Prove that, if \(x_1\) and \(x_2\) are connected by the relation $$ax_1x_2 + bx_1 + cx_2 + d = 0,$$ there are, in general, two unequal values \(m\), \(n\) of \(x_1\) for which \(x_2 = x_1\); and that the relation is equivalent to $$\frac{x_2 - m}{x_2 - n} = k \frac{x_1 - m}{x_1 - n},$$ where \(k\) is a root of the equation $$(bc - ad)(k^2 + 1) + (b^2 + c^2 - 2ad)k = 0.$$ Find the relation between \(a\), \(b\), \(c\), \(d\) in order that the equations \begin{align} ax_1x_2 + bx_1 + cx_2 + d &= 0,\\ ax_1x_3 + bx_2 + cx_3 + d &= 0,\\ ax_2x_4 + bx_3 + cx_4 + d &= 0,\\ ax_3x_4 + bx_4 + cx_1 + d &= 0 \end{align} may be satisfied by values of \(x_1\), \(x_2\), \(x_3\), \(x_4\) which are all different.
Write down the (complex) factors of \(x^2 + y^2 + z^2 - yz - zx - xy\). If \(x\), \(y\), \(z\), \(a\), \(b\), \(c\) are real and \(ax + by + cz = 0\), prove that the product of $$\frac{x^2 + y^2 + z^2 - yz - zx - xy}{(x + y + z)^3} \text{ and } \frac{a^3 + b^3 + c^3 - bc - ca - ab}{(a + b + c)^3}$$ cannot be less than \(\frac{1}{4}\). Find the ratios \(x:y:z\) in terms of \(a\), \(b\), \(c\) if the product is equal to \(\frac{1}{4}\).
Express \(\tan n\theta\) in terms of \(\tan \theta\), where \(n\) is a positive integer. If \(n\) is odd, prove that $$n \tan n\theta = 1 + 2 \sum_{r=1}^{(n-1)/2} \frac{\sec^2(2r-1)\alpha}{\tan^3(2r-1)\alpha - \tan^3 \theta},$$ where \(2n\alpha = \pi\).
A flagstaff leaning due north at an angle \(\alpha\) to the vertical subtends angles \(\phi_1\) and \(\phi_2\) respectively, from two points \(P_1\) and \(P_2\) on a horizontal road leading north-west from its base. Prove that the length of the flagstaff is $$\frac{\pm \sqrt{2b} \sin \phi_1 \sin \phi_2}{\sin(\phi_1 - \phi_2)(2 - \sin^2 \alpha)^{1/2}},$$ where \(b\) is the distance \(P_1P_2\).
Prove that the area of the greatest equilateral triangle which can be drawn with its three sides passing respectively through three given points \(A\), \(B\), \(C\) is $$2\Delta + \frac{a^2 + b^2 + c^2}{2\sqrt{3}},$$ where \(a\), \(b\), \(c\) are the sides of the triangle \(ABC\), and \(\Delta\) is its area.
If \(ax^2 + 2hxy + by^2 + 2gx + 2fy + c = 0\), prove that $$y_3 = \frac{d^2y}{dx^2} = \frac{k}{(hx + by + f)^3},$$ where \(k\) is a constant. Prove also that $$\frac{d}{dx}\left(\frac{1}{y_3}\frac{d^2}{dx^2}(y_3^{-1})\right) = 0,$$ and express this result rationally in terms of derivatives of \(y\) with respect to \(x\).
Evaluate $$\int_0^1 \frac{dx}{(1 + x + x^2)^{3/2}} \quad \int_0^{2\pi} \frac{\sin^2 \theta}{a - b\cos \theta} d\theta \quad (a > b > 0).$$ By remarking that, when \(0 \leq x \leq 1\), we have \(0 \leq x^3 \leq x^2\), prove that $$0.35 < \int_0^1 \frac{dx}{(9 - 4x^3 + x^6)^{1/2}} < 0.37.$$
The rectangular cartesian coordinates \(x\), \(y\) of a point \(P\) on a closed oval curve are given as functions of the arc \(s\) measured from a fixed point of the curve in such a direction that the inclination of the tangent to the \(x\)-axis increases with \(s\). Prove that if the coordinates of the point \(Q\) at a distance \(t\) from \(P\) along the outward drawn normal are $$X = x + t\sin \psi, \quad Y = y - t\cos \psi,$$ where \(\cos \psi = dx/ds\), \(\sin \psi = dy/ds\). Prove that, if \(t\) is a function of \(s\), $$X\frac{dY}{ds} - Y\frac{dX}{ds} = x\frac{dy}{ds} - y\frac{dx}{ds} + \{t(x\cos \psi + y\sin \psi)\} + 2t + \rho\kappa,$$ where \(\kappa = d\psi/ds\) is the curvature of the given curve at \(P\). Deduce that, if \(t = 1/\kappa\), the area enclosed by the curve described by \(Q\) is $$A + \frac{3}{2}\int \frac{ds}{\kappa},$$ where \(A\) is the area enclosed by the original curve and the integral is taken round it.
The equations \begin{align} x^3 + 2x^2 + ax - 1 &= 0, \\ x^3 + 3x^2 + x + b &= 0 \end{align} have two common roots. Find all possible pairs of values of the constants \(a\), \(b\), and find the remaining roots in each case.
By the use of complex numbers or otherwise, evaluate the sums \(\sum_{n=0}^{\infty} r^n \cos n\theta\) where \(0 < r < 1\). Hence write \[ \frac{(1 - r^2 \cos 2\theta) r \cos \theta - r^3 \sin \theta \sin 2\theta}{1 - 2r^2 \cos 2\theta + r^4} \] in the form \(\sum a_n r^n \cos n\theta\), where the \(a_n\) are constants to be determined.