State and prove the conditions of equilibrium of any number of forces acting on a body in one plane. A heavy uniform rod \(AB\) has a string \(ACB\) attached to its ends; the rod hangs with the end \(A\) touching a smooth vertical wall and the string passing through a ring fixed in the wall at a point \(C\) vertically above \(A\). Prove that \(\tan\frac{1}{2}(ACB) = 2\cot BAC\).
What is meant by the total reaction between two bodies in contact? Shew that the total reaction makes an angle with the normal at the point of contact, which is never greater than the angle of friction between the two bodies. A uniform rod rests inside a rough vertical circle with its highest point on a level with the centre of the circle. If the friction is just sufficient to prevent sliding, shew that the angle of friction \(= \frac{1}{2}\{\alpha - \sin^{-1}(\sin\alpha\cos 2\alpha)\}\), where \(\alpha\) is the inclination of the rod to the horizontal.
Define velocity and acceleration. A particle starts from rest with acceleration \(f\), and the acceleration diminishes uniformly with the time. Shew that the space described before the particle acquires its maximum velocity \(u\) is \(\frac{4}{3}\frac{u^2}{f}\).
Shew that there are in general two directions in which a particle may be projected with a given velocity from a point \(A\) so as to pass through a point \(B\). Shew that, if two particles are projected simultaneously from \(A\) with velocity \(u\) in these two directions, they will be travelling in parallel directions after a time \(\frac{u}{g} \cos\frac{\theta}{2} \text{cosec } \frac{\alpha}{2}\), where \(\theta\) is the angle between the two directions and \(\alpha\) the inclination of \(AB\) to the vertical.
Prove that, if a particle describes a circle of radius \(r\) with uniform velocity \(v\), it has an acceleration \(\frac{v^2}{r}\) towards the centre of the circle. A heavy ring can slide on a string of length \(l\), which is attached to two points in a vertical line at a distance \(a\) apart. If the ring describes a horizontal circle with angular velocity \(\omega\), shew that it is nearer to the lower than the upper end of the string by a distance \(\frac{2alg}{\omega^2(l^2-a^2)}\).
State what is meant by an involution on a given base and prove that it is determined by two pairs of corresponding points. Show that quadrics through eight points cut any given line in points in involution.
Define the terms focus, corresponding directrix for any given quadric. Have confocal quadrics necessarily the same foci? Determine the foci of the quadric \[ x^2+2y^2+4z^2=1 \] showing which are real and which imaginary.
Show that any irrational number can be represented in one, and only one, way by a continued fraction of the form \[ a_0 + \frac{1}{a_1+} \frac{1}{a_2+} \dots \] where \(a_0, a_1, a_2, \dots\) are integers, all save possibly the first being positive. Prove that, if the series \[ \frac{1}{c_1} - \frac{1}{c_2} + \frac{1}{c_3} - \frac{1}{c_4} + \dots \] converges, so does the continued fraction \[ \frac{1}{c_1+} \frac{c_2^2}{c_2-c_1+} \frac{c_3^2}{c_3-c_2+} \dots \] and the two represent the same number. Hence show that \[ \frac{1}{1+} \frac{1^2}{2+} \frac{3^2}{2+} \frac{5^2}{2+} \dots = \frac{\pi}{4}. \]
Prove that, if \(f(x,y)\) and \(\phi(x,y)\) are one valued, continuous and possess continuous first order derivatives at and near the point \((a,b)\), then the equations \[ \xi=f(x,y), \quad \eta=\phi(x,y) \] determine \(x\) and \(y\) uniquely in terms of \(\xi\) and \(\eta\) in the neighbourhood of the point \(\xi=\alpha, \eta=\beta\) which corresponds to \(x=a, y=b\) if the Jacobian \(\frac{\partial(\xi, \eta)}{\partial(x,y)}\) does not vanish. Prove further that \(x\) and \(y\) are exactly the same kind of functions (i.e. continuous, with continuous first order derivatives) of \(\xi\) and \(\eta\) that \(\xi\) and \(\eta\) are of \(x\) and \(y\); and find expressions for \[ \frac{\partial x}{\partial \xi}, \frac{\partial x}{\partial \eta}, \frac{\partial y}{\partial \xi}, \frac{\partial y}{\partial \eta}. \]
Obtain a reduction formula for \(\int \frac{P}{Q^n}dx\) where \(P\) and \(Q\) are given polynomials in \(x\), the latter having no repeated factors. It may be assumed that \(P\) is prime to and of lower degree than \(Q\). Examine whether \(\int \frac{x^3-2x}{(x^6+3x^2+1)^3}dx\) is rational or not, and evaluate it as far as you can.