Let \(G\) be the group of symmetries of the equilateral triangle \(ABC\). Express all the symmetries of the triangle under reflection and rotation by the permutations which they induce on the letters \(A\), \(B\), \(C\). Let \(R\) in \(G\) correspond to a rotation of the triangle by \(\frac{2\pi}{3}\), and \(M\) in \(G\) to a reflection of the triangle about an altitude. Show that all elements of \(G\) may be expressed as \(R^i\) or \(R^i M\) for \(i = 0, 1, 2\). Given a subgroup \(H\) of \(G\) and an element \(g\) of \(G\), we define \(gH\) to be the set \(\{gh : h \in H\}\); similarly \(Hg = \{hg : h \in H\}\). \(H\) is then said to be a normal subgroup if \(gH = Hg\) for all \(g\) in \(G\). Show that the subgroup \(\{E, R, R^2\}\), where \(E\) is the identity permutation of \(G\), is normal. Find a subgroup of \(G\) which is not normal, and justify your answer.
A square matrix \(B\) has an inverse \(B^{-1}\); \(B\) satisfies \[BX = \lambda X\] for some scalar \(\lambda\) and non-zero column vector \(X\). Show that the inverse of \(B\) satisfies \[B^{-1}X = \lambda^{-1}X.\] For \(B = \begin{pmatrix} 3 & 1 \\ 1 & 3 \end{pmatrix}\) show there are exactly two values \(\lambda_1\), \(\lambda_2\) such that (*) has a solution for \(X\), and find corresponding normalized vectors \(X_1\), \(X_2\) (a column vector \(\begin{pmatrix} x \\ y \end{pmatrix}\) is normalized if \(x^2 + y^2 = 1\)). Show that \(\lambda_1 X_1 X_1^T + \lambda_2 X_2 X_2^T = B\), where \(X_i^T\) is the row vector transpose of \(X_i\), \(i = 1, 2\). Assuming a similar representation for \(B^{-1}\), determine \(B^{-1}\).
Five points \(A\), \(B\), \(C\), \(D\) and \(E\) lie in that order on a circle. The lengths \(AB\) and \(DE\) are equal, and the lengths \(BC\) and \(CD\) are equal. The tangents to the circle at \(C\) and \(A\) meet at \(F\), and the line \(AB\) is extended to meet \(FC\) at \(G\). Prove that triangles \(AFG\) and \(EBA\) are similar.
Let a convex quadrilateral \(Q\) have sides \(a\), \(b\), \(c\), \(d\). Let \(a\) and \(b\) include the angle \(\alpha\), \(c\) and \(d\) the angle \(\beta\), and define \(\gamma = \alpha + \beta\). If \(A\) denotes the area of \(Q\), show that \begin{align} \text{(i)} \quad & 2A = ab\sin\alpha + cd\sin\beta,\\ \text{(ii)} \quad & a^2 + b^2 - 2ab\cos\alpha = c^2 + d^2 - 2cd\cos\beta, \end{align} and deduce that \begin{align} \text{(iii)} \quad & 16A^2+(a^2+b^2-c^2-d^2)^2 = 4(ab+cd)^2-16abcd\cos^2(\tfrac{1}{2}\gamma),\\ \text{(iv)} \quad & A^2 = (s-a)(s-b)(s-c)(s-d) - abcd\cos^2(\tfrac{1}{2}\gamma), \end{align} where \(2s = a+b+c+d\). By using the inequality relating geometric and arithmetic means, deduce that, among all quadrilaterals of given perimeter, the square has the greatest area. [Standard formulae for triangles may be quoted without proof.]
Show that \[\cos 3\theta = 4\cos^3\theta - 3\cos\theta\] (thus expressing \(\cos 3\theta\) as a cubic in \(\cos\theta\)). Show that if we can express \(\cos m\theta\) and \(\sin\theta\sin(m-1)\theta\) as polynomials of degree at most \(m\) in \(\cos\theta\) for all \(m\) with \(1 \leq m \leq n\), then we can express \(\cos(n+1)\theta\) and \(\sin\theta\sin n\theta\) as polynomials of degree at most \(n+1\) in \(\cos\theta\). Deduce that \[\cos n\theta = \sum_{r=0}^{n} a_{nr}(\cos\theta)^r\] for suitable real numbers \(a_{n0}, a_{n1}, \ldots, a_{nn}\). If we write \[T_n(x) = \sum_{r=0}^{n} a_{nr}x^r,\] show, using the fact that \(T_n(x) = \cos(n\cos^{-1}x)\) for \(|x| \leq 1\), or otherwise, that \begin{align} \text{(i)} \quad & |T_n(x)| \leq 1 \text{ for } |x| \leq 1,\\ \text{yet (ii)} \quad & |T_n'(1)| = n^2. \end{align} [Hint for (ii): If \(f\) is continuous then, automatically, \(f(1) = \lim_{x \to 1}f(x)\).]
Find a solution to the differential equation \[\frac{dy}{dt} = 2(2y^{\frac{1}{2}} - y^2)^{\frac{1}{2}}\] for which \(y = 1\) when \(t = 0\).
Prove that the curves \(y = \frac{3x}{2}\) and \(y = \sin^{-1}x\) intersect precisely once in the range \(0 < x \leq 1\); \(\sin^{-1}x\) is to be interpreted as the value of \(\theta\) between 0 and \(\frac{1}{2}\pi\) for which \(\sin\theta = x\). Sketch, on the same axes, these two functions for this range of \(x\). Use this sketch to illustrate graphically the sequence of numbers \(q_n\) governed by \[q_{n+1} = \sin\left(\frac{3q_n}{2}\right), \quad q_0 = \frac{1}{2},\] and deduce from the picture that the sequence converges as \(n \to \infty\) to a number less than 1.
Evaluate the indefinite integral \[\int \frac{d\theta}{a + \cos\theta},\] where \(a > 1\), using the substitution \(t = \tan\frac{1}{2}\theta\) or otherwise. What is the value of \[\int_0^{2\pi}\frac{d\theta}{a+\cos\theta}?\] What happens to the latter integral as \(a \to 1\) from above?
Let \((a,b)\) be a fixed point, and \((x,y)\) a variable point, on the curve \(y = f(x)\) (where \(z > a\), \(f'(x) \geq 0\)). The curve divides the rectangle with vertices \((a,b)\), \((a,y)\), \((x,y)\) and \((x,b)\) into two portions, the lower of which has always half the area of the upper. Show that the curve is a parabola with its vertex at \((a,b)\).
By using diagrams or otherwise, explain why \[\sum_{r=n}^{\infty} r^{-2} > \int_{n}^{\infty} x^{-2}dx > \sum_{r=n+1}^{\infty} r^{-2}.\] If we write \(A = \sum_{r=1}^{\infty} r^{-2}\), show that \[n^{-1} > A - \sum_{r=1}^{n} r^{-2} > (n+1)^{-1}.\] How large must we take \(n\) to ensure that \(\sum_{r=1}^{n} r^{-2}\) approximates \(A\) with an error of less than \(10^{-4}\)? Show that, for the same \(n\), \[(n+1)^{-1} + \sum_{r=1}^{n} r^{-2}\] approximates \(A\) with an error of less than about \(10^{-8}\).