Problems

(i) A smooth tube \(AB\) of length \(\frac{1}{2}\pi a\) and of small cross-section is bent in the form of a circular arc of radius \(a\) and is fixed in a vertical plane with the end \(A\) uppermost, the tangent to the axis of the tube at \(A\) being horizontal and the tangent at \(B\) being vertical. The tube contains a uniform flexible inextensible chain of length \(\frac{1}{2}\pi a\) which just extends throughout the length of the tube. The chain is released from rest and slides out of the tube under gravity. Find its speed when it is just clear of the lower end of the tube. (ii) The same tube is now held fixed in a horizontal plane and a stream of identical particles is fired into the tube at \(A\), emerging from the tube at \(B\). Assuming that each particle loses half its energy in its passage through the tube, and that there is no accumulation of particles in the tube, find the magnitude and direction of the force experienced by the tube in terms of the mass \(m\) of each particle, the number \(N\) per unit time entering the tube at \(A\), and the speed \(v\) of the particles at \(A\). [You are not asked to find the torque or the line of action of the force.]

Let \(\alpha\), \(\beta\), \(\gamma\) be real constants and \(\mathbf{a}\) a real vector in three dimensions. Show that if the equations \[\alpha\mathbf{x}+\beta\mathbf{y} = \mathbf{a},\] \[\mathbf{x}.\mathbf{y}=\gamma\] have real solutions for the vectors \(\mathbf{x}\), \(\mathbf{y}\), then \[\mathbf{a} \cdot \mathbf{a} \geq 4\alpha\beta\gamma.\] Find the general solution of the equations when this inequality is satisfied.

Prove the Binomial Theorem, that \begin{equation*} (1+x)^n = \sum_{r=0}^{n} \binom{n}{r} x^r \end{equation*} (where \(n\) is a positive integer). Let \(m, k\) be positive integers. By considering the powers of 2 occurring in the numerator and denominator, or otherwise, show that \(\binom{2^k m}{2^k}\) and \(m\) are either both even or both odd. Deduce that if the coefficients of \(x^r\) in \((1+x)^n\) are all odd, then \(n+1 = 2^k\) for some \(k\).

Let \(u_1\) be an odd positive integer greater than 1. For \(n > 1\), \(u_n\) is defined by the relation \begin{equation*} u_n = u_{n-1}^2 - 2. \end{equation*} Show that, for \(n > 1\), 1 is the highest common factor of \(u_n\) and \(u_m\) for \(1 \leq m \leq n-1\). Show further that 1 is the highest common factor of \(u_n\) and \(u_m - 1\) for \(1 \leq m \leq n\).

Discuss the symmetry group of the plane which preserves the following pattern, considered to extend to infinity in both directions (the 'ends' of the pattern). Define two particular symmetries \(T\) (a translation to the right by one unit) and \(R\) (a rotation through \(180^{\circ}\) about the mid-point of one of the Z's), and show that: (i) any symmetry which leaves fixed the two ends of the pattern is of the form \(T^r\) with \(r\) a positive or negative integer. (ii) \(R\) interchanges the two ends, and \(R^2\) is the identity. (iii) \(RTR = T^{-1}\). (iv) Any symmetry \(S\) which interchanges the two ends of the pattern is of the form \(T^r R\) for some \(r\). Show also that in case (iv) \(S\) has a single fixed point in the plane, and that \(S^2\) is the identity.

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\(P\) is a point, and \(l\) and \(m\) are lines, in 3-dimensional space. Show that if \(l, m\) and \(P\) are general enough, there is a unique line passing through \(P\) and meeting both \(l\) and \(m\). Suppose that, in terms of coordinates \((x, y, z)\), \(l\) is given by \(x = 1, y = 0\), and \(m\) by \(x = -1, z = 0\); prove that the line through the point \((0, \alpha, \alpha)\) which meets \(l\) and \(m\) is given by \begin{equation*} \alpha(x-1)+y = 0, \quad \alpha(x+1)-z = 0. \end{equation*} Deduce that, if \(n\) is a third line given by \(x = 0, y = z\), then the point \(P\) with coordinates \((x, y, z)\) lies on a line which meets \(l, m\) and \(n\) if and only if \(z(x-1)+y(x+1) = 0\).

Two coplanar circles \(S\) and \(S'\) are exterior to one another and have different radii. A line is called special if it intersects \(S\) and \(S'\) in four points which are at the ends of four distinct diameters having the property that they are parallel in pairs. Determine all the distinct diameters through which pass at least three special lines. Determine also the points through which pass two, but not three, special lines.

The cubic curve \(C\) in the \((x, y)\)-plane is defined by \(y^2 = x^3-x\). Sketch the curve. Let \(P\) be the point \((1, 0)\), and let \(Q\) be a point \((x_0, y_0)\), lying on \(C\) and distinct from \(P\). Show that the line \(PQ\) touches \(C\) at \(Q\) if and only if \(x_0^2 - 2x_0 - 1 = 0\). Let \(P'\) be the point \((1-\epsilon, 0)\), where \(\epsilon\) is small and positive. Sketch all the tangents from \(P'\) to \(C\).

\(f(x)\) is a real function that satisfies, for all \(x, y\), \begin{equation*} f(x+y)+f(x-y) = 2f(x)f(y). \tag{*} \end{equation*} Prove that either \(f(0) = 0\), or \(f(0) = 1\) and \(f'(0) = 0\). Prove that in the former case \(f(x)\) is identically zero, and that in the latter case \begin{equation*} f'(x) = f''(0)f(x). \end{equation*} Hence find all solutions of (*). [You may assume that \(f\) is twice differentiable.]

A disc \(D\) of radius \(b\), whose centre is initially at a point with rectangular cartesian coordinates \((1+b, 0)\), rolls without slipping round a disc with radius 1 and centre the origin; its point of contact at time \(t\) is \((\cos t, \sin t)\). A point \(P\) is embedded in the disc \(D\), and is initially at \((1+a+b, 0)\); at time \(t\), the coordinates of \(P\) are \((x(t), y(t))\). Determine \(x(t), y(t)\), and check your answers by considering the case \(a = 0\). In the case \(a > 0\), show that \(P\) will return to its initial position if and only if \(b\) is rational (i.e. \(b = p/q\), where \(p\) and \(q\) are integers and \(q \neq 0\)). Show that the area enclosed by the curve traced by \(P\) when \(a = b = 1\) is \(6\pi\).