Problems

Four articles are distributed to four persons, with no restriction as to how many any person may receive. Shew that the probability that two will be given to one person and one to each of two others is \(\frac{9}{32}\). Find the probability of the same distribution if there are four articles but five persons.

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 \[ ab \le \frac{a^p}{p} + \frac{b^q}{q}. \]

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.

Shew that the equation of the normal at a point \((\alpha, \beta)\) of the curve \(f(x,y)=0\) is \[ (x-\alpha) \frac{\partial}{\partial \beta} f(\alpha, \beta) = (y-\beta) \frac{\partial}{\partial \alpha} f(\alpha, \beta). \] Hence shew that from any point \((\xi, \eta)\) three normals, of which either one or all three must be real, can be drawn to the parabola \(y^2 = 4ax\). Prove that the area of the triangle formed by joining the feet of these three normals is \[ \{4a(\xi-2a)^3 - 27a^2\eta^2\}^{\frac{1}{2}} \] and deduce the equation to the locus of centres of curvature of the parabola.

Shew that for all real values of \(x\) and \(\theta\) the expression \(\dfrac{x^2+x\sin\theta+1}{x^2+x\cos\theta+1}\) lies between \(\dfrac{4-\sqrt{7}}{3}\) and \(\dfrac{4+\sqrt{7}}{3}\).

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 \(\frac{9}{4}\).

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.

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.

Sum the series, \(n\) being a positive integer:

1. [(i)] \(\dfrac{1}{(2n)!^2} + \dfrac{1}{(2n-2)!(2n+2)!} + \dfrac{1}{(2n-4)!(2n+4)!} + \dots + \dfrac{1}{2!(4n-2)!} + \dfrac{1}{(4n)!}\),
2. [(ii)] \(1 + x\cos\theta + x^2\cos 2\theta + \dots + x^n\cos n\theta\).

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 \(-1 < x < 1\), \[ \log_e(1+x) = x - \frac{x^2}{2} + \frac{x^3}{3} - \dots \text{ ad inf.} \]