Problems

If \(\alpha, \beta, \gamma\) are the roots of \(x^3 + px + q = 0\), prove that \[ \frac{\alpha^5 + \beta^5 + \gamma^5}{5} = \frac{\alpha^3 + \beta^3 + \gamma^3}{3} \frac{\alpha^2 + \beta^2 + \gamma^2}{2}. \]

Give an account of methods by which the \(n\)th differential coefficient of certain functions can be found, giving illustrations. Prove that the method of partial fractions enables us to find the \(n\)th differential coefficient of any rational algebraic fraction. Find the \(n\)th differential coefficients of \[ \text{(i) } e^{ax}\cos bx, \quad \text{(ii) } (2+x)^2/(1-x^3). \] Prove that, if \(\sin^{-1}y = a+b\sin^{-1}x\), the values when \(x=0\) of the successive differential coefficients of \(y\) satisfy \[ \frac{d^{n+2}y}{dx^{n+2}} = (n^2-b^2)\frac{d^ny}{dx^n}. \]

A uniform cylinder rests on two fixed planes as shewn in the figure; the plane \(AB\) is smooth and the coefficient of friction between the cylinder and the plane \(AC\) is \(\mu\). A horizontal force equal to the weight of the cylinder acts at \(D\), the middle point of the highest generator of the cylinder. Shew that equilibrium is impossible unless \(\alpha\) is greater than \(\frac{\pi}{4}\), and that if \(\alpha = \tan^{-1} 2.4\) there will be equilibrium if \(\mu\) is not less than \(\frac{1}{4}\).

By taking the asymptotes as axes, the equation of a rectangular hyperbola \[ x^2 - y^2 + 2hxy + 2fy + 2gx = 0 \] is reduced to \[ x'y' - k^2 = 0. \] Prove, by the use of invariants, or otherwise, that \[ k^2 = \pm \tfrac{1}{2} (-f^2 + 2fgh + g^2) (1+h^2)^{-\frac{3}{2}}. \]

Prove the Binomial Theorem for a positive integral index. If the binomial expansion of \((1+x)^m\), where \(m\) is a positive integer, be written \[ (1+x)^m = 1 + p_1x + p_2x^2 + \dots + p_mx^m, \] shew that \begin{align*} 1 + p_1 + p_2 + \dots + p_m &= 2^m, \\ p_1 + 2p_2 + 3p_3 + \dots + mp_m &= m2^{m-1}, \end{align*} and find the value of \[ 1+p_1^2+p_2^2+\dots+p_m^2. \]

Define the centre of mass of a system of particles. Prove that, if \(G\) be the centre of mass of a set of \(n\) particles of masses \(m_1, m_2, \dots m_n\) at the points \(A_1, A_2, \dots A_n\), and \(O\) be any point:

1. [(i)] \(\sum_{r=1}^n m_r \vec{OA_r} = \vec{OG} \sum_{r=1}^n m_r,\) where the lengths are added vectorially; and
2. [(ii)] \(\sum_{r=1}^n m_r OA_r^2 = \sum_{r=1}^n m_r GA_r^2 + OG^2 \sum_{r=1}^n m_r.\)

A light equilateral triangular frame is loaded at the corners with weights \(w_1, w_2, w_3\) and suspended from a fixed point by strings of lengths \(l_1, l_2, l_3\) attached to its corners. Prove that the tensions \(T_1, T_2, T_3\) in the strings are given by \[ \frac{w_1l_1}{T_1} = \frac{w_2l_2}{T_2} = \frac{w_3l_3}{T_3} = \frac{\{(w_1+w_2+w_3)(w_1l_1^2+w_2l_2^2+w_3l_3^2) - (w_2w_3+w_3w_1+w_1w_2)a^2\}^{\frac{1}{2}}}{w_1+w_2+w_3}, \] where \(a\) is the length of a side of the triangle.

A horizontal portion of a toboggan run is worn into a series of sine-curve undulations 20 ft. from crest to crest, with a maximum height from crest to trough of 1 in. Shew that a toboggan will jump at the crest of an undulation when its speed exceeds about 60 miles an hour.

Prove that the equation of the rectangular hyperbola of closest (i.e. four-point) contact with the parabola \(y^2 = 4ax\) at the point \((a \tan^2\phi, 2a \tan\phi)\) is \[ (x \cos\phi - y \sin\phi + a \sin\phi \tan\phi)^2 = y^2-4ax. \] Prove that in the limit when two of these hyperbolas approach coincidence, they intersect at the point \[ x = -4a - 3a \tan^2\phi, \quad y = 6a \tan\phi + 4a \tan^3\phi. \]

Write down the most general values of \(x\) which satisfy the equations

1. [(i)] \(\sin x = \sin \alpha\),
2. [(ii)] \(\cos x = \cos \alpha\),
3. [(iii)] \(\tan x = \tan \alpha\).

Find all sets of values of \(x, y, z\) which satisfy the simultaneous equations \begin{align*} \tan x + \tan 2y &= 0, \\ \tan y + \tan 2z &= 0, \\ \tan z + \tan 2x &= 0, \end{align*} subject to the restriction that each of \(x, y, z\) is positive and less than \(\pi\).

Explain the application of the method of the "funicular polygon" to determine the resultant of a system of coplanar forces. Shew that as applied to a system of vertical coplanar forces \(P, Q, R, \dots\) the method may be employed to construct the bending moment diagram for a straight horizontal beam supported at two given points and loaded by \(P, Q, R, \dots\). Establish the relations which obtain between transverse loading, shear and bending moment in a beam which is subjected to distributed transverse loads. Shew that, if the beam is supported at its ends, the bending moment diagram is parabolic when the loading is uniformly distributed. What is the form of the bending moment diagram when it is similar to the diagram of transverse loading?