Show that the result of eliminating \(y\) and \(z\) between the three equations \begin{align} y^2+2ay+b=0, \tag{1} \\ z^2+2cz+d=0, \tag{2} \\ x=yz, \nonumber \end{align} is the quartic equation whose roots are \(\alpha\beta, \alpha\beta', \alpha'\beta, \alpha'\beta'\), where \(\alpha, \alpha'\) are the roots of the quadratic equation \((1)\) and \(\beta, \beta'\) are the roots of the quadratic equation \((2)\). Find the quartic equation whose roots are \[ (\alpha-\beta), (\alpha'-\beta), (\alpha-\beta'), (\alpha'-\beta'), \] and hence write down the equation whose roots are \[ (\alpha-\beta)^2, (\alpha'-\beta)^2, (\alpha-\beta')^2, (\alpha'-\beta')^2. \]
Show that, when \(a,b\) and \(c\) are real and positive, the system of equations \begin{equation} \tag{1} xyz = a(y+z) = b(z+x) = c(x+y) \end{equation} has real non-zero solutions \(x,y,z\) if and only if there is a proper triangle whose sides are proportional to \((a^{-1}, b^{-1}, c^{-1})\), and find the complete solution of the system in this case. Show also that, when \(b=1\) and \(c=-1\), the system \((1)\) has real non-zero solutions if and only if either \(a > \frac{1}{2}\) or \(-\frac{1}{2} < a < 0\).
By expanding the expression \((e^x-1)^n\) in two different ways, or otherwise, evaluate the sum \[ n^{n+2} - \binom{n}{1}(n-1)^{n+2} + \binom{n}{2}(n-2)^{n+2} - \dots + (-1)^{n-1} \binom{n}{n-1}, \] where \(\binom{n}{r} = \frac{n!}{r!(n-r)!}\). Verify your result by direct computation for the case \(n=4\).
(i) Prove that \(n^7-n\) is divisible by 42 for every positive integer \(n\). (ii) Prove that a number of the form \(n^2-n+1\), where \(n\) is a positive integer greater than 1, can never be a perfect square.
By considering an isosceles triangle with base angles \(\pi/5\), or otherwise, show that \[ \cos \frac{\pi}{5} = \frac{\sqrt{5}+1}{4}. \] Show how this result can be used to evaluate \(\cos(\pi/10)\) and \(\sin(\pi/10)\), and hence to obtain the sines and cosines of every angle of the form \(n\pi/60\), where \(n\) is a positive integer.
Prove that if, in measuring the three sides of a triangle, small errors \(x, y\) and \(z\) are made in the lengths of the sides \(a, b\) and \(c\) respectively, the resulting error in the calculated angle \(A\) will be \[ \frac{x}{c}\csc B - \frac{y}{c}\cot C - \frac{z}{b}\cot B \] approximately.
An isosceles triangle is circumscribed about a circle of given radius \(R\). Express the perimeter of the triangle as a function of its altitude, and find the altitude when the perimeter is a minimum. Suppose that the above figure is revolved about the altitude of the triangle, thus generating a right circular cone circumscribed about a sphere of radius \(R\). Find the altitude of the cone when the area of its curved surface is a minimum, and show that the minimum area is \(\pi(3+2\sqrt{2})R^2\).
Let \[ y=f(x) = \frac{\sinh^{-1} x}{\sqrt{1+x^2}}. \] Prove that \[ (1+x^2)\frac{dy}{dx} + xy = 1. \] Show that the Maclaurin series for \(f(x)\) is \[ x - \frac{2^2}{3!}x^3 + \frac{2^4(2!)^2}{5!}x^5 - \frac{2^6(3!)^2}{7!}x^7 + \dots. \]
(i) Find an indefinite integral of the function \[ \frac{1}{2+\sin x - \cos x}. \] (ii) Evaluate the integrals \[ \int_0^a \frac{a^2dx}{(x^2+a^2)^3}, \quad \int_0^a \frac{axdx}{(x^2+a^2)^3}, \quad \int_0^a \frac{x^2dx}{(x^2+a^2)^3}. \]
Let \((r,\theta)\) denote polar coordinates in the plane. (i) Find the area lying within both the circle \(r=1\) and the cardioid \(r=1+\sin\theta\). (ii) Find the area lying within both the circle \(r=\sqrt{2}\sin\theta\) and the lemniscate \(r^2=\cos 2\theta\).