Problems

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*}

Explain the application of the integral calculus to the computation of areas (i) in Cartesian, (ii) in polar coordinates. P and Q are two points on a rectangular hyperbola whose centre is C and PL, QM are perpendiculars on an asymptote. Prove that the area bounded by the lines CP, CQ and the arc PQ is equal to that bounded by the lines PL, LM, MQ and the arc PQ.