A uniform cylinder of mass \(m\) and radius \(a\) is hung from a fixed point by a very long light string fastened to a point on it. The cylinder is released from rest with the string wound half a turn round it, as in the left-hand diagram, and descends with its axis remaining horizontal and parallel to its original position. What points in the cylinder have zero velocity when it has reached the position shown in the second diagram? Find the angular velocity and the tension in the string when the axis reaches its lowest point.
Find the most general solution of the system of equations \begin{align} 3x + 2y + z &= 7,\\ x + y + az &= 3,\\ (2a + 1)x + 4y + 5z &= 11, \end{align} where \(a\) is a given real number. Examine carefully any exceptional cases.
Let \(Z\), \(W\) be points with rectangular cartesian coordinates \((x, y)\), \((u, v)\) respectively, and suppose that the complex numbers \(z = x + iy\), \(w = u + iv\) are related by the equation \[ w = \frac{1-z}{1+z}. \] Show that, as \(Z\) varies on a general straight line \(l\) in the \((x, y)\) plane, \(W\) describes a circle in the \((u, v)\) plane. Identify geometrically those lines \(l\) which are exceptional in this respect. Let \(l_1\), \(l_2\), \(\ldots\), \(l_n\) be concurrent (non-exceptional) lines in the \((x, y)\) plane. Show that the corresponding circles \(C_1\), \(C_2\), \(\ldots\), \(C_n\) belong to a coaxial system. What condition must be satisfied by \(l_1\), \(l_2\) in order that \(C_1\), \(C_2\) should be orthogonal circles?
Prove that the geometric mean of a finite set of positive real numbers does not exceed their arithmetic mean. Prove that \(k = \frac{2\sqrt{2}}{3}\) is the smallest constant which has the following property: if \(a\), \(b\) are real numbers such that \(a \geq 2b > 0\), then \[ \sqrt{(ab)} \leq k\left(\frac{a+b}{2}\right). \] Show that, if \(a_1\), \(\ldots\), \(a_n\), \(b_1\), \(\ldots\), \(b_n\) are real numbers such that \[ a_i \geq 2b_i > 0 \quad (i = 1, \ldots, n), \] then \[ (a_1a_2\ldots a_nb_1b_2\ldots b_n)^{\frac{1}{2n}} \leq \frac{2\sqrt{2}}{3}\left(\frac{a_1 + a_2 + \ldots + a_n + b_1 + b_2 + \ldots + b_n}{2n}\right). \]
If \(f\), \(g\) are real-valued functions of a real variable, let \(f*g\) denote the function whose value at \(x\) is \(f(g(x))\). (i) Show that there is exactly one function \(u\) such that, for every \(f\), \[ f*u = u*f = f. \] (ii) Find the form of the most general function \(v\) such that, for every \(f\), \[ v*f = v. \] (iii) Show that a function \(w\) satisfies the condition \[ w*f + f \] if and only if, the equation \(w(t) = t\) has no solutions. Give an example of such a function \(w\).
Let \(x\) be a real number such that \(0 < x < 1\). Find all the maxima and minima of the function \[ f(x) = xx - \cos x. \] Show how to determine the number of distinct positive roots of the equation \(\cos x = xx\). Show that this number is even if, and only if, \[ \frac{x\sin^{-1}x + \sqrt{(1-x^2)}}{2\pi x} \] is an integer (the value of \(\sin^{-1}x\) being chosen between \(0\) and \(\frac{1}{2}\pi\)).
A man observes that the summit of a nearby hill is in a direction \(x\) radians east of north, and at an inclination \(\theta\) above the horizontal. He then walks due north, down a slope of uniform inclination \(\tan^{-1}k\) below the horizontal, a distance \(x\) yards (measured along the slope), and finds that the direction and inclination of the summit are now (respectively) \(\beta\) east of north, \(\phi\) above the horizontal. Show that \[ k\sin(\beta-\alpha) = \sin\alpha\tan\phi - \sin\beta\tan\theta. \] Calculate the height of the summit above the man's initial position.
(i) Evaluate the sum \[ \sum_{r=1}^{n} \sin^3 rx. \] (ii) By using an algebraic relation between \(\tan x\) and \(\tan 2x\), or otherwise, evaluate the sum \[ \sum_{r=0}^{n} 2^r\tan\left(\frac{y}{2^r}\right)\tan^3\left(\frac{y}{2^{r+1}}\right). \] Show that \[ \sum_{r=0}^{\infty} 2^r\tan\left(\frac{y}{2^r}\right)\tan^3\left(\frac{y}{2^{r+1}}\right) = \tan y - y. \]
Suppose that the function \(f(x)\) has derivatives of all orders. Show by induction that \[ \frac{d^n}{dx^n}\{f(\frac{1}{2}x^2)\} = \sum_{r=0}^{[\frac{1}{2}n]} a(n,r)x^{n-2r}f^{(n-r)}(\frac{1}{2}x^2), \] where \([\frac{1}{2}n]\) denotes the greatest integer not exceeding \(\frac{1}{2}n\), and the constants \(a(n,r)\) satisfy \begin{align} a(n,0) &= 1 \quad (n = 0, 1, 2, \ldots),\\ a(2r,r) &= a(2r-1,r-1) \quad (r = 1, 2, \ldots),\\ a(n+1,r) &= a(n,r) + (n-2r+2)a(n,r-1) \quad (n = 2r, 2r+1, \ldots; r = 1, 2, \ldots). \end{align}
(i) Integrate the function \[ \frac{1}{1+\sqrt{(1+e^x)}}. \] (ii) Show that the definite integrals \[ \int_0^1 \frac{\sin^{\frac{1}{2}}\pi x}{\sqrt{(x-x^2)}} dx, \quad \int_0^1 \frac{\cos^{\frac{1}{2}}\pi x}{\sqrt{(x-x^2)}} dx \] are equal. Hence, or otherwise, evaluate these integrals.