If a rational integral function of \(x\) vanishes when \(x\) is equal to \(a\), prove that it is divisible by \(x-a\). Resolve into factors:
Find the sum of the squares of the first \(n\) natural numbers. Find the sum of all possible products two at a time of the first \(n\) natural numbers.
Prove the rule for the formation of successive convergents to the continued fraction \[ a + \frac{a_1}{b_1+} \frac{a_2}{b_2+\dots}. \] Find the value of the \(n\)th convergent to the continued fraction \[ 2+\frac{1}{3+} \frac{1}{2+} \frac{1}{3+\dots}. \]
Differentiate \(\log(\sin x), \tan^{-1}\frac{x}{1+\sqrt{1+x^2}}\). If \(y=\sqrt{\frac{1-x^2}{1+x^2}}\), prove that \(\frac{dy}{dx}=-\sqrt{\frac{1-y^4}{1-x^4}}\). Find the \(n\)th differential coefficient of \(\frac{1+3x}{(x-1)^2(x+2)}\).
Prove that for a plane curve, with the usual notation, \[ \sin\phi = r\frac{d\theta}{ds}, \quad \cos\phi=\frac{dr}{ds}. \] Prove that \[ \frac{d^2\phi}{ds^2} = \frac{\sin 2\phi}{r^2} - \frac{\cos\phi}{pr} - \frac{1}{\rho}\frac{dp}{ds}, \] where \(\rho\) is the radius of curvature at the point.
Find the condition that three forces acting on a body should keep it in equilibrium. Two equal smooth cylinders, each of radius \(a\), rest in parallel positions on a horizontal plane. On them there rests an equilateral triangular prism in a symmetrical position touching the cylinders with two of its faces and with the third face horizontal. The cylinders are prevented from moving outwards by means of two stops in the plane. Prove that the height of each stop must be at least \[ a\left\{1-\frac{2\kappa+1}{2(\kappa^2+\kappa+1)^{\frac{1}{2}}}\right\}, \] where \(\kappa\) is the ratio of the weight of a cylinder to that of the prism.
State the principle of virtual work; and shew that when gravity is the only external force acting, the depth of the centre of mass of the system is a maximum or minimum when the system is in equilibrium. Three uniform rods, of equal weights and lengths (\(2a\)), are placed over two smooth pegs in the same horizontal line, distant \(2c\) apart. Find the positions of equilibrium, and shew that no position is possible if \(c>5a/3\).
Prove that the path of a particle projected from a point under gravity is a parabola. A particle is projected from a point at a distance \(c\) from an inclined plane of angle \(\alpha\), in a direction up the plane and making an angle \(\beta\) with the plane. Prove that if it strikes the plane perpendicularly the velocity of projection \(U\) is given by \[ U^2 = 4gc\sin^2\alpha\sec\beta/(3\cos\overline{\alpha+\beta}-\cos\overline{\alpha-\beta}). \]
State and prove the principle of the conservation of linear momentum for a system of particles. A smooth inclined plane of angle \(\alpha\) and mass \(M'\) can slide on a smooth horizontal plane. Another smooth inclined plane of the same angle \(\alpha\) and of mass \(M\) is placed on the first so that the upper face of \(M\) is horizontal. A particle of mass \(m\) is placed on this horizontal face and the whole system is then released from rest. Find the acceleration of \(m\).
Prove that \(v^2/\rho\) is the acceleration inwards along the normal, when a particle describes a plane curve with velocity \(v\). A smooth parabolic tube has its axis vertical and vertex upwards. A particle inside the tube is projected from the vertex with a velocity \(\sqrt{2g(a+h)}\). When the particle is at a depth \(z\) below the directrix prove that the pressure on the tube is \(mgh\sqrt{az^{-3}}\), where \(m\) is the mass of the particle and \(4a\) is the latus rectum of the parabola.