A, B and C are three points in the plane of a conic S; the pole of BC with respect to S is A', the pole of CA is B', and the pole of AB is C'. Prove that AA', BB', CC' meet in a point P. \par If B and C are fixed, and A moves in a straight line \(l\), prove that the locus of P is a conic through B, C and A'. What happens if \(l\) passes through A'?
The base of a cone is bounded by a circle of radius \(a\) lying in a horizontal plane. The centre of the circle is O, and A is a fixed point on the circle. The vertex V of the cone lies in the vertical plane through OA, VO is of length \(b\) (greater than \(a\)), and VOA is an acute angle \(\alpha\). P is a variable point on the circle, and the angle AOP is denoted by \(\phi\). Prove that \[ \cos\text{VOP} = \cos\alpha\cos\phi. \] Discuss the maximum value of the angle between VO and the generator VP for different positions of P on the circle. Prove that, if \(b < a \sec\alpha\), the maximum is \(\beta\), where \[ \tan\beta = \frac{a\sin\alpha}{b-a\cos\alpha}, \] and that, if \(b>a\sec\alpha\), the maximum is \(\sin^{-1}(a/b)\).
The variables \(x,y\) are connected by the homographic relation \[ y = \frac{ax+b}{cx+d} \quad (c\neq 0, ad-bc \neq 0). \] Prove (i) that the cross-ratio of any four \(x\)'s is equal to the cross-ratio of the corresponding four \(y\)'s, (ii) that, if the self-corresponding points \(\alpha, \beta\) are distinct, \[ \frac{y-\alpha}{y-\beta} / \frac{x-\alpha}{x-\beta} \] has a constant value independent of \(x\). \par A sequence of numbers \(u_1, u_2, u_3, \dots\) is determined by the relations \[ u_1 = 2a, \quad u_{n+1} = 2a + \frac{b^2}{u_n} \quad (n\ge 1), \] where \(a, b\) are positive constants. Determine \(u_n\) in terms of \(n\) and the constants \(a, b\).
Express \((x^{2n+1}+1)/(x+1)\) as a product of real quadratic factors. \par If \(k\) is an odd integer greater than 1, prove that \[ \cos\frac{\pi}{k}\cos\frac{3\pi}{k}\cos\frac{5\pi}{k}\dots\cos\frac{(k-2)\pi}{k} = (-)^m 2^{-(k-1)/2}, \] where \(k\) is of the form \(4m+1\) or \(4m+3\); and that \[ \tan\frac{\pi}{2k}\tan\frac{3\pi}{2k}\tan\frac{5\pi}{2k}\dots\tan\frac{(k-2)\pi}{2k} = \frac{1}{\sqrt{k}}. \]
The function \(f(x)\) has a continuous second derivative \(f''(x)\) in the interval \([a,b]\); prove that, if \(a
If \(f(x)\) is continuous for all real values of \(x\), prove that \[ \frac{d}{dx}\int_0^x f(t)dt = f(x). \] The function \(\theta(x)\) has continuous derivatives \(\theta'(x), \theta''(x), \theta'''(x)\), for all \(x>0\), and \(\theta(x), \theta'(x)\) are positive. The functions \(\phi(x), \psi(x)\) are defined, for \(x>0\), by the equations \[ \phi(x) = \frac{1}{x}\int_0^x \theta'(t)dt, \quad \psi(x) = \theta(x)\phi(x). \] Prove that \(\phi(x), \psi(x)\) are positive when \(x\) is positive, and that \[ \frac{\psi''(x)}{\psi(x)} = \frac{\theta''(x)}{\theta(x)} + \frac{\phi''(x)}{\phi(x)}. \] If \(\theta'(x)\) tends to a positive limit as \(x\to\infty\), prove that \(\phi(x)/x^2 \to \frac{1}{2}\) as \(x\to\infty\).
A regular pentagon ABCDE is formed of five uniform heavy rods each of weight \(w\) smoothly jointed at their ends. It is suspended from A and its form is maintained by two light struts BD and CE. Prove that the thrust in each strut is \(\frac{1}{2}w \cot 18^\circ \operatorname{cosec} 18^\circ\).
Three equal particles attract one another so that the potential energy between two of the particles at a distance \(r\) apart is \(-Ar^{-m}+Br^{-n}\), where A, B, m and n are positive constants, and \(m
A particle is projected vertically upwards in air, which produces a resistance \(g\mu^2 v^2\) per unit mass, where \(v\) is the velocity of the particle. Find the maximum height attained, and show that the particle returns to its initial position after a time \[ \frac{1}{\mu g}[\tan^{-1}\mu u + \log\{\mu u + \sqrt{(1+\mu^2 u^2)}\}], \] where \(u\) is the initial velocity.
A particle rests in equilibrium on the outer surface of a rough uniform cylindrical shell of radius \(a\), which is free to turn about its axis, which is horizontal. The particle and the shell have equal masses, and the coefficient of friction is \(\mu\). The equilibrium is disturbed by giving the cylinder a small angular velocity. Show that, if the particle slips, it does so when the cylinder has turned through an angle \(\theta\) given by \(4\mu\cos\theta - \sin\theta = 2\mu\). \par Investigate whether the particle can leave the surface before slipping occurs.