A piece of paper has the shape of a triangle \(ABC\), where \(\angle BCA = \frac{1}{5}\pi\), \(\angle CAB = \frac{2}{5}\pi\), \(AB = c\). It is folded so that \(C\) coincides with a point of \(AB\), and the crease meets \(CA\) at \(Y\). Show that the minimum area of the triangle \(XYC\) is $$\frac{c^2 \sin^2 x \cos^2 x}{4 \sin \frac{3}{5}(\pi - x) \sin^2 \frac{1}{5}(\pi + 2x)}.$$
The sequence \(x_0, x_1, x_2, \ldots\) satisfies the relation $$2n^2 x_{n+1} = x_n (3n^2 - x_n^2),$$ where \(n = 0, 1, 2, \ldots\) Show that, if \(0 < x_0 < a\), then (i) \(0 < x_n < a\); (ii) \(x_n < x_{n+1}\); (iii) \(\lim_{n \to \infty} x_n\). Show also that, for \(n \geq 1\), $$a - x_n < \frac{2a}{3} \left[\frac{3(a-x_0)}{2a}\right]^{2^n}.$$
Two variable complex numbers \(z\) and \(w\) are connected by $$w = \frac{z + i}{1 + iz}.$$ The point in the complex plane (Argand diagram) represented by \(z\) describes a circle with centre \(z_0\). Find \(z - z_0\) as a function of \(w\), and hence show that the point \(w\) also describes a circle, which is orthogonal to the line joining \(-i\) to \((z_0 + i)/(1 + iz_0)\).
Sketch the curve $$x^3 + y^2 = 3xy.$$ By rotating the axes through \(45^\circ\), or otherwise, find the area of its loop.
Find:
Obtain a recurrence relation connecting \(F(p)\) and \(F(p+1)\), where \(F(p) = \int_0^1 x^p (1-x)^{-1/4} dx,\) Hence, or otherwise, evaluate \(F(2)\) and \(F(\frac{3}{2})\).
Show that \(y = (x + (x^2 + 1)^{1/2})^k\) satisfies the differential equation \((x^2 + 1)y'' + xy' - k^2y = 0,\) Derive an equation connecting \(y^{(n)}(x)\), \(y^{(n+1)}(x)\) and \(y^{(n+2)}(x)\). Hence show that, if \(k\) is an integer, then \(y^{(k+1)}(x) = A(x^2 + 1)^{-k-\frac{1}{2}},\) where \(A\) is a constant, and find \(A\).
Prove that, if \(y > x > 0\) and \(k > 0\), then \(x^k (y-x) < \int_x^y t^k dt < y^k (y-x).\) Hence show that \(\lim_{n \to \infty} \{n^{k-1}(1^k + 2^k + \ldots + n^k)\} = \frac{1}{k+1}.\)
Show that, if \(u = r + x\), \(v = r - x\), where \(r = (x^2 + y^2)^{1/2}\), and \(f(x,y) = g(u,v)\), then \(r\left(\frac{\partial^2 f}{\partial x^2} + \frac{\partial^2 f}{\partial y^2}\right) = 2u \frac{\partial^2 g}{\partial u^2} + 2v \frac{\partial^2 g}{\partial v^2} + \frac{\partial g}{\partial u} + \frac{\partial g}{\partial v}.\)
Prove that the solution of the differential equation \(\frac{dy}{dx} + ay = f(x),\) where \(a\) is constant, is \(y = y_0 e^{-ax} + \int_0^x f(u) e^{a(u-x)} du,\) where \(y_0 = y(0)\). Hence, or otherwise, solve \(\frac{d^2 y}{dx^2} + (a+b) \frac{dy}{dx} + aby = \begin{cases} 1, & (0 < x < 1) \\ 0, & (x > 1), \end{cases}\) where \(a\) and \(b\) are constants \((a \neq b)\), given that \(y = 0\), \(dy/dx = 0\) for \(x = 0\), and that \(y\) and \(dy/dx\) are continuous.