A uniform heavy chain rests on a smooth cycloidal curve in a vertical plane, the base of the cycloid being horizontal and its vertex uppermost and the chain extending from one cusp to the next. Prove that the pressure on the curve at any point is proportional to the radius of curvature.
Prove that the geometric mean of \(n\) positive numbers does not exceed their arithmetic mean. Shew that if \(a, b\) are positive, and \(p, q\) are positive rational numbers satisfying \(\dfrac{1}{p}+\dfrac{1}{q}=1\), then \[ ab \le \frac{a^p}{p} + \frac{b^q}{q}. \]
Sum the series, \(n\) being a positive integer:
If \(M, N\) are the double (self-corresponding) points of a homography on a line and \(A, A'\); \(B, B'\) are any two pairs of corresponding points, prove that the cross-ratios \((MN, AA')\) and \((MN, BB')\) are equal, and that, if \(x, x'\) are the coordinates of any pair of corresponding points, then, provided \(M\) and \(N\) do not coincide and are not at infinity, the formula for the homography may be written \[ \frac{x'-m}{x'-n} = k \frac{x-m}{x-n}, \] and interpret \(k\). If the two double points coincide, shew that the formula may then be written \[ \frac{1}{x'-m} = \frac{1}{x-m} + d, \] where \(d\) is a constant.
A uniform rod of mass \(M\) and length \(l\) rests on a rough horizontal plane. A gradually increasing horizontal force is applied to the rod at one end at right angles to its length. Assuming the pressure of the plane on the rod to be uniformly distributed along it, prove that the rod begins to turn about a point at a distance \(l/\sqrt{2}\) from one end and find the least force required to move it if the coefficient of friction is \(\mu\). Find expressions for the shearing force and bending moment at each point of the rod when it is just about to move; represent them graphically and show that there is no discontinuity either in the bending moment or in its gradient.
Explain how to determine the maximum and minimum values of a function of a single real variable by means of its differential coefficient. Illustrate by considering the function \((x-2)^5(x-3)^{10}\). Shew that \(\tan 3x \cot 2x\) takes all values except those between \(\frac{2}{3}\) and \(\frac{9}{4}\).
\(ABC\) is an equilateral triangle inscribed in a circle of radius \(a\); \(P\) is any point on a concentric circle of radius \(r\). Shew that \(PA^2+PB^2+PC^2\) is constant, and that \(PA \cdot PB \cdot PC\) lies between \(r^3 \sim a^3\) and \(r^3+a^3\).
Shew that if a focus be taken as pole and the axis as initial line, then the equation to a conic may be written in the form \[ \frac{l}{r} = 1 + e \cos\theta. \] Derive the equation to the chord joining the points on the conic whose vectorial angles are \(\alpha+\beta\) and \(\alpha-\beta\) and deduce the equation to the tangent at the point whose vectorial angle is \(\alpha\). Hence or otherwise, prove that chords of a conic which subtend a constant angle at a focus touch a fixed conic.
State the principle of virtual work. A smooth circular cylinder of radius \(a\) is fixed with its axis horizontal. Two equal cylinders of radius \(b\) and weight \(w\) rest inside the first cylinder with their axes horizontal, and another smooth cylinder of radius \(c\) and the same weight \(w\) rests upon the last two, with its axis parallel to theirs, separating them without touching the fixed cylinder. Prove that, if the planes through the axes of each of the equal cylinders and the axis of the fixed cylinder make an angle \(\theta\) with the vertical, then \[ 8(a-b)^2 \cos^2\theta = 9\{(a-b)^2 - (b+c)^2\}. \] Prove also that this position is one of unstable equilibrium.
By repeated integration by parts, or otherwise, shew that \[ f(x) = f(0) + \frac{x}{1!}f'(0) + \dots + \frac{x^n}{n!}f^{(n)}(0) + \int_0^x \frac{(x-t)^n}{n!} f^{(n+1)}(t)dt. \] Hence prove that, for \(-1 < x < 1\), \[ \log_e(1+x) = x - \frac{x^2}{2} + \frac{x^3}{3} - \dots \text{ ad inf.} \]