The region \(A_n\) of the \((x,y)\)-plane is bounded by the portions of the curves \(y = 0\) and \(y = \sin^n x\) given by \(0 \leq x \leq \pi\), where \(n\) is a positive integer. Show that, if \(n > 2\), the area \(a_n\) of \(A_n\) satisfies \[na_n = (n-1)a_{n-2},\] and hence find \(a_n\) for all \(n \geq 1\). Deduce that \(a_{2n}/a_{2n+1} \to 1\) as \(n \to \infty\), and hence show that \[\frac{2}{2n+1} \frac{2^{4n}(n!)^4}{[(2n)!]^2} \to \pi \quad \text{as} \quad n \to \infty.\]
By use of the identity \[(1+y)(1-y+y^2-\ldots+(-y)^n) \equiv 1-(-y)^{n+1},\] or otherwise, prove that, for any \(n > 0\), the value of \(\tan^{-1} x\) in the range \((-\frac{1}{2}\pi, \frac{1}{2}\pi)\) lies between \(s_n\) and \(s_{n+1}\), where \[s_n = \sum_{r=1}^n \frac{(-1)^{r+1} x^{2r-1}}{2r-1}.\] Show that \[\sum_{n=1}^\infty \frac{(-1)^{n+1}}{(2n-1) \cdot 3^n}\] converges to \(\pi/(6\sqrt{3})\).
Define the curvature \(\kappa\) at a point of a curve having a smoothly-turning tangent. Show that, if the rectangular Cartesian coordinates \((x,y)\) of the general point of the curve are given as functions of a parameter \(\theta\), then \[\kappa = \frac{x'y'' - y'x''}{[(x')^2 + (y')^2]^{3/2}},\] where dashes denote differentiations with respect to \(\theta\). Find the curvature at the general point of the curve \begin{align} x &= a \cos \theta + a\theta \sin \theta,\\ y &= a \sin \theta - a\theta \cos \theta. \end{align} Verify your result by showing that the normal at any point of the curve touches a circle \(x^2 + y^2 = a^2\). Deduce a mechanical method of drawing the curve, and sketch that part corresponding to values of \(\theta\) in the range \([0, 2\pi]\).
A triangular lamina is given, and instruments capable of measuring lengths and angles to within known small margins of error (absolute, not percentage). If you were asked to calculate the radius of the circumcircle of the triangle from measurements of one side and one angle of the triangle, which would you choose in order to obtain the most reliable answer? Why? A number of triangular laminas of widely varying shapes are given, all of whose perimeters lie between 15 and 16 inches. The instruments provided can measure lengths to within 0.1 inch and angles to within 0.001 radian. For which shape would you expect the percentage error in your estimate (by the above method) of the radius of the circumcircle to be greatest? Why?
The following are three properties that may or may not belong to a sequence \((a_n)\) of strictly positive real numbers: \begin{align} (P_1) \quad & a_n \to 0 \text{ as } n \to \infty.\\ (P_2) \quad & \sum_{n=1}^\infty a_n \text{ converges}.\\ (P_3) \quad & \text{There is a constant } C \text{ such that } na_n < C \text{ for all values of } n. \end{align} For each pair of integers \((i,j)\) with \(1 \leq i \leq 3\), \(1 \leq j \leq 3\), \(i \neq j\), establish whether the statement `\(P_i\) implies \(P_j\)' is true or false. [You may quote without proof the behaviour of standard series.]
Explain carefully what is meant by the statement that a function of a real variable \(x\) is continuous at a particular value \(x_0\) of \(x\). The function \(f(x)\) of the real variable \(x\) takes the value 0 whenever \(x\) is irrational, and the value \(x^2(1-x^2)\) whenever \(x\) is rational. Find all values of \(x\) at which \(f(x)\) is continuous. Determine which (if either) of the following statements defines a real number, and find each number so defined:
A submarine travelling east at 16 km/hr sights a ship at a distance of 2.6 km to the E.S.E. Three minutes later the ship is seen to be straight ahead and 1.6 km away. Given that the angle between E.S.E. and E. is 22.5° and that \(\tan 22.5^\circ = \frac{2}{5}\) approximately, find the velocity of the ship. The submarine immediately alters its course so as to pass as near as possible to the ship. Find the magnitude of the relative velocity of the two, and the time the submarine will take to reach the position of closest approach.
A heavy uniform disc, with centre \(O\) and mass \(m\), rests on a rough floor. It is supported by three small feet under its circumference, forming an equilateral triangle \(ABC\), and does not touch the floor elsewhere. A tangential force \(T\) is applied to its circumference at a point \(P\), such that the angle \(DOP\) (= \(\theta\)) between \(OP\) and the diameter \(AOD\) is less than \(30^\circ\); and the force is increased until the disc begins to move. Given that only the nearest two feet \(B\) and \(C\) will begin to slip, find the magnitude of the force \(T\) when slipping begins, and prove that it does not exceed \((8\sqrt{3} - 12)F\), where \(F\) denotes the limiting friction (\(\frac{1}{2}\mu mg\)). Show also that the horizontal component of the reaction at \(A\) when slipping begins is $$\frac{1}{2}T\sqrt{5 - 2\cos \theta - 3\cos^2 \theta A}.$$
A uniform solid parabolic cylinder, whose cross-section consists of the area in the \((x,y)\) plane defined by the inequalities $$y^2 \leq 4ax,$$ $$x \leq h,$$ where \(a\) and \(h\) are positive constants, rests with its curved surface in contact with a rough horizontal plane. Show that if \(h < 10a/3\) it can rest in stable equilibrium with its plane surface horizontal. What are the possible positions of equilibrium when \(h > 10a/3\)? State (with reasons) which of them are stable.
A uniform cubical block of wood of edge \(a\) and mass \(M\) rests with one of its faces in contact with a smooth horizontal plane. A small bullet of mass \(m\) travelling with high velocity \(U\) impinges normally on one of the vertical faces of the block. It hits the centre of this face at time \(t = 0\) and penetrates the wood, in which it is subject to a retarding force of magnitude \(kv\), where \(v\) is the velocity of the bullet relative to the block. Show that if $$U > \frac{(M + m)kA}{Mm}$$ the bullet will go right through the block, emerging with velocity $$U - \frac{kA}{m}$$ at time $$t = -\frac{Mm}{(M + m)k}\log\left(1 - \frac{(M + m)kA}{MmU}\right).$$ [Gravity may be ignored.]