\(k\) integers are selected from the integers 1, 2, ..., \(n\). In how many ways is it possible if
Describe the path traced out by the point \(w = z+ 1/z\) in the Argand diagram as the point \(z\) traces out the circle \(|z| = r > 0\). The sequence \(w_0\), \(w_1\), \(w_2\), ... is defined by the recurrence relation \(w_n = w_{n-1}^2 - 2\). Show that if \(w_0\) is real and satisfies \(-2 \leq w_0 \leq 2\), then the same is true for all \(w_n\), and that for all other real or complex values of \(w_0\), \(|w_n| \to \infty\).
Show that the set of complex valued \(2 \times 2\) matrices of the form $\begin{pmatrix} z & w\\ -\overline{w} & \overline{z} \end{pmatrix}$ satisfying \(|z|^2+ |w|^2 = 1\) forms a group \(G\) under matrix multiplication. Determine the subsets \(G_2\) consisting of all elements of \(G\) whose square is the identity matrix, and \(G_4\) consisting of all elements of \(G\) whose fourth power is the identity matrix. Do they form subgroups of \(G\)?
State precisely, without proof, the arithmetic-geometric mean inequality. The equation \(f(x) = x^n+a_1x^{n-1}+a_2x^{n-2}+ ... + a_n = 0\) has \(n\) distinct positive roots. Writing \(a_i = (-1)^i\binom{n}{i}b_i\), where \(\binom{n}{i}\) denotes the usual binomial coefficient, prove that \(b_{n-1} > b_n\). By considering \(f'(x)\), or otherwise, prove further that \(b_1 > b_2 > ... > b_{n-1} > b_n\).
Let \(C_1\), \(C_2\) and \(C_3\) be circles in the plane, each pair of which intersect in two points. The common tangents to \(C_2\) and \(C_3\) meet at \(P_1\), and points \(P_2\) and \(P_3\) are defined similarly. Prove that \(P_1\), \(P_2\) and \(P_3\) are collinear. What is the analogous result if the circles are mutually disjoint?
Two adjacent corners \(A\), \(B\) of a rigid rectangular lamina \(ABCD\) slide on the \(x\)-axis and the \(y\)-axis respectively, and all the motion is in one plane. Prove that the locus of \(C\) is an ellipse, and find the area of the ellipse in terms of \(a = AD\) and \(b = AB\). [The area of an ellipse is \(\pi \times\) the product of the lengths of the semi-axes.]
\(P\) and \(Q\) are the intersections of the line \(lx + my + n = 0\) with the parabola \(y^2 = 4ax\). The circle on \(PQ\) as diameter meets the parabola again in \(R\) and \(S\). Find the equation of \(RS\).
A triangle \(ABC\) has area \(\Delta\), and \(P\) is an interior point. The line through \(P\) parallel to \(BC\) cuts \(AB\) in \(W\) and \(AC\) in \(T\); the line through \(P\) parallel to \(CA\) cuts \(AB\) in \(V\) and \(BC\) in \(S\); and the line through \(P\) parallel to \(AB\) cuts \(AC\) in \(U\) and \(BC\) in \(R\). The triangles \(PBC\), \(PCA\), \(PAB\) have areas \(\alpha\), \(\beta\), \(\gamma\) respectively and the triangles \(AVU\), \(BWR\), \(CST\) have areas \(\alpha'\), \(\beta'\), \(\gamma'\) respectively. By first showing that $$\frac{\beta'}{\gamma} = \frac{\alpha}{\Delta},$$ prove that the area of the hexagon \(RSTUVW\) is at least \(\frac{9}{8}\Delta\). [You may assume that if positive numbers \(p\), \(q\), \(r\) add up to 1, then \(pq +qr+rp\) cannot exceed \(\frac{1}{3}\).]
Let \(I_n = \int_0^{\pi/4} \tan^n\theta d\theta\). Obtain an expression for \(I_n\) in terms of \(I_{n-2}\), and hence evaluate \(I_4\) and \(I_5\). Show that for all \(n \geq 1\), \(0 \leq I_{4n} \leq \frac{1}{4}n - \frac{3}{8}\), and \(0 \leq I_{4n-2} \leq 1- \frac{1}{4}n\).
By considering the integral \(\int_1^x \frac{dt}{t}\) or otherwise, prove that \(0 < \log x < x\) for all \(x > 1\). Hence show that for fixed \(k > 0\), \(\frac{\log x}{x^k}\) tends towards 0 as \(x\) tends towards infinity. (You may find it helpful to use the substitution \(y = x^n\) in the first inequality.) Deduce that \(x^k \log x\) tends towards 0 as \(x\) tends towards 0 through positive values. Use this theory to investigate the behaviour of the function \(y = x^x\) (\(x > 0\)) when \(x\) is near to 0. Sketch the graph of \(y = x^x\) for values of \(x > 0\).