Shew that the Arithmetic mean of a number of positive quantities is never less than their Geometric mean. If \(x, y, z\) are positive quantities such that \(x+y+z=1\), prove that
(i) Denoting the roots of the equation \(x^4-x+1=0\) by \(x_1, x_2, x_3, x_4\), shew that, if \(y_r = x_r^3+x_r\), [\(r=1,2,3,4\)], then \(y_r\) satisfies the equation \(y^4-3y^3+7y^2-7y+5=0\). (ii) Given that the sum of two of its roots is zero, solve completely the equation \[ 4x^4+8x^3+13x^2+2x+3=0. \]
(i) Find the sum to \(n\) terms of the series: \[ \frac{1.2.12}{4.5.6} + \frac{2.3.13}{5.6.7} + \frac{3.4.14}{6.7.8} + \dots. \] (ii) By expanding \(\log_e(1+x^2+x^4)\), or otherwise, shew that \[ \sum_{r=\frac{n}{2}}^n \frac{(-1)^r|r-1|}{(n-r)|2r-n|} = 2\sum_{r=n}^{2n} \frac{(-1)^r|r-1|}{(2n-r)|2r-2n|}, \] where \(n\) and \(r\) are positive integers.
Explain a general method of finding the Highest Common Factor of two polynomials \(f(x), \phi(x)\). Shew how this method may be used to find polynomials \(F(x), \Phi(x)\), such that \(f(x).F(x) + \phi(x).\Phi(x) = 1\), when \(f(x)\) and \(\phi(x)\) have no common algebraic factor.
Taking \((1/u, \theta)\) as the polar coordinates of a point of a plane curve, obtain an expression for the curvature in terms of \(u, \frac{du}{d\theta}, \text{and } \frac{d^2u}{d\theta^2}\). Shew that the curvature of the curve \(au = \cosh n\theta\) has a stationary value provided \(3n^2\) is not less than 1. Determine whether this value is a maximum or minimum.
Define the envelope of a family of plane curves. If circles are described on focal chords of a parabola as diameters, prove that their envelope consists of the directrix and a circle. Prove also that the points of contact of any circle of the system with the envelope are collinear with the vertex of the parabola.
If \(u=x+y\) and \(v=x-y\), express \(\frac{\partial f}{\partial x}\) and \(\frac{\partial f}{\partial y}\) in terms of \(\frac{\partial f}{\partial u}\) and \(\frac{\partial f}{\partial v}\) where \(f\) is any function of the variables considered. Deduce that the necessary and sufficient condition that \(f\) should be expressible as a function of \(x+y\) only is that \(\frac{\partial f}{\partial x} = \frac{\partial f}{\partial y}\). If \begin{align*} f_0(x) &= 1+\frac{x^4}{4}+\frac{x^8}{8}+\dots, \\ f_1(x) &= x+\frac{x^5}{5}+\frac{x^9}{9}+\dots, \\ f_2(x) &= \frac{x^2}{2}+\frac{x^6}{6}+\frac{x^{10}}{10}+\dots, \\ f_3(x) &= \frac{x^3}{3}+\frac{x^7}{7}+\frac{x^{11}}{11}+\dots, \end{align*} prove that \[ f_0(x)f_3(y)+f_1(x)f_2(y)+f_2(x)f_1(y)+f_3(x)f_0(y) = f_3(x+y). \]
If \(I(r,s) = \int_a^\infty \frac{(x-a)^s}{x^r}dx\), \(s>0, r>s+2\), express \(I(r,s)\) in terms of (a) \(I(r,s-1)\), (b) \(I(r-1,s)\). Assuming \(r\) and \(s\) to be integers satisfying the above inequalities, find the value of \(r-1 I(r,s)\).
If the coordinates \((x,y)\) of any point on a plane curve are expressed as functions of a parameter \(\theta\), interpret the expression \(\frac{1}{2}\int \left(x\frac{dy}{d\theta}-y\frac{dx}{d\theta}\right)d\theta\). Sketch the curve given by \(x=2a(\sin^3\theta+\cos^3\theta)\), \(y=2b(\sin^3\theta-\cos^3\theta)\), and prove that its area is \(3\pi ab\).
Prove that the length of the arc of the curve whose pedal \((p,r)\) equation is \(p=r-d\) between the points \(r=a, r=2a\) is \(a(\sqrt{3}-\frac{\pi}{3})\). Shew that the polar equation of this curve may be written in the form \[ 2r = a\sec^2\left(\frac{\sqrt{2ar-a^2}+\frac{a\pi}{2}-a\theta-a}{2a}\right). \]