Problems

What is meant by the statements (i) that a sequence \(s_n\) tends to a limit as \(n \to \infty\), (ii) that an infinite series is convergent? Prove

1. [(i)] \(x^n \to 0\) as \(n \to \infty\), if \(-1 < x < 1\),
2. [(ii)] \(\sum_0^\infty x^n\) is convergent, if \(-1 < x < 1\).

Prove that \(\sum_1^\infty \frac{x^n}{n}\) is convergent for \(-1 \le x < 1\), and is divergent for all other real values of \(x\).

Give a definition of \(e^x\), and from your definition deduce (i) that \(\frac{e^x}{x^n} \to \infty\) as \(x \to \infty\), where \(n\) is a fixed positive integer, (ii) that \(1+x > xe^{1/x}\) for sufficiently large values of \(x\). Prove that, if \(a>1\) and \(b>0\) and \(y=x^b\), then \[ \frac{a^y}{x^n} \to \infty \text{ as } x \to \infty, \] and \[ \left(1+\frac{1}{x}\right)^{xy} \to \infty \text{ as } x \to \infty. \]

Shew that the locus of a point \(P\), such that the tangents from \(P\) to the two conics \[ S \equiv x^2+y^2+z^2=0, \quad S' \equiv ax^2+by^2+cz^2=0 \] form a harmonic pencil, is the conic \[ F \equiv a(b+c)x^2 + b(c+a)y^2 + c(a+b)z^2=0. \] Shew that \(\lambda^2 S + \lambda F + abc S' = 0\), where \(\lambda\) is a parameter, is the equation of a conic touching the four common tangents of \(S\) and \(S'\). Hence shew that the equation of the four common tangents is \(F^2 - 4abcSS' = 0\).

Prove the following sequence of results:

1. [(i)] The envelope of the join of two points \(P, P'\), which are conjugate with respect to a conic \(S\), and which lie respectively on two fixed lines \(l,l'\), is a conic \(S'\) which touches \(l,l'\).
2. [(ii)] If two lines \(OAA'\) and \(OBB'\), through the point \(O\) of intersection of \(l,l'\), are harmonically separated by \(l\) and \(l'\) and meet the conic \(S\) in \(A, A'\) and \(B, B'\), then the conic \(S'\) touches the four joins \(AB, AB', A'B, A'B'\).
3. [(iii)] The four points \(\alpha, \beta, \gamma, \delta,\) lie on a conic \(S\), and the four lines \(\alpha\beta, \beta\gamma, \gamma\delta, \delta\alpha\) touch a conic \(\Sigma\). Any tangent to \(\Sigma\) cuts \(S\) in \(L,M\). The joins of \(L,M\) to \(O\), the point of intersection of \(\alpha\gamma\) and \(\beta\delta\), meet \(S\) again in \(L', M'\). By considering the involution of line-pairs through \(O\), of which \(O\alpha, O\beta\) and \(OL, OM\) are two pairs, prove that the conic \(\Sigma\) also touches \(ML', L'M', M'L\).
4. [(iv)] If one quadrilateral is inscribed in a conic \(S\) and circumscribed to another conic \(\Sigma\), then there is an infinite number of such quadrilaterals.

If \(s_n\) denotes the sum of the \(n\)th powers of the roots \(\alpha, \beta, \gamma, \delta\) of the equation \[ x^4 + a_1x^3 + a_2x^2 + a_3x+a_4 = 0, \] shew that, if \(n \ge 4\), \[ s_n + a_1s_{n-1} + a_2s_{n-2} + a_3s_{n-3} + a_4s_{n-4} = 0. \] Shew that, if \[ A_n = \begin{vmatrix} s_n & s_{n+1} & s_{n+2} & s_{n+3} \\ s_{n+1} & s_{n+2} & s_{n+3} & s_{n+4} \\ s_{n+2} & s_{n+3} & s_{n+4} & s_{n+5} \\ s_{n+3} & s_{n+4} & s_{n+5} & s_{n+6} \end{vmatrix}, \] then \(A_n = a_4^n A_0\). By squaring the determinant \[ \begin{vmatrix} 1 & 1 & 1 & 1 \\ \alpha & \beta & \gamma & \delta \\ \alpha^2 & \beta^2 & \gamma^2 & \delta^2 \\ \alpha^3 & \beta^3 & \gamma^3 & \delta^3 \end{vmatrix} \] shew that \(A_0\) is the square of the product of the differences of the roots.

