The roots of the cubic equation \(x^3-px+q=0\) are \(\alpha, \beta, \gamma\). Evaluate \(\alpha^7+\beta^7+\gamma^7\) in terms of \(p\) and \(q\). Hence, or otherwise, solve the equation \[ 64\sin^7\theta + \sin 7\theta = 0. \]
Solution: Let \(s_i = \alpha^i + \beta^i + \gamma^i\) then \(s_n = ps_{n-2} - qs_{n-3}\) for \(n \geq 3\), so \begin{align*} && s_0 &= 3 \\ && s_1 &= 0 \\ && s_2 &= 0^2 - 2(-p) = 2p \\ && s_3 &= -3q \\ &&s_4 &= p(2p) - q(0) = 2p^2 \\ &&s_5 &= p(-3q) - q(2p) = -5pq \\ && s_6 &= p(2p^2) - q(-3q) = 2p^3 + 3q^2 \\ && s_7 &= p(-5pq) - q(2p^2) = -7p^2q \end{align*} \begin{align*} && \sin 7 \theta + 64 \sin^7 \theta &= \frac{e^{7i\theta} -e^{-7i \theta}}{2i} + \frac12 \left ( 2 \sin \theta\right)^7 \\ &&&= \frac12 \left ( -i(e^{i \theta})^7 + i(e^{-i\theta})^7 + (2 \sin \theta)^7\right) \\ &&&= \frac12\left ( (ie^{i \theta})^7 + (-ie^{-i\theta})^7 + (2 \sin \theta)^7 \right) \\ \end{align*} since \(2 \sin \theta - i(e^{i \theta} - e^{-i\theta}) = 0\) we must have \begin{align*} && \sin 7 \theta + 64 \sin^7 \theta &= \frac12\left ( (ie^{i \theta})^7 + (-ie^{-i\theta})^7 + (2 \sin \theta)^7 \right) \\ &&&= -\frac{7}{2} \left (ie^{i \theta}(-ie^{-i\theta}) + ie^{i \theta}(2 \sin \theta) + (-ie^{-i\theta})(2 \sin \theta) \right)^2 \left (-ie^{i \theta}(-ie^{-i\theta})(2 \sin \theta) \right) \\ &&&= -\frac72 \left (1 - 4\sin^2 \theta \right)^2(-2 \sin \theta) \\ &&&= 7 \sin \theta(1- 4\sin^2 \theta)^2 \end{align*} Therefore for our equation to equal zero we need \(\sin \theta = 0\) or \(\sin \theta = \pm \frac12\), ie \(\theta = n \pi , \frac{\pi}{6} + n \pi, \frac{5 \pi}{6} + n \pi\)
Prove that \(|z_1+z_2| \le |z_1|+|z_2|\) where \(z_1, z_2\) are complex numbers. Show that if \(|a_n|<2\) for \(1 \le n \le N\) then the equation \[ 1+a_1z+\dots+a_Nz^N = 0 \] has no solution such that \(|z|<\frac{1}{3}\). Is the converse true?
The function \(f(x)\) is ``bounded as \(x\to 0\) through positive values'' if and only if there exist positive constants \(K, \delta\) such that \(|f(x)| < K\) for \(0< x< \delta\). Show that if \(f(x), g(x)\) are bounded as \(x\to 0\) through positive values then so are \(f(x)+g(x)\), \(f(x)g(x)\), \(\int_x^\delta f(t)\,dt\). Show that \[ \int_x^1 \frac{e^{-t}}{t}dt + \log x \] is bounded as \(x\to 0\) through positive values.
