A quadratic equation is of the form \[ x^2 + ax + b = 0, \] where \(a\) and \(b\) are integers (including zero), and its roots are complex and have their moduli equal to 1. Shew that the roots must be third or fourth roots of unity.
Two uniform rods \(AB\), \(BC\), each of length \(2a\), and rigidly connected at right angles at \(B\), are placed astride a fixed rough circular cylinder of radius \(a\), whose axis is horizontal. Shew that in limiting equilibrium the radius from the centre to \(B\) makes an angle \(2\epsilon\) with the vertical, provided that the angle of friction, \(\epsilon\), is less than \(\pi/4\).
The planes of two intersecting circles of radii \(a\) and \(b\) are inclined at an angle \(\alpha\), and the length of their common chord is \(2c\), shew that the radius \(R\) of the sphere on which the circles lie is given by the equation \[ (R^2-c^2)\sin^2\alpha = a^2-c^2+2\sqrt{(a^2-c^2)(b^2-c^2)}\cos\alpha + b^2-c^2. \]
Obtain equations for the centre, the foci, and the asymptotes of the conic given by the general equation, \(ax^2+2hxy+by^2+2gx+2fy+c=0\), the axes of reference being rectangular. From the property that the axes of the conic are the diameters perpendicular to the chords they bisect, or otherwise, prove that the equation of the axes is \[ h(X^2-Y^2) - (a-b)XY = 0, \] where \(X=ax+hy+g, Y=hx+by+f\); and shew that the equation of the asymptotes is \[ bX^2-2hXY+aY^2=0. \]
Radii \(OQ_1, OQ_2 \dots\) are drawn to represent the velocities of points \(P_1, P_2 \dots\) of a thin plate moving in its own plane; prove that the velocity diagram formed by the points \(Q_1, Q_2 \dots\) is similar to the diagram of the points \(P_1, P_2 \dots\). Shew also that if radii \(OR_1, OR_2 \dots\) represent the accelerations of the same points, the acceleration diagram formed by the points \(R_1, R_2 \dots\) has the same property; so that, for example, a series of points of the plate whose accelerations are equal in magnitude lie on a circle.
Solve the equations \[ \frac{1}{y} - \frac{1}{z} = a - \frac{1}{a}, \quad y - \frac{1}{z} = b - \frac{1}{c}, \quad z - \frac{1}{x} = c - \frac{1}{a}. \]
A suspension bridge of 40 ft. span has a post erected at each end so that 15 ft. of it projects above ground; from the top of each post run two light wire ropes, one to the middle point of the roadway and the other to the point midway between the middle and the corresponding end. If these three points, taken in order from one end, carry loads of 1, 2 and 3 tons respectively, find the tension in each rope, and calculate the upsetting couple on each post. (Neglect the weight of the roadway in comparison with the above loads, and assume it to consist of four flexibly jointed sections.)
If \[ (x+y+z)\left(\frac{1}{x}+\frac{1}{y}+\frac{1}{z}\right) = 1 \] shew that \[ (y+z)(z+x)(x+y) = 0. \]
Write a short essay on the theory of the convergence of series of positive terms, starting from the beginning and proceeding far enough for the proof of the following propositions:
Taking as rectangular coordinates the kinetic energy of a particle moving in a straight line and the reciprocal of the resultant force acting on it, and drawing a curve to represent the motion, shew that the distance travelled is represented by the area under this curve. Sketch the curve for the case of a vehicle driven by an engine which is working at constant horse-power, the only resistance being a force proportional to the square of the velocity. Shew that the velocity has a certain limiting value; and calculate the distance travelled while the velocity increases from a half to three quarters of this limiting value, taking the mass of the vehicle as 2000 pounds, the horse-power as 100, and the limiting velocity as 150 feet per second.