Prove that if \(x_1, x_2, \ldots, x_n\) and \(y_1, y_2, \ldots, y_n\) are two sets of positive quantities, both in increasing order of magnitude, then \[\frac{1}{n} \sum_{r=1}^{n} x_r y_r > \left(\frac{1}{n} \sum_{r=1}^{n} x_r\right) \left(\frac{1}{n} \sum_{r=1}^{n} y_r\right).\] Prove that if \(a\), \(b\), \(c\) are three unequal positive quantities, then \[a^3 + b^3 + c^3 > abc(a^2 + b^2 + c^2).\]
(i) Find the equation whose roots are the cubes of the roots of the equation \[x^3 + ax^2 + bx + c = 0.\] (ii) Show how to obtain the equation whose roots are the roots of a given algebraic equation multiplied by the same constant quantity. Hence, or otherwise, prove that an algebraic equation with integer coefficients and unit coefficient for the highest power cannot have a real rational root which is not integral.
Discuss the recurring series which is such that each term above the second is equal to the sum of the previous term and twice the term previous to that. Write down the scale of relation and determine the general term if the values of the first two terms are given. Examine the case when \(u_0 = 1\) and \(u_1 = 2\).
Stating without proof any properties of determinants used, express as a product of two linear terms and one quadratic term the determinant: \[\begin{vmatrix} x+a & b & c & d \\ b & x+c & d & a \\ c & d & x+a & b \\ d & a & b & x+c \end{vmatrix}.\]
Given that the equation \[x^6 - 5x^5 + 5x^4 + 9x^3 - 14x^2 - 4x + 8 = 0\] has three coincident roots, find the value of this multiple root, and hence, or otherwise, solve the equation completely.
Find the limits of the following expressions \[\frac{x - \sin x}{x^3} \quad \text{and} \quad \frac{1 - \frac{1}{2}x^2 - \cos x}{x^4}\] as \(x \to 0\). Find also the limits of \[\frac{\cos \frac{1}{2}x}{x^2 - \pi^2}\] as \(x \to \pi\), and as \(x \to \infty\).
Establish Leibnitz' theorem for the \(n\)th derivative of a product of two functions. If \[f(x) = \frac{px + q}{ax^2 + 2bx + c},\] and \(f_n\) denotes the \(n\)th derivative of \(f(x)\), prove that \[(ax^2 + 2bx + c) f_{n+2} + 2(ax + b)(n + 2) f_{n+1} + a(n + 1)(n + 2) f_n = 0.\]
Evaluate the integrals: \[\int_0^{\pi/2} (a^2 \cos^2 \theta + b^2 \sin^2 \theta)^{-1} d\theta,\] \[\int_0^{\pi/4} (\cos \theta + \sin \theta)(9 + 32 \cos \theta \sin \theta)^{-1} d\theta,\] \[\int_1^{\infty} \frac{dx}{x^2 + x}.\]
A rigid parabola rolls without slipping on a fixed straight line. Find the locus described by its focus.
The sides \(AB\), \(BC\), \(CD\), \(DA\) of a deformable but plane quadrilateral are of fixed lengths \(a\), \(b\), \(c\), \(d\) respectively. Show that its area is greatest when the shape is such that \(A\), \(B\), \(C\), \(D\) are concyclic.