Problems

A heavy uniform chain of length \(l\) hangs in equilibrium over the edge of a smooth horizontal table, and the end of the chain which is upon the table is attached to a spring of strength such that a force equal to the weight of the chain would stretch it through a length \(a\) (\(

Shew that if A and B are two polynomials in \(x\) with no common factor then polynomials X, Y can be found such that \[ AX+BY=1. \] Find a solution in case \(A=x^4+x^3-1, B=x^2+1\) and indicate the general solution.

Draw graphs of the functions \[ \frac{(x-2)(x-4)}{(x-1)(x-3)}, \quad \left\{ \frac{(x-2)(x-4)}{(x-1)(x-3)} \right\}^{\frac{1}{2}}, \quad \frac{x^3}{x^2+1}, \] and shew how to approximate to them for large values of \(x\).

Form an equation with integer coefficients which has

1. [(i)] \(\sqrt{2}+\sqrt{3}\),
2. [(ii)] \(\sqrt{2}+\sqrt{3}+\sqrt{5}\),

for a root. State what the other roots are in each case.

Write an account of the notation, the elementary properties, and the utility of determinants. Shew that \[ \begin{vmatrix} 1 & bc & bc(b+c) \\ 1 & ca & ca(c+a) \\ 1 & ab & ab(a+b) \end{vmatrix} = 0. \]

Explain and illustrate the concept of convergence in connexion with infinite series. Discuss the convergence or otherwise of the series

1. [(i)] \(1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\dots\),
2. [(ii)] \(\frac{1}{1^2}+\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{4^2}+\dots\),
3. [(iii)] \(\frac{1}{1} + \frac{1}{1+2} + \frac{1}{1+2+3} + \frac{1}{1+2+3+4} + \dots\).

Define a differential coefficient and find from first principles the differential coefficients of \(e^x\) and \(\cos x\). Differentiate \[ \tan^{-1}\frac{x}{1+x^2}, \quad \log(\log x), \quad x^2\sin\frac{1}{x}, \] and consider especially the last for \(x=0\).

Obtain the expansions of \(\tan^{-1}x\) and \(\sin^{-1}x\) in ascending powers of \(x\) and discuss their range of validity. Shew that if \[ y = \frac{\sin^{-1}x}{(1-x^2)^{\frac{1}{2}}}, \] then \[ (1-x^2)\frac{dy}{dx} = 1+xy, \] and thence find a series for y.

Give an account of the application of the calculus to the discovery of, and the discrimination between, the maxima and minima of functions of one variable. Examine whether \[ \frac{\sin^3 x}{x^3\cos x} \] is a maximum or minimum for \(x=0\).

Establish the following results: \begin{align*} \int_0^{\pi} \frac{dx}{a\cos^2 x + b\sin^2 x} &= \frac{\pi}{\sqrt{ab}}, \quad (a,b>0) \\ \int_0^{\pi} \frac{dx}{(a\cos^2 x + b\sin^2 x)^2} &= \frac{\pi(a+b)}{2(ab)^{\frac{3}{2}}}, \\ \int_0^1 x^n \log x \,dx &= -\frac{1}{(n+1)^2}, \\ \int_0^\infty \frac{x^2}{(1+x^2)^2} \,dx &= \frac{\pi}{4}. \end{align*}