Problems

A picture (which may be regarded as a uniform rectangular sheet) 48 inches high and 24 inches broad is suspended against a smooth wall by two equal parallel strings 25 inches long fixed at one end to two points on the wall at the same height 24 inches apart and at the other to two hooks that project backwards 7 inches from the middle points of the side edges of the picture at right angles to its plane. Shew that the strings make with the wall an angle of nearly \(14^\circ\).

Shew how to obtain the formulae \begin{align*} 2 \cos n\theta &= (2\cos\theta)^n - n(2\cos\theta)^{n-2} + \frac{n(n-3)}{2!}(2\cos\theta)^{n-4} - \dots, \\ (-1)^{\frac{n}{2}} \cos n\theta &= 1 - \frac{n^2}{2!}\cos^2\theta + \frac{n^2(n^2-2^2)}{4!}\cos^4\theta - \dots, \end{align*} where \(n\) is a positive integer, and is even in the second formula. Discuss the case corresponding to the second formula in which \(n\) is odd. Discuss also similar formulae for \(\sin n\theta/\sin\theta\) in terms of \(\cos\theta\). Shew conversely that any polynomial in \(\cos\theta\) and \(\sin\theta\) may be expressed as a finite sum of the type \[ \sum (a_n \cos n\theta + b_n \sin n\theta).\]

Find numerical values of \(a,b,c\) such that the expansions of \[(1+x)^n + b\left(1+\frac{x}{4}\right)^{4n} \text{ and } a\left(1+\frac{x}{2}\right)^{2n}+c\left(1+\frac{x}{8}\right)^{8n}\] may be identical for the first four terms.

Find the Cartesian equation of the director-circle of the conic given by the general tangential equation of the second degree. Two conics have an asymptote in common. By using tangential equations or otherwise, shew that their two other common tangents intersect on the radical axis of their director-circles.

A solid homogeneous circular cylinder of radius \(r\) is bisected by a plane passing through its axis and on one half as base is constructed a triangular prism of isosceles section and of the same substance: the whole is placed in equilibrium on the top of a fixed circular cylinder of radius \(2r\) with axis horizontal -- the axes of the cylinders being parallel and the curved surfaces in contact. Shew that the greatest height of the prism consistent with stability for a small rolling displacement is \[r\frac{\sqrt{9-2\pi}-1}{2}.\] [N.B. The centre of gravity of a semi-circle of radius \(r\) is distant \(4r/3\pi\) from the centre of the semi-circle.]

Discuss the integration of rational functions. Illustrate your account by evaluating \[ \int \frac{x^2 dx}{(x+1)^2(x^2+1)}, \quad \int \frac{dx}{(x^4-1)^2}.\]

From a bag containing 9 red and 9 blue balls 9 are drawn at random, the balls being replaced; shew that the probability that 4 balls of each colour will be included is a little less than \(\frac{1}{4}\).

If \begin{align*} x^2+y^2+z^2 &= \xi^2+\eta^2+\zeta^2 = 1, \\ x+y+z &= \xi+\eta+\zeta = x\xi+y\eta+z\zeta = 0, \end{align*} shew that \[3x^2 = (\eta-\zeta)^2,\] and that \[\xi^2+\eta^2 = 2/3.\]

A smooth right circular cone, of semi-vertical angle \(\alpha\), has its axis vertical and vertex upwards; a heavy elastic string of weight \(W\), modulus of elasticity \(\lambda\), and natural length \(2\pi l\) is placed round it and allowed to sink gradually to rest. Find the position of equilibrium. A second string of equal weight and equal natural length, but of modulus \(\lambda'\), \(<\lambda\), is placed, without stretching, round the cone and the two strings are allowed to sink gradually to rest: shew that in the position of equilibrium both strings will be at a depth \(h\) below the vertex if \[\pi(\lambda+\lambda')(h\tan\alpha - l)\tan\alpha = Wl.\]

Establish formulae for the curvature at any point of a plane curve, in the cases when the curve is defined (a) by an intrinsic equation, (b) by a Cartesian equation, (c) by equations of the type \(x=\phi(t), y=\psi(t)\), (d) by a \(p,r\) equation, and (e) as the envelope of a line whose equation contains a variable parameter. Apply such of these formulae as are suitable to the cases of the parabola and the equiangular spiral.