From the centre O of an ellipse whose foci are S, H, a line is drawn perpendicular to the tangent at any point P on the curve meeting SP in Q and HP in R. Prove that SQ and HR are each equal to the semi-major axis. Prove also, that if Y, Z are the projections of S, H on the tangent at P, then YQZR is a rhombus having its sides equal to SO.
Shew that \(x^2-yz\) is a factor of the expression \[ (pyz+zx+xy)^2 - xyz(px+y+z)^2; \] and determine the other factors when \(p=1\) and when \(p=-1\).
Find in its simplest form the coefficient of \(x^n\) in the expansion of \((1-x)^{-p}\). Prove that, if \((1-x)^{-p} = 1+c_1x+c_2x^2+\dots\) \[ 1+c_1+c_2+\dots+c_n=(n+1)c_{n+1}/p. \]
Prove that, provided \(n>1\), \[ \log_e n - \log_e(n-1) = \frac{1}{n} + \frac{1}{2n^2} + \frac{1}{3n^3} + \dots. \] Having given that \(\log_{10}e = \cdot 4343\), determine the value of \(\log_{10}999\) to seven places of decimals.
Three smooth equal cylinders of radius \(r\) are placed symmetrically inside a hollow cylinder of radius \(R\), such that two of the smaller cylinders are in contact with the larger one, and the axes of all the cylinders are horizontal. Prove that, for equilibrium to be possible, \(R\) must not exceed \(r(1+\sqrt{28})\).
Three uniform rods of similar material are jointed to form an isosceles triangle ABC, in which each of the angles B and C is equal to \(\alpha\). The triangle is supported in a vertical plane with BC in contact with a rough peg. Prove that, if the coefficient of friction \(>2\cot\alpha(1+\cos\alpha)\), the triangle will rest with any point of BC in contact with the peg.
A motor car weighing one ton attains a speed of 40 miles per hour when running down an incline of 1 in 20 with the engine cut off. It can attain the speed of 30 miles up the same incline when the engine is working. Assuming that the resistance varies as the square of the velocity, find the horse-power developed by the engine.
A particle moves in a circle of radius \(r\), and has a velocity \(v\) after time \(t\). Prove that it has an acceleration whose resolved parts are \(v^2/r\) towards the centre and \(\frac{dv}{dt}\) along the tangent. A mass of 1 lb. is attached to the end of a string which is 20 inches long and is tied to a fixed point A. Initially the string is horizontal and the mass is allowed to fall. Determine the tension in the string when the mass is vertically below A. If the string catches against a peg B vertically below A so that the mass begins to describe a circle about B, find the least depth of B below A in order that the mass may describe a complete circle about B.