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 \(2a/3v\).
A stream of particles, of mass \(\rho\) per unit volume and moving with velocity \(v\), impinges on a fixed plane \(S\), the normal to which makes an angle \(\alpha\) with the initial velocity. The impact is frictionless and the coefficient of restitution is \(e\). Find the pressure exerted on \(S\) by the stream. Show that the loss of kinetic energy per unit volume of the impinging stream is \(\frac{1}{2}(1-e)\rho\).
Particles of a system move in one plane under forces between the particles and external forces in the plane. Prove that the rate of change of angular momentum about their centroid is equal to the resultant moment of the external forces about the centroid. A compound pendulum of radius of gyration \(k\) about the centroid \(G\) hangs from a point \(P\) at distance \(h\) from \(G\). \(P\) is forced to move along a horizontal line in the plane of the pendulum, its displacement \(x\) being a known function of time \(t\), and the inclination of \(PG\) to the downward vertical being \(\theta\). Show that $$h\cos\theta \ddot{x} + (h^2 + k^2)\ddot{\theta} + hg\sin\theta = 0.$$ Find the value of \(\ddot{x}\) that is needed to make the pendulum maintain a constant inclination \(\alpha\). Show that if \(\ddot{x}\) has this value the period of small oscillations in inclination is \(\frac{2\pi}{n}\), where \(n^2 = (hg\sec\alpha)/(h^2 + k^2)\).