Problems

Find the sum of the series \[ c\sin(\alpha+\beta) + \frac{c^2}{2!}\sin(\alpha+2\beta) + \frac{c^3}{3!}\sin(\alpha+3\beta) + \dots. \]

Find all pairs of values of \(a\) and \(b\) for which the equation whose roots are the squares of the roots of the cubic \(x^3-ax^2+bx-1=0\) is identical with this cubic.

If \[ u = (a-b)^n + (b-c)^n + (c-a)^n, \] where \(n\) is a positive integer, prove that

1. [(i)] If \(u\) is divisible by \(\Sigma a^2 - \Sigma bc\) then \(n\) is of the form \(3k \pm 1\), where \(k\) is an integer.
2. [(ii)] If \(u\) is divisible by \((\Sigma a^2 - \Sigma bc)^2\) then \(n\) is of the form \(3k+1\).

A convex quadrilateral of sides \(a,b,c,d\) is inscribed in a circle of radius \(R\). Prove that \[ R = \frac{1}{4} \sqrt{\frac{(ab+cd)(ac+bd)(ad+bc)}{(s-a)(s-b)(s-c)(s-d)}}, \] where \[ s = \frac{1}{2}(a+b+c+d). \]

The equation \(\phi(x,y,z)=0\) defines \(z\) as a function of \(x,y\). Writing \[ \frac{\partial z}{\partial x} = p, \quad \frac{\partial z}{\partial y} = q, \quad \frac{\partial^2 z}{\partial x^2} = r, \quad \frac{\partial^2 z}{\partial x \partial y} = s, \quad \frac{\partial^2 z}{\partial y^2} = t, \] prove that, if the same equation is considered as defining \(x\) as a function of \(y,z\), then \[ \frac{\partial x}{\partial z} = \frac{1}{p}, \quad \frac{\partial x}{\partial y} = -\frac{q}{p}, \quad \frac{\partial^2 x}{\partial z^2} = -\frac{r}{p^3}, \quad \frac{\partial^2 x}{\partial y^2} = -\frac{t p^2 - 2spq + rq^2}{p^3}, \quad \frac{\partial^2 x}{\partial y \partial z} = \frac{sp-rq}{p^3}. \]

Find the limits, as \(n \to \infty\), of

1. [(i)] \(\frac{\log a_1 + \log a_2 + \dots + \log a_n}{n \log n}\) where \(a_1, a_2, \dots, a_n\) are any \(n\) numbers included between \(n\) and \(2n\);
2. [(ii)] \(\frac{\sqrt{n^2+1} + \sqrt{n^2+2} + \dots + \sqrt{n^2+n}}{n^2}\).

Evaluate \(\int \sin^m\theta \,d\theta\) for positive and negative integral values of \(m\).

If \(m_1, m_2, m_3\) are three points of a circle \(C\) of radius \(R\), find the limiting value of the radius of the circumcircle of the triangle formed by the tangents to the circle \(C\) at the points \(m_1, m_2, m_3\), as \(m_2, m_3\) approach \(m_1\).

Prove the Leibniz formula for the \(n\)th derivative of the product of two functions. Find the \(n\)th derivatives of the functions \[ \frac{x^2}{(1+5x)^4}, \quad \arcsin x. \]

Sketch the curve \[ y(y+1)(y+2)-(x-2)x(x+2) = 0 \] and prove that the point \((0,-1)\) is a point of inflexion.