A reel of thread of radius \(a\) is unwound by moving the end of the thread in a plane \(p\) perpendicular to the axis of the reel in such a way that the free part of the thread is straight and moves with constant angular velocity \(\omega\); the reel is kept fixed, and it may be assumed that all the thread on the reel is in the plane \(p\). Find the magnitude and direction of the acceleration of the end of the thread when the free part of the thread is of length \(l\).
Prove that the geometric mean of a number of positive quantities can never exceed their arithmetic mean. Prove that if \(a_1, a_2, \dots, a_n\) are essentially positive but not all equal then \(\sum_{r \ne s} a_r/a_s > n(n-1)\).
Prove that \[ a^3+b^3+c^3-3abc = \tfrac{1}{2}(a+b+c)[(b-c)^2+(c-a)^2+(a-b)^2]. \] Hence, or otherwise, establish the identity \[ (a^3+b^3+c^3-3abc)^2 = (a^2-bc)^3 + (b^2-ca)^3 + (c^2-ab)^3 - 3(a^2-bc)(b^2-ca)(c^2-ab). \]
In the permutation (denoted by \(p\)) obtained by rearranging the integers 1 to \(n\) in any manner, the "number of inversions with respect to one of the given integers," say \(r\), is defined as the number of integers greater than \(r\) which precede it in the permutation \(p\), and is denoted by \(k_r\). The total number of inversions in the permutation is given by \(\sum_1^n k_r\) and is denoted by \(k\). Prove that if the permutation \(p\) is modified by the simple interchange of two integers, say \(r\) and \(s\), the change in the total number of inversions is \(2q+1\), where \(q\) is the number of integers that lie between \(r\) and \(s\) both in the original order of integers and in the permutation \(p\). Hence, or otherwise, show that if the permutation \(p\) had been effected by a number of simple interchanges, the total number of such interchanges is always odd or even with the total number of inversions.
Prove that if the two equations \begin{align*} ax^2+2bx+c &= 0 \\ a'x^2+2b'x+c' &= 0 \end{align*} have a single common root, then \[ 4(b^2-ac)(b'^2-a'c') - (ac'+ca'-2bb')^2=0. \] Show that the condition \[ 4(b^2-ac)(b'^2-a'c') - (ac'+ca'-2bb')^2 \ge 0, \] is necessary for the fraction \[ \frac{ax^2+2bx+c}{a'x^2+2b'x+c'} \] to assume all real values for real values of \(x\), and that if this condition is not fulfilled, the range of inadmissible values of the fraction will either be entirely between or entirely outside the roots of the equation \[ x^2(b'^2-a'c') + x(ac'+ca'-2bb') + b^2-ac = 0. \]
Show that the conditions that an algebraic equation \(f(x)=0\) has a double root at \(x=a\) are that \(f(a)=f'(a)=0\). If the equation \[ x^4 - (a+b)x^3 + (a-b)x - 1 = 0 \] has a double root, prove that \[ a^\frac{2}{3} - b^\frac{2}{3} = 2^\frac{2}{3}. \]
Prove Leibniz' theorem for the \(n\)th differential coefficient of the product of two functions. By using this theorem, or otherwise, prove that if \(n\) is a positive integer the polynomial \[ \frac{d^n}{dx^n}(x^2-1)^n \] is a solution of the differential equation \[ (1-x^2)\frac{d^2y}{dx^2} - 2x\frac{dy}{dx} + n(n+1)y=0. \]
(i) Evaluate the integral \(\displaystyle\int \sec^3\theta \,d\theta\). (ii) The mass per unit area \(\sigma\) at any point of a square lamina is proportional to its distance \(r\) from one corner, that is \(\sigma=kr\), where \(k\) is a constant. Find the total mass of the lamina in terms of \(k\) and \(a\) the length of side of the square.
Show that the function \[ \frac{x^3+x^2+1}{x^2-1} \] of the real variable \(x\) has only two critical values one of which is at \(x=0\) and the other at a certain value of \(x\) lying between 2 and 3. Establish which of these is a maximum and which a minimum value.
Prove that if \[ I_{p,q} = \int_0^{\pi/2} \sin^p\theta \cos^q\theta \,d\theta, \] where \(p>1, q>1\), then \[ (p+q)I_{p,q} = (p-1)I_{p-2,q}, \] and find the corresponding reduction formula involving \(I_{p,q-2}\). If the function \(\Gamma(n)\) is defined to have the following properties \[ \Gamma(n+1)=n\Gamma(n), \quad \Gamma(1)=1, \quad \text{and} \quad \Gamma(\tfrac{1}{2})=\sqrt{\pi}, \] verify that for all positive integral values of \(p\) and \(q\) greater than unity \[ I_{p,q} = \Gamma\left(\frac{p+1}{2}\right)\Gamma\left(\frac{q+1}{2}\right) / 2\Gamma\left(\frac{p+q+2}{2}\right). \]