Two flywheels, whose radii of gyration are in the ratio of their radii, are free to rotate in the same plane, a belt passing around both. Initially one, of mass \(m_1\) and radius \(a_1\), is rotating with angular velocity \(\Omega\), and the other, of mass \(m_2\) and radius \(a_2\), is at rest. Suddenly the belt is tightened, so that there is no more slipping at either wheel. Show that the second wheel begins to rotate with angular velocity $$\frac{m_1 a_1 \Omega}{(m_1 + m_2)a_2}.$$
A large flat circular disc, of moment of inertia \(mk^2\), is free to rotate in a horizontal plane about an axis through its centre. A particle of mass \(m\) is projected with velocity \(V\), from a point distant \(a\) from the centre of the disc, along a smooth groove cut into its upper surface, the disc being initially at rest. The groove is in the form of an equiangular spiral, whose equation in polar coordinates is given by $$\log r = \theta \tan \alpha.$$ Show that, for the particle to come to rest with respect to the disc, \(a^2 \cos^2 \alpha\) must be greater than \(k^2\) and the particle must be projected towards the centre of the disc.
A point \(P\) is taken at random inside an ellipse of eccentricity \(e\). Calculate, in terms of \(e\), the probability that the sum of the focal distances of \(P\) should be not greater than the distance from a focus to the opposite end of the major axis.
(i) Solve the differential equation $$\frac{d^2y}{dx^2} - \frac{dy}{dx} = e^x$$ subject to the conditions that \(y = d^2y/dx^2 = 2\) when \(x = 0\). (ii) Find the general solution of the differential equation $$\frac{1-x^2}{y^2}\frac{dy}{dx} + \frac{x}{y} + \sin^{-1}x = 0 \quad (-1 \leq x \leq 1)$$ by taking as a new variable a suitably chosen power of \(y\), or otherwise.
The function \(f(x)\) is defined by $$f(x) = \begin{cases} -\pi - x & \text{if } -\pi \leq x < -\frac{1}{2}\pi, \\ x & \text{if } -\frac{1}{2}\pi \leq x < \frac{1}{2}\pi, \\ \pi - x & \text{if } \frac{1}{2}\pi \leq x \leq \pi. \end{cases}$$ Show that the value of \(A\) that makes the maximum of \(|f(x) - A\sin x|\) for \(-\pi \leq x \leq \pi\) as small as possible is a root of the equation $$A + (A^2 - 1)^{\frac{1}{2}} - \cos^{-1}(A^{-1}) = \frac{1}{4}\pi.$$
\(AB\) is the segment \(0 \leq x \leq 1\); at each point \(P\) of \(AB\) whose distance from \(A\) is of the form \(m/2^n\) (where \(n\) is a non-negative integer \(> 1\), and \(m\) is an odd integer) a line \(QPQ'\) is drawn, perpendicular to \(AB\), with \(QP = PQ' = \lambda^n\) (\(0 < \lambda < 1\)). Calculate, in terms of \(\lambda\), the area of the smallest convex figure containing \(A\), \(B\), and all the points \(Q\), \(Q'\). [A figure is convex if whenever it contains two points it also contains all the points on the straight line segment between them.]
The polynomial \(p(x)\) is real and non-negative for all real values of \(x\). Prove that it is possible to write $$p(x) = \{q(x)\}^2 + \{r(x)\}^2,$$ where \(q(x)\) and \(r(x)\) are polynomials with real coefficients. [It may be helpful to establish (i) \(p(x)\) has real coefficients; (ii) if \(p(x)\) has any real linear factors, they must be of even multiplicity. The function that is identically zero is to be regarded as a polynomial.]
A finite set \(S\) of elements \(x\), \(y\), \(z\), ... (all different) has the following properties:
A solid fills the region common to two equal circular cylinders whose axes meet at right angles. Prove that its volume is \(4/\pi\) times the volume of a sphere with radius equal to that of the cylinders.
An aircraft flies due east from a point \(A\) at speed \(v\). A homing missile, starting at the same time from a point \(B\) at distance \(a\) due south of \(A\), flies at speed \(2v\) always in the direction of the aircraft. Neglecting the curvature of the earth, show that \(\psi\), the angle made by the instantaneous direction of flight of the missile with a line pointing north, obeys the equation $$\frac{d}{dt}\left(\log\frac{d\psi}{dt}\right) = \frac{2(1-\sin\psi)d\psi}{\cos\psi \cdot dt}.$$ Using \(\phi = \frac{1}{4}\pi - \psi\) and \(\int\textrm{cosec}\phi d\phi = -\log(\textrm{cosec}\phi + \cot\phi)\) or otherwise, show that the time taken for the missile to reach the aircraft is \(2a/3v\).