\(Ox\) and \(Oy\) are two perpendicular horizontal axes through the centre \(O\) of a uniform sphere of radius \(a\). Together with the upward vertical \(Oz\) they form a right-handed set. The sphere is given a spin \(\omega_0\) about \(Ox\) and a velocity \(v_0\) in the direction \(Oy\) and placed on a rough horizontal table. Show that the sphere's motion is such that under certain conditions it may return subsequently to its initial position. Show in particular that it will just begin to roll when it has returned to its initial position if $$a\omega_0 = 6v_0.$$
(i) Evaluate the determinant $$\begin{vmatrix} a^3 & 3a^2 & 3a & 1 \\ a^2 & a^2+2a & 2a+1 & 1 \\ a & 2a+1 & a+2 & 1 \\ 1 & 3 & 3 & 1 \end{vmatrix}.$$ (ii) Show that if $$\begin{vmatrix} a & a^3 & a^4-1 \\ b & b^3 & b^4-1 \\ c & c^3 & c^4-1 \end{vmatrix} = 0,$$ and \(a\), \(b\), \(c\) are all different, then $$bc + ca + ab = \frac{1}{bc} + \frac{1}{ca} + \frac{1}{ab}.$$
If in the equation $$x^{3-\lambda} = a^3$$ the number \(\lambda\) is very small, show that an approximate root is given by $$x = a(1 + \frac{1}{3}\lambda\log a).$$ Continue the approximation process and find the root correct to order \(\lambda^2\).
A rectangular American city consists of \(p\) streets running east--west and \(q\) avenues running north--south. Find the number of different routes by which a man could travel from the south-west corner to the north-east corner of the city, it being supposed that he always proceeds either north or east.
Show that $$1^2 - 2^2 + 3^2 - \ldots + (-)^{n-1}n^2 = (-)^{n-1}(n^2 + n)/2.$$ Find also the sum of $$1^3 - 2^3 + 3^3 - \ldots + (-)^{n-1}n^3,$$ distinguishing between the cases when \(n\) is odd and even.
Explain Newton's method for approximation to the real roots of an equation, namely, that in \emph{certain circumstances} if \(a\) is a first approximation to a root of the equation \(f(x) = 0\), then $$a - \frac{f(a)}{f'(a)}$$ is a better one. Apply this to the equation \(\sin x = \lambda x\), where \(\lambda\) is a small positive quantity, and show that \(\pi[1 - \lambda + \lambda^2 - \lambda^3(1 + \frac{1}{6}\pi^2)]\) is a better approximation to the root near \(\pi\) than \(\pi\) itself.
Two functions \(P(x)\) and \(Q(x)\) have the following properties: $$P(0) = 1, \quad P'(x) = Q(x),$$ $$Q(0) = 0, \quad Q'(x) = P(x).$$ Deduce the following properties: \begin{align} P(x)^2 - Q^2(x) &= 1, \quad P(x)P(x+a) - Q(x)Q(x+a) = P(a), \\ P(x)Q(x+a) - Q(x)P(x+a) &= Q(a), \\ (P(x) + Q(x))(P(y) + Q(y)) &= P(x+y) + Q(x+y), \\ P(-x) &= P(x), \quad Q(-x) = -Q(x). \end{align}
Find the following limits: $$\lim_{x \to 0} \frac{2\sin x - \sin 2x}{x^3}, \quad \lim_{x \to 0} x \sin\left(\frac{1}{x}\right), \quad \lim_{x \to 0} \frac{\cos ec x - \cot x}{x}, \quad \lim_{x \to 1} \frac{\cos \frac{1}{2}\pi x}{\log x}.$$
Evaluate the integrals: $$\int_0^1 \frac{\sin^{-1} x}{(1+x)^2} dx, \quad \int_0^a \frac{x dx}{x + \sqrt{a^2 - x^2}}, \quad \int_1^2 \frac{(x-1) dx}{\sqrt{x+1}} \cdot \frac{1}{x}, \quad \int_0^{\pi/4} \frac{dx}{\sec x + \cos x}.$$
Sketch roughly the possible forms of the curve given by the equation $$y(ax^2 + 2bx + c) = a'x^2 + 2b'x + c',$$ where \(a\), \(b\), \(c\), \(a'\), \(b'\), and \(c'\) are real, and \(a\) and \(a'\) are non-zero. Prove that a necessary condition for \(y\) to take every real value at least once as \(x\) takes all real values can be put in the form $$(ca' - ac')^2 \leqslant 4(ab' - ba')(bc' - cb').$$