Show that the curve in rectangular coordinates of parametric equations $$x = at^2 + 2bt + c, \quad y = a't^2 + 2b't + c',$$ is a parabola of latus rectum \(2(a'b - ab')/(a^2 + a'^2)^{1/2}\). Find also the equation of the directrix of the parabola.
If for a triangle \(ABC\) the circumcentre is \(O\) and the orthocentre is \(H\), show that $$OH^2 = R^2(1 - 8\cos A \cos B \cos C),$$ where \(R\) is the radius of the circumcircle. Hence show that the circumcircle and nine-points circle of a triangle intersect in distinct real points only if the triangle is obtuse, and find an expression for the angle at which they then intersect.
Prove that \(\tan^2(\pi/11)\) is a root of the equation $$x^5 - 55x^4 + 330x^3 - 462x^2 + 165x - 11 = 0,$$ and state what are its other roots. By expressing the left-hand side in terms of \(\tan(\pi/11)\), or otherwise, prove that $$\tan(3\pi/11) + 4\sin(2\pi/11) = \sqrt{11}.$$
Let $$f(x) = k\cos x - \cos 2x,$$ where \(k\) is a constant, \(k > 0\). By considering the sign of \(f'(x)\), or otherwise, find the greatest and least values taken by \(f(x)\) for \(0 \leq x \leq \frac{1}{2}\pi\), distinguishing the various cases that arise according to the value of \(k\). Sketch the graph of \(y = f(x)\) for \(0 \leq x \leq \frac{1}{2}\pi\) in each case.
Two numbers \(a\) and \(b\) are given such that \(a > b > 0\). Two sequences \(a_n\) and \(b_n\) (\(n = 0, 1, 2, \ldots\)) are defined by the rules:
Complex numbers \(z = re^{i\theta}\) (\(r > 0\), \(\theta\) real) and \(w = u + iv\) (\(u\), \(v\) real) are connected by the relation $$2w = z + \frac{1}{z},$$ and \(z\) and \(w\) are represented by points in complex planes. Find the loci described by \(w\) when \(z\) describes the following curves:
Show that the increment in the radius \(R\) of the circumcircle of a triangle \(ABC\) due to small increments in the sides \(a\), \(b\), \(c\) is given by $$\delta R = \Sigma \frac{\delta a}{a} \cot B \cot C.$$ \(R\) is calculated from measurements of \(a\), \(b\), \(c\), and each measurement is liable to a small relative error \(\epsilon\) (so that, for example, \(\delta a\) can lie anywhere between \(\pm \epsilon a\)). Show that, when \(A\), \(B\), \(C\) are all acute, the calculated value of \(R\) is likewise liable to a relative error \(\epsilon\). How must this result be modified when \(A\) is obtuse?
Prove that $$\log \frac{n}{n-1} - \frac{1}{n} = \int_0^1 \frac{t}{(n-t)^n} dt \quad (n = 2, 3, \ldots).$$ Denoting the right-hand side by \(u_n\), prove that $$0 < u_n < \frac{1}{2(n-1)^n},$$ and that the series \(\sum_{n=2}^\infty u_n\) is convergent, with a sum \(U\) satisfying \(0 < U < \frac{1}{2}\). Deduce (or prove otherwise) that $$\sum_{n=1}^N \frac{1}{n} - \log N$$ tends to a limit \(\gamma\) as \(N \to \infty\), and that \(\frac{1}{2} < \gamma < 1\).
The function \(f(x)\) is defined, for \(x > 0\), by the formula $$f(x) = \int_0^{\pi/2} \frac{d\theta}{x + \cos\theta}.$$ Evaluate \(f(x)\), distinguishing the cases (i) \(0 < x < 1\), (ii) \(x = 1\), (iii) \(x > 1\), and expressing the results in cases (i) and (iii) in terms of the variable \(u\) defined by $$u^2 = \frac{|x-1|}{x+1}, \quad u > 0.$$ Prove from your results, or from the original definition, that \(f(x) \to f(1)\) when \(x \to 1\) from below and from above.
Let $$S_r = \int_0^{\pi/2} \sin^r\theta \, d\theta \quad (r \geq 0),$$ $$P_r = rS_rS_{r-1} \quad (r \geq 1),$$ where \(r\) is not necessarily an integer. Prove that