The end \(P\) of a straight rod \(PQ\) describes with uniform angular velocity a circle of centre \(O\), while the other end moves on a fixed line through \(O\) in the plane of the circle. The end \(Q'\) of an equal straight rod \(P'Q'\) moves on the same fixed line through \(O\). Prove that the velocities of \(Q, Q'\) are in the ratio \(QO:OQ'\).
A cyclist works at the constant rate of \(P\) horse-power. When there is no wind he can ride at 22 feet per second on level ground, and at 11 feet per second up a hill making an angle \(\sin^{-1}\frac{1}{5}\) with the horizon. The total mass of man and cycle is 180 lb. The resistance of the air is \(kv^2\) lb. weight, when the velocity of the man relative to the air is \(v\) feet per second; the other frictional forces are negligible. Find \(P\), and show that the speed of the cyclist when riding on level ground against a wind of 22 feet per second is between 10 and 10.5 feet per second.
Two equal smooth spheres moving along parallel lines in opposite directions with velocities \(u, v\) collide, the line of centres making an angle \(\alpha\) with the direction of motion. If, after the impact, their lines of motion are perpendicular, show that \[ \left(\frac{u-v}{u+v}\right)^2 = \sin^2\alpha+e^2\cos^2\alpha, \] where \(e\) is the coefficient of elasticity.
Find the equation whose roots are the squares of the differences of the roots taken in pairs of the cubic equation \(x^3+bx+c=0\). Hence or otherwise shew that if \(b\) and \(c\) are real the equation \(x^3+bx+c=0\) will have three real roots or only one real root according as \(4b^3+27c^2\) is negative or positive. Consider also the case in which \(4b^3+27c^2=0\).
(a) Find by the method of differences or otherwise the \(n\)th term and the sum to \(n\) terms of the series: \(1+4+11+26+57+120+\dots\). (b) Find the sum of the infinite series whose \(n\)th term is \(\displaystyle\frac{3n^2+3n+1}{n^3(n+1)^3}\).
Prove that the Arithmetic Mean of a number of positive quantities is not less than their Geometric Mean. If \(a_1, a_2 \dots a_N\) be \(N\) given positive constants and \(n_1, n_2 \dots n_N\) be any \(N\) positive integers, shew that \[ a_1^{n_1}a_2^{n_2}\dots a_N^{n_N} \text{ is not greater than } n_1^{n_1}n_2^{n_2}\dots n_N^{n_N}\left(\frac{a_1+a_2+\dots+a_N}{n_1+n_2+\dots+n_N}\right)^{n_1+n_2+\dots+n_N}. \] (Note: This inequality appears to be transcribed correctly from the paper, but may contain a misprint in the original document.)
If \(a,b,c,d\) are four real quantities whose sum is zero, shew that \[ \frac{a^5+b^5+c^5+d^5}{5} = \frac{a^3+b^3+c^3+d^3}{3}\frac{a^2+b^2+c^2+d^2}{2}. \] If \(d\) is zero, shew further that \[ \frac{a^7+b^7+c^7}{7} = \frac{a^5+b^5+c^5}{5}\frac{a^2+b^2+c^2}{2} = \frac{a^4+b^4+c^4}{2}\frac{a^3+b^3+c^3}{3}. \]
If \(\omega\) is one of the imaginary \(n\)th roots of unity, shew that \[ \sum_{r=1}^{n-1}\frac{1-\omega^r}{y-\omega^r} = \frac{n(y^{n-1}-1)}{y^n-1}. \] By the use of the calculus, or otherwise, prove that if \(x>1\), then \[ (n+1)^2(x+3)(x-1) > 4n^2\{x^{n+1}+(n+1)x^n-n-2\} - 4n(n+1)\sum_{r=1}^{n-1}(1-\omega^r)\log\frac{1}{1-\omega^r x^{-1}} \] \[ > 4(n+1)^2(x-1), \] where \(n-1\) is a positive integer, and \(x^n\) is real.
State and prove Leibnitz's theorem on the \(n\)th differential coefficient of the product of two functions. If \(y=(x+a)\cot^{-1}\frac{x}{a}\) satisfies identically the equation \[ P\frac{dy}{dx}+Qy+R=0, \] where \(P, Q, R\) are polynomials in \(x\), find the simplest possible forms for \(P, Q,\) and \(R\), and shew that if \(\frac{p_n}{n!}\) is the coefficient of \(x^n\) in the Maclaurin expansion of \(y\) in a series of ascending powers of \(x\) then \[ a^3p_n+(n-3)a^2p_{n-1}+(n-1)(n-2)ap_{n-2}+(n-1)(n-2)(n-5)p_{n-3}=0, \] provided \(n>4\).
Write an account of the theory of rectilinear asymptotes of a plane curve whose equation is given either in rectangular cartesian form or in polar form. The lines whose equations are \(x=y, x+y=0, x=2y\) are the asymptotes of a cubic curve which touches the axis of \(x\) at the origin and which passes through the point \((0,b)\). What is the equation of the curve?