Problems

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.

Two intersecting forces act on a rigid body along the lines \(OP, OQ\) respectively and are of magnitude \(\lambda OP, \mu OQ\). Shew that the magnitude and direction of their resultant are given by the form \((\lambda+\mu)OR\) along \(OR\), \(R\) being a certain point of \(PQ\), and extend the result to the cases of (a) two parallel forces, (b) several concurrent forces. Forces of given magnitudes and directions act in a plane at the points \(A_1, A_2, \dots, A_n\) respectively. Shew that if the system of points \((A_1, \dots, A_n)\) is rotated rigidly about any axis perpendicular to its plane, the forces remaining unaltered in magnitude and direction, the resultant always passes through a point fixed relatively to the system \((A_1, \dots, A_n)\).

Four rough uniform spheres equal in every respect are placed with three of them resting on a horizontal plane with their centres at a distance apart equal to three times their radius, and with the fourth resting symmetrically on top in contact with the other three. Find the limitations on the coefficients of friction between the spheres and between the spheres and plane, necessary for equilibrium.

A uniform inextensible rough string hangs over a fixed circular cylinder of radius \(a\) and horizontal axis, in a vertical plane perpendicular to this axis. \(\mu\) is the coefficient of friction between string and cylinder. If one free end of the string is at one point of the circular cross section where the tangent is vertical, shew that the greatest length of string which can hang vertically on the other side without causing slipping is \[ \frac{2\mu(1+e^{\mu\pi})}{\mu^2+1}a. \]