A set of functions \(J_n(x)\), \(n=0, \pm 1, \pm 2, \dots\), satisfy the following equations: \begin{align*} J_{n-1}(x)+J_{n+1}(x) &= \frac{2n}{x}J_n(x), \\ J_{n-1}(x)-J_{n+1}(x) &= 2\frac{d}{dx}J_n(x). \end{align*} Show that \begin{align*} \left(\frac{1}{x}\frac{d}{dx}\right)^m x^n J_n(x) &= x^{n-m}J_{n-m}(x), \\ \left(\frac{1}{x}\frac{d}{dx}\right)^m x^{-n} J_n(x) &= (-)^m x^{-n-m}J_{n+m}(x). \end{align*} Also prove that \(x^2\dfrac{d^2J_n(x)}{dx^2} + x\dfrac{dJ_n(x)}{dx} + (x^2-n^2)J_n(x)=0\).
Prove that the point whose rectangular cartesian coordinates are \[ x=\frac{2t}{a(1+t^2)}, \quad y=\frac{1-t^2}{b(1+t^2)}, \] where \(t\) is a variable parameter, describes an ellipse. Find the equation of the chord joining the points where \(t\) has the values \(t_1, t_2\), and the equation of the tangent at the point \(t_1\). Prove that, if the four points \(t_1, t_2, t_3, t_4\) are concyclic, then \[ \Sigma t_1 = \Sigma t_1 t_2 t_3, \] and that, if they lie on a rectangular hyperbola through the origin, then \[ a^2(1+\Sigma t_1 t_2 + t_1 t_2 t_3 t_4) + 2b^2(1+t_1t_2t_3t_4)=0. \]
Show that the equation of the pair of tangents to the conic whose equation, in homogeneous coordinates, is \[ S \equiv ax^2+by^2+cz^2+2fyz+2gzx+2hxy=0, \] at the points where it is met by the line \[ L \equiv \lambda x + \mu y + \nu z = 0, \] is \[ S\Sigma - \Delta L^2 = 0, \] where, in the usual notation, \(\Sigma = A\lambda^2+B\mu^2+C\nu^2+2F\mu\nu+2G\nu\lambda+2H\lambda\mu\), \(A=bc-f^2\), \(F=gh-af\), etc., and \(\Delta=abc+2fgh-af^2-bg^2-ch^2\). If \(x, y\) are rectangular cartesian coordinates, prove that, if the line \(\lambda x+\mu y+\nu=0\) is a directrix of the conic \(ax^2+by^2+c+2fy+2gx+2hxy=0\), then \[ \frac{\lambda^2-\mu^2}{a-b} = \frac{\lambda\mu}{h} = \frac{\Sigma}{\Delta}. \] \subsubsection*{SECTION B}
A lamina is in equilibrium under the joint action of two systems of forces in its plane, all of given magnitudes and applied at given points. All the forces of the first system are then turned anticlockwise through an angle \(\theta\) about their respective points of application, and all those of the second system are turned clockwise through the same angle \(\theta\). Show that in general the resultant of the new set of forces is a single force whose line of action is independent of \(\theta\); but that if, and only if, each of the two original systems, taken separately, reduces to a couple or is in equilibrium, then the resultant is a couple (or exceptionally the new set of forces may be in equilibrium).
The ends of a uniform rod of length \(8a\) are free to move on a fixed smooth wire bent in the form of a parabola of latus rectum \(4a\) with axis vertical and vertex downwards. Express the potential energy of the rod in terms of its inclination to the horizontal; find the inclinations of the rod to the horizontal in all the possible positions of equilibrium and determine whether each position is stable or unstable. Show that the period of small oscillations about a stable position of equilibrium is \[ 2\pi\sqrt{\left(\frac{1a}{3g}\right)}. \]
The centre of mass of a car, moving in a straight line on level ground, is at height \(h\) above ground level and at a distance \(a\) from the vertical plane through the rear axle and \(b\) from the vertical plane through the front axles. Show that if braking is applied equally to the two rear wheels only, excessive braking may cause a skid but cannot cause the wheels to leave the ground; but that if braking is applied to the front wheels the rear wheels may leave the ground if the coefficient of friction \(\mu\) is great enough; and find the condition for this to happen, neglecting the rotatory inertia of the revolving parts. If the braking force is divided between the front and back wheels, determine whether it is possible to get more effective braking than with front-wheel brakes only, and whether the result is any different if the car is running downhill. