(i) Use de Moivre's theorem to express \(\cos 6\theta\) and \(\sin 6\theta\) in terms of powers of \(\cos\theta\) and \(\sin\theta\). (ii) Let the roots of the equation \(z^4 - 1 = i\sqrt{3}\) be \(z_r\) (\(r = 1, 2, 3, 4\)), where \(z_r\) lies in the \(r\)th quadrant of the complex plane. Show that \[(z_1 + z_3) = -(z_3 + z_4) = 2^{-\frac{1}{2}}(1 + i\sqrt{3}).\]
Prove that, if \(x\) and \(y\) are real numbers, and \(\max(x, y)\) denotes the greater of \(x\) and \(y\) when \(x \neq y\), and their common value when \(x = y\), then \[\max(x, y) = \frac{1}{2}(x + y) + \frac{1}{2}|x - y|.\] Explain what is meant by saying that a real-valued function \(f\), defined on an interval of the real line, is continuous at a point of the interval. Suppose that \(f\) and \(g\) are defined in the same interval \(I\), and that \(h(x) = \max[f(x), g(x)]\) for all \(x\) in \(I\); prove that, if \(f\) and \(g\) are both continuous at a point \(x_0\) in \(I\), then \(h\) is also continuous at \(x_0\). Give an example to show that if \(f\) and \(g\) are both differentiable at \(x_0\), then \(h\) is not necessarily differentiable at \(x_0\); and another example to show that \(h\) can be differentiable at \(x_0\).
Two sequences \((x_0, x_1, x_2, \ldots)\) and \((y_0, y_1, y_2, \ldots)\) of positive integers are defined inductively by taking \(x_0 = 2\), \(y_0 = 1\), and counting rational and irrational parts in the equations \[x_n + y_n\sqrt{3} = (x_{n-1} + y_{n-1}\sqrt{3})^2 \quad (n = 1, 2, 3, \ldots).\] Prove that \[x_n^2 - 3y_n^2 = 1 \quad (n = 1, 2, 3, \ldots),\] and that when \(n \to \infty\), the sequences \(x_n/y_n\) and \(3y_n/x_n\) tend to the limits \(\sqrt{3}\) from above and below respectively. By carrying this process far enough, obtain two rational numbers enclosing \(\sqrt{3}\) and differing from one another by less than \(5 \times 10^{-9}\).
The triangle \(ABC\) is inscribed in a circle \(K\) of radius \(R\), and its angles are all acute. If small changes \(\delta a\), \(\delta b\), \(\delta c\) are made in the sides \(a\), \(b\), \(c\) of the triangle in such a way that it remains inscribed in \(K\), prove that \[\frac{\delta a}{\cos A} + \frac{\delta b}{\cos B} + \frac{\delta c}{\cos C} = 0\] approximately. Discuss what happens when \(C\) is a right angle. Show also that, if \(S\) is the area of the triangle, then the small change \(\delta S\) in \(S\) under the same conditions is given approximately by the equation \[\frac{\delta S}{S} = \frac{\delta a}{a} + \frac{\delta b}{b} + \frac{\delta c}{c}.\] [The formula \(a = 2R\sin A\) may be assumed.]
A leaf of a book is of width \(a\) and height \(b\), where \(3a \leq 2\sqrt{2}b\); the lower corner of the leaf is folded over so that the corner just reaches the inner edge of the page. Find the minimum length of the resulting crease. Explain why the condition \(3a \leq 2\sqrt{2}b\) is relevant.
Prove that, if \(|x| < 1\), then \[x - \frac{1}{2}\left(\frac{2x}{1+x^2}\right) = \frac{1}{2.4}\left(\frac{2x}{1+x^2}\right)^3 + \frac{1.3}{2.4.6}\left(\frac{2x}{1+x^2}\right)^5 + \frac{1.3.5}{2.4.6.8}\left(\frac{2x}{1+x^2}\right)^7 + \cdots\] What function is represented by the infinite series when \(|x| > 1\)?
Solution: Consider \begin{align*} \frac{1 - \sqrt{1-y^2}}{y} &= \frac{1 - \sum_{k=0}^{\infty} \binom{-1/2}{k}(-y^2)^k}{y} \\ &= \frac{1}{y} \left (-\sum_{k=1}^{\infty} \binom{1/2}{k}(-y^2)^k \right) \\ &= -\sum_{k=1}^\infty \frac{\left(\frac12 \right)\left(-\frac12 \right)\left(-\frac32 \right)\cdots \left(-\frac{2k-3}2 \right)}{k!} (-1)^ky^{2k-1} \\ &= \sum_{k=1}^\infty \frac{\left (\frac12 \right)\left(\frac12 \right)\left(\frac32 \right)\cdots \left(\frac{2k-3}2 \right)}{k!} y^{2k-1} \\ &= \sum_{k=1}^\infty \frac{1 \cdot 3 \cdots (2k-3)}{2 \cdot 4 \cdots (2k)} y^{2k-1} \\ \end{align*} But since \(\frac{1-\sqrt{1-y^2}}{2}\) is the inverse for \(\frac{2x}{1+x^2}\) for \(|x| < 1\) this computes the inverse. When \(|x| > 1\) it will compute \begin{align*} && \frac{1-\sqrt{1- \left (\frac{2x}{1+x^2} \right)^2}}{\frac{2x}{1+x^2}} &= \frac{1 - \sqrt{\frac{(1-x^2)^2}{(1+x^2)^2}}}{\frac{2x}{1+x^2}} \\ &&&= \frac{1 - \frac{x^2-1}{1+x^2}}{\frac{2x}{1+x^2}} \\ &&&= \frac{1}{x} \end{align*} and in particular the series is equal to \(\frac{1}{x} - \frac{1}{2}\left(\frac{2x}{1+x^2}\right)\)
By means of the calculus or otherwise, prove that if \(p > q > 0\) and \(x > 0\), then \[q(x^p - 1) > p(x^q - 1).\] Hence or otherwise prove that, under the same conditions, \[\frac{1}{p}\left(\frac{x^p}{(p+1)^p} - 1\right) > \frac{1}{q}\left(\frac{x^q}{(q+1)^q} - 1\right)\] for every positive integer \(n\).
Prove that, if \(x > 0\) and \(N\) is a positive integer, then \[\frac{1}{2^x} + \frac{1}{3^x} + \cdots + \frac{1}{(N+1)^x} < \int_1^{N+1} \frac{dx}{x^x} < 1 + \frac{1}{2^x} + \cdots + \frac{1}{N^x}.\] Deduce, or prove otherwise, that \(\sum_{n=1}^{\infty} n^{-x}\) is convergent when \(x > 1\) and divergent when \(x < 1\). Find the set of values of the real number \(\beta\) for which the infinite series \[\sum_{n=1}^{\infty} \frac{n^{\beta}}{n^{2\beta} - n^{\beta} + 1}\] is convergent, and the set of values of \(\beta\) for which it is divergent.
(i) A groove of semicircular section, of radius \(b\), is cut round a right circular cylinder of radius \(a\), where \(a > b\); find the surface area of the groove. (ii) Suppose that the region \(R\) of area \(A\) in the first quadrant of the \((x, y)\)-plane generates a solid of revolution of volume \(U\) when it is revolved about the \(x\)-axis, and a solid of revolution of volume \(V\) when it is revolved about the \(y\)-axis. Find the volume generated by \(R\) when it is revolved about the straight line whose equation is \[x\cos\alpha + y\sin\alpha = p,\] assuming that this line does not meet \(R\). Explain why the sign of the expression obtained for the volume appears to be negative for certain positions of the line, and describe the positions for which this happens.
A certain statistical procedure to be applied to the numbers \(x_1, x_2, \ldots, x_n\) requires the calculation of the median of the numbers \(x_r\). Construct a flow diagram for the solution of this problem, where \(n\) is odd and is included in the data, and \(x_1, x_2, \ldots, x_n\) are available in that order. Carry out all the steps and obtain the solution when \(n = 5\) and the numbers \(x_1, \ldots, x_5\) are \(5, 1, 2, 4, 3\) respectively.