In a tetrahedron \(ABCD\) the edges \(AD\) and \(BC\) are perpendicular, \(AB = CD\), and \(AC = BD\). Prove that \(AB = AC\).
Show that there exists a circle that intersects a conic in the four points in which it meets its directrices. Show further that a quadrilateral can be drawn with one vertex at any point of the circle and inscribed in the circle and circumscribed about the conic. State any peculiarity of such a quadrilateral.
Prove Pascal's Theorem, that the intersections of opposite sides of a hexagon inscribed in a conic are collinear. Use this theorem to devise a construction to find the second point in which any given straight line through a point \(A\) meets the conic through \(A\) and touching two given straight lines \(OB\) and \(OC\) at \(B\) and \(C\) respectively.
Prove that the polars of a point \(P\) with respect to the conics through four fixed points will meet in another point \(Q\). Prove that as \(P\) moves on a given straight line \(l\), the locus of \(Q\) is a conic.
The area of a triangle is to be determined by the measurement of its sides. If the maximum small percentage error in the measurement of the sides is \(e\), prove that if the triangle is acute angled, the maximum percentage error in the calculated value of the area is approximately \(2e\) per cent. Explain briefly how the percentage error could be calculated when the triangle is obtuse.
Prove that \(\cot \theta - 2 \cot 2\theta = \tan \theta\). Hence or otherwise prove that: \[\frac{1}{2} \tan \frac{\theta}{2} + \frac{1}{2^2} \tan \frac{\theta}{2^2} + \ldots + \frac{1}{2^n} \tan \frac{\theta}{2^n} = \frac{1}{2^n} \cot \frac{\theta}{2^n} - \cot \theta.\] Deduce the result \[\frac{1}{\theta} = \cot \theta + \sum_{r=1}^{\infty} \frac{1}{2^r} \tan \frac{\theta}{2^r}.\]
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.