The sum of the lengths of the twelve edges of a rectangular box is \(5l\), and the sum of the areas of its six faces is \(l^2\). Find the limits between which the shortest edge must lie. A solid cube is fitted into one corner of the box, the edge of the cube being equal to the shortest edge of the box. Prove that the volume of the space inside the box not occupied by the cube lies between \(l^3/36\) and \(3l^3/64\).

A particle moves in a plane field of force so that the force which acts on the particle depends only on its position. The plane is referred to rectangular cartesian axes, the components of the force at \((x,y)\) being denoted by \((X,Y)\). If the field is conservative (i.e. the work done by the field when the particle moves from one point of the field to another is independent of the path), shew that there exists a single valued function \(V(x,y)\) such that \[ X = -\frac{\partial V}{\partial x}, \quad Y = -\frac{\partial V}{\partial y}. \] What conditions are satisfied by \(V\) at the points of equilibrium? By taking the origin at one of the positions of equilibrium of the particle and by choosing suitable directions for the coordinate axes, shew that, neglecting terms of the third and higher degrees in \(x\) and \(y\), the value of \(V\) in the immediate neighbourhood of the origin may, in general, be taken as \[ V = V_0 + \frac{1}{2}(ax^2+by^2), \] where \(V_0, a, b\) are constants. Assuming that neither \(a\) nor \(b\) is zero obtain equations of motion of the particle for small oscillations about the position of equilibrium and deduce necessary and sufficient conditions for the stability of the position of equilibrium. The value of \(V\) at the point \(P(x,y)\) is given by \[ V = \frac{k}{r} + \frac{4k}{r'}, \] where \(k\) is a constant and \(r\) and \(r'\) are the distances of \(P\) from the points \((a,0)\) and \((2a,0)\) respectively. Shew that the origin is a point of equilibrium for a unit mass which is free to move in the plane and discuss the stability or instability of this point of equilibrium.

A small sphere is projected from a point \(P\) in a horizontal plane so that it rebounds from a smooth vertical wall which is normal to the plane of motion and at a distance \(a\) from the point of projection, the coefficient of restitution between the sphere and the wall being \(e\). Shew that the path of the sphere after hitting the wall is the same as that obtained by projecting with a suitable velocity from a point \(P'\) in the horizontal plane on the other side of the wall, when the wall is absent, and determine the position of \(P'\) and the conditions of projection from \(P'\) in terms of those from \(P\). The sphere after impact against the wall hits another smooth vertical wall which is parallel to the first wall and at a distance \(a\) on the other side of \(P\), and then, after rebounding once from the horizontal plane, returns to \(P\), the coefficient of restitution at all impacts being \(e\). Shew that, if the walls are absent and the sphere is projected with the same initial conditions as before, its range on the horizontal plane is \[ \frac{a(1+e)}{e^2}. \]

A right-angled isosceles wedge of mass \(M'\) carrying a small smooth pulley at its vertex is placed with its largest face on the face of another wedge of mass \(M\) and angle \(45^\circ\) which is standing on a horizontal table. Two particles, each of mass \(m\), are attached to the ends of a light inextensible string which passes over the pulley so that one mass hangs vertically and the other mass rests on the horizontal face of the first wedge as indicated in the diagram. Assuming that all the surfaces in contact are smooth, and neglecting the mass of the pulley, shew that, if the system is released from rest with the string tight and motion takes place in a vertical plane, the string remains at rest relative to the pulley, and find the acceleration of the wedge \(M\). \includegraphics[width=0.5\textwidth]{image.png} % Placeholder for diagram

Two masses, \(3m\) and \(m\), are connected by a light inextensible string of length \(2l\) which passes through a small hole in a smooth horizontal table on which the mass \(3m\) can move while \(m\) hangs vertically. Initially \(m\) is released from rest at the hole and simultaneously \(3m\) is projected with a horizontal velocity \(\sqrt{(gl/3)}\) at a distance \(l/\sqrt{3}\) from the hole and at right angles to the line joining the mass to the hole. Shew that just after the string becomes tight the mass \(m\) is instantaneously at rest, and, by using the principles of the conservation of energy and of angular momentum, shew that subsequently it oscillates through a distance \(l/2\).