A plank of breadth \(2b\) and thickness \(2c\) rests inside a horizontal cylinder of radius \(a\) with its long edges parallel to the axis of the cylinder and at such a height that it is just about to slip down. Show that the plank makes an angle \(\theta\) with the horizontal given by \[ a \sin \lambda \cos(\theta-\lambda) = (a \cos\alpha - c) \sin\theta \cos\alpha, \] where \(\lambda\) is the angle of friction and \(\sin\alpha = b/a\).
\(x_1, x_2, \dots, x_n\); \(a_1, a_2, \dots, a_n\) are two systems of positive numbers with the same sum. Shew that, the \(a\)'s being individually fixed and the \(x\)'s variable, \[ \frac{x_1^p}{a_1^{p-1}} + \frac{x_2^p}{a_2^{p-1}} + \dots + \frac{x_n^p}{a_n^{p-1}}, \] where \(p\) (not necessarily integral) is greater than 1, is least when \[ x_1 = a_1, \quad x_2 = a_2, \quad \dots, \quad x_n = a_n, \] so that \[ \frac{x_1^p}{a_1^{p-1}} + \frac{x_2^p}{a_2^{p-1}} + \dots + \frac{x_n^p}{a_n^{p-1}} \ge a_1 + a_2 + \dots + a_n. \] Deduce that, if \(a_1, a_2, \dots, a_n\) are any positive numbers whatsoever, then \[ \frac{a_1^p}{a_1^{p-1}} + \frac{a_2^p}{a_2^{p-1}} + \dots + \frac{a_n^p}{a_n^{p-1}} \ge \frac{(a_1+a_2+\dots+a_n)^p}{(a_1+a_2+\dots+a_n)^{p-1}}. \] By taking \(a_n = A_n B_n\), shew that with a suitable choice of \(a_n\) the above gives \[ \Sigma A_n B_n \le (\Sigma A_n^p)^{\frac{1}{p}} (\Sigma B_n^q)^{\frac{1}{q}}, \] where \(q\) is determined by the relation \[ \frac{1}{p} + \frac{1}{q} = 1. \]
Prove that, if \(\alpha, \beta, \gamma\) are the roots of \[ x^3 + qx + r = 0, \] then \[ \alpha^2 (\beta + \gamma) + \beta^2 (\gamma + \alpha) + \gamma^2 (\alpha + \beta) = 3r \] and \[ \alpha^3 (\beta + \gamma) + \beta^3 (\gamma + \alpha) + \gamma^3 (\alpha + \beta) = -2q^2. \]
A tangent to a rectangular hyperbola meets the asymptotes in \(T, T'\). Prove that \(T, T'\) are concyclic with the foci, and that \(TT'\) subtends angles \(\pi/4, 3\pi/4\) at the foci.
Two uniform planks \(AB, AC\) (not necessarily of the same length) are smoothly hinged together at \(A\) and stand with the ends \(B\) and \(C\) on a smooth horizontal plane so that the angle \(BAC\) is a right angle, equilibrium being preserved by a string connecting the ends \(B, C\). Find the tension of the string in terms of the weights of the planks, the height \(h\) of the hinge \(A\) and the length \(l\) of the string. Also show that if relative motion of the hinge is opposed by a friction couple whose limiting value is \(L\) the tension of the string may be diminished or increased by as much as \(L/h\).
Prove carefully that, if \[ f(x) = a_0 x^m + a_1 x^{m-1} + \dots + a_m \] vanishes for \(m\) distinct values \(x_1, x_2, \dots, x_m\) of \(x\), then \[ f(x) = a_0 (x-x_1)(x-x_2)\dots(x-x_m). \] Shew that \[ \cos n\theta - \cos n\phi = 2^{n-1} \prod_{r=0}^{r=n-1} \left\{\cos\theta - \cos\left(\phi + \frac{2r\pi}{n}\right)\right\}, \] and that \[ \sin n\theta = 2^{n-1} \sin\theta \sin\left(\theta+\frac{\pi}{n}\right) \sin\left(\theta+\frac{2\pi}{n}\right) \dots \sin\left(\theta+\frac{(n-1)\pi}{n}\right). \] \[ \sqrt{n} = 2^{\frac{n-1}{2}} \sin\frac{\pi}{n} \sin\frac{2\pi}{n} \dots \] the last factor being \(\sin\frac{(n-2)\pi}{2n}\) or \(\sin\frac{(n-1)\pi}{2n}\) according as \(n\) is even or odd. Criticise the argument: \(\sin\theta\) vanishes for \(\theta = \pm r\pi\) and for no other values of \(\theta\), therefore \[ \sin\theta = \theta\left(1-\frac{\theta^2}{\pi^2}\right)\left(1-\frac{\theta^2}{2^2\pi^2}\right)\dots\left(1-\frac{\theta^2}{r^2\pi^2}\right)\dots. \]
Prove, by taking logarithms or otherwise, that if \(k, l, m, n, p, q, r\) are positive numbers of the form \(n-3, n-2, n-1, n, n+1, n+2, n+3\), the ratio of \(l^6 n^9 q^6\) to \(k m^7 p^{15} r\) is \(1 + 120n^{-6} + 1260n^{-8} + \dots\).
A conic \(S\) is the polar reciprocal of itself with respect to another conic \(S'\). Prove that the conics touch at two distinct points \(P, Q\); that any chord of \(S\) through the pole of \(PQ\) is divided harmonically by \(S'\); and that \(S'\) is the polar reciprocal of itself with respect to \(S\).
Four equal rods each of length \(l\), freely jointed at their opposite corners, form a rhombus \(ABCD\). The opposite corners \(A, C\) are connected by an elastic string whose unstretched length is \(b (< 2l)\). The system is kept in equilibrium by forces at \(B\) and \(D\) acting inwards along the line \(BD\). Show that these forces have a maximum value when \[ \frac{AC}{BD} = \frac{\left(\frac{b}{2l}\right)^{\frac{1}{3}}}{\left\{1 - \left(\frac{b}{2l}\right)^{\frac{2}{3}}\right\}^{\frac{1}{2}}}. \]
Explain the usual process for finding the H.C.F. of two polynomials \(U(x), V(x)\) and shew that, if they have no common factor, then polynomials \(L(x), M(x)\) can be found such that \[ L(x)U(x) + M(x)V(x) = W(x), \] where \(W(x)\) is any other given polynomial. Discuss the case \[ U(x)=Q(x), \quad V(x) = \frac{d}{dx}Q(x), \quad W(x)=P(x), \] where \(Q(x)\) has no repeated factors and the degree of \(P(x)\) is less than that of \(Q(x)\), and shew how a reduction formula for the integral \[ \int \frac{P(x)}{[Q(x)]^n} dx \] may be found. Reduce, and thus evaluate \[ \int \frac{2x^3-1}{(x^3-3x+1)^2} dx. \]