Resolve the expression \[ y = \frac{2(1-x)}{(x^2+1)^2(x+1)} \] into real partial fractions. Show that \[ \int_0^\infty y\,dx = \frac{\pi}{2}-1. \]
If \(\alpha, \beta, \gamma\) are three real angles, and if the equations \begin{align*} x\cos\alpha + y\sin\alpha &= 1, \\ x\cos\beta + y\sin\beta &= 1, \\ x\cos\gamma + y\sin\gamma &= 1 \end{align*} hold simultaneously, show that \(\cos 2\alpha, \cos 2\beta\) and \(\cos 2\gamma\) cannot all be different.
The altitudes of an obtuse-angled triangle \(ABC\) intersect at a point \(H\). Prove that the circumcircle subtends at \(H\) an angle \(\theta\) whose cosine is \[ \frac{8\cos A\cos B\cos C+1}{8\cos A\cos B\cos C-1}, \] and show that \(\theta\) is always greater than \(2\sin^{-1}\frac{1}{3}\).
Obtain an expression for \(\tan 7\theta\) in terms of \(\tan\theta\), and find the value of \[ \cot\frac{\pi}{7}\cot\frac{2\pi}{7}\cot\frac{3\pi}{7}. \] Prove that \[ \cot\frac{\pi}{7}+\cot\frac{2\pi}{7}-\cot\frac{3\pi}{7} = \sqrt{7}. \]
Find the sum of the first \(n\) terms of each of the following series
A circular field of unit radius is divided in two by a straight fence. In the smaller segment a circular enclosure as large as possible is fenced off. Show that the total area of the two remaining pieces of the segment can at most be \(\psi-\sin\psi\), where \(\psi=2\tan^{-1}(4/\pi)\).
Find the asymptotes of the curve \[ x^3+2x^2+x = xy^2 - 2xy + y. \] Show that there are values which \(y-x\) never takes, and sketch the curve.
Evaluate the following integrals (in which \(\sqrt{\phantom{x}}\) means the positive square root):
The roots of the cubic equation \[ ax^3+bx^2+cx+d=0 \quad (a\neq 0, d\neq 0) \] are \(\alpha, \beta, \gamma\). Find the equations whose roots are (i) \(\displaystyle\frac{1}{\alpha}, \frac{1}{\beta}, \frac{1}{\gamma}\); (ii) \(\alpha^2, \beta^2, \gamma^2\); (iii) \(\beta\gamma/\alpha, \gamma\alpha/\beta, \alpha\beta/\gamma\). Deduce a necessary and sufficient condition that the product of two of the roots of the original equation should be equal to the third root.
The polynomials \(f(x), g(x)\) are of degrees \(m,n\) respectively, where \(m\ge n\ge 1\), and have real coefficients. Show that polynomials \(q(x), r(x)\) exist such that \(f(x)=g(x)q(x)+r(x)\), and such that the degree of \(r(x)\) is less than \(n\). Show also that \(q(x), r(x)\) are unique, and have real coefficients. If \(g(x) = x^2 - (\alpha+\beta)x + \alpha\beta\), prove that, if \(\alpha\neq\beta\), \[ r(x) = \frac{f(\alpha)-f(\beta)}{\alpha-\beta}x + \frac{\alpha f(\beta) - \beta f(\alpha)}{\alpha-\beta}. \] Verify that the coefficients of \(r(x)\) are real if \(\alpha, \beta\) are conjugate complex numbers. Find also the form of \(r(x)\) if \(\alpha=\beta\).