Explain what is meant by the angle of friction. Find the direction in which the least force necessary must be applied to move a body up a rough plane inclined at an angle \(\alpha\) to the horizon, the angle of friction being \(\lambda\). The end \(A\) of a uniform rod \(AB\) rests on a rough horizontal plane, and the end \(B\) is connected by a string to a point \(C\) above it. When \(A\) is as far as possible from \(C\) for equilibrium, \(AB\) and \(BC\) make angles \(\alpha\) and \(\beta\) with the vertical. Prove that the angle of friction between the rod and the plane is \(\cot^{-1}(\cot\beta-2\cot\alpha)\).
Shew that two couples in the same plane balance each other if their moments are equal and opposite. If \(r\) and \(R\) are the radii of the inscribed and circumscribed circles of a triangle \(ABC\) and \(D, E, F\) are the points at which the sides are touched by the inscribed circle, shew that forces represented by \(r.BC, r.CA\) and \(r.AB\) are balanced by forces represented by \(2R.FE, 2R.DF\) and \(2R.ED\).
Define work and power, and shew that, when a force \(F\) is moving its point of application with velocity \(v\), the power is measured by \(Fv\). An engine of 300 horse-power pulls a train of 200 tons mass up an incline of 1 in 120, the resistance of wind and rails being 10 lb. weight per ton. Find the maximum velocity acquired, correct to one place of decimals in miles per hour.
Prove that the path of a particle projected from a given point with a given velocity is a parabola, and that the velocity at any point is equal to the velocity acquired by a particle falling freely from the directrix to that point. A particle is projected from a point at a height \(h\) above a horizontal plane with velocity \(\sqrt{gh}\); shew that the farthest point in the plane which the particle can reach is at a distance \(2h\) from the point of projection.
Find the equation whose roots are the squares of the reciprocals of the roots of the equation \[ ax^2+2bx+c=0. \]
Prove that \[ {}_nC_r = \frac{n!}{r! (n-r)!}. \] In a certain examination 20 papers are set. These are divided into five groups A, B, C, D, E: A, B, C contain five papers each, D and E contain three and two respectively. A candidate may take the examination in one of the following ways:
Prove that \(e\) is an incommensurable number, and that \(e^x\) tends to infinity with \(x\) more rapidly than any power of \(x\).
Show that any surd can be converted into a continued fraction and prove that if \(a\) is positive \[ m + \cfrac{a}{m + \cfrac{a}{m + \dots}} = \frac{\sqrt{m^2+4a}+m}{2}. \]
Prove that \[ \sum_{s=1}^{n-1} \sin \frac{s\pi}{n} = \cot\frac{\pi}{2n}. \] Hence calculate \(\cot 7^\circ 30'\) in terms of the trigonometrical ratios of 45\(^\circ\) and 30\(^\circ\).
From both ends of a measured base \(AB\) the bearings \(CAB, CBA, C'AB, C'BA\) of two points \(C, C'\) are measured; the four points \(C, C', A, B\) lie in a horizontal plane. Find \(CC'\) in terms of the measured quantities. If \(AB=2\) miles, \(CAB=CBA=45^\circ\), \(C'AB=30^\circ\) and \(C'BA=60^\circ\), find \(CC'\).