first impact will be parallel to the original velocity of projection. Hence show that the distance between \(P\) and the highest point of the plane reached is \(\frac{2}{9}\) times the distance between \(P\) and the first point of impact.
A particle is free to move on the smooth inner surface of a sphere of radius \(a\). It is projected with velocity \(V\) along the surface from its lowest point. Show that during the subsequent motion it will lose contact with the surface if and only if \(2ag < V^2 < 5ag\). If this condition is satisfied find the height above the point of projection at which contact is lost.
The cross-section of a uniform solid cylinder is an ellipse of major and minor semi-axes \(a,b\) respectively. A generator at the end of the major axis is fixed horizontally and the cylinder is free to rotate about it. Show that the period of small oscillations about the equilibrium position is \(\pi \sqrt{\left(\frac{5a^2+b^2}{ag}\right)}\).
A particle of mass \(m\) is projected horizontally with velocity \(V\) from the top of a tower which stands on a horizontal plain. The air resistance is \(mk\) times the velocity of the particle. What is the height of the tower if the particle hits the plain at a distance \(V/2k\) from the base of the tower?
The point of suspension of a simple pendulum of length \(l\) is made to move in a horizontal straight line so that its displacement from a fixed point \(O\) of the line is \(a \sin mt\) at time \(t\). The pendulum oscillates in a vertical plane through the line and makes a small angle with the vertical throughout the motion. Prove that \(x\), the horizontal component of the displacement at time \(t\) of the bob from \(O\), satisfies the equation \[ \frac{d^2x}{dt^2} + n^2x = n^2a \sin mt, \] where \(ln^2=g\); prove that if \(m \neq n\), a solution of this equation is \[ x = (A-\frac{1}{2}nat)\cos nt + B \sin nt, \] where \(A\) and \(B\) are arbitrary constants, and hence find the solution such that \(x=dx/dt=0\) when \(t=0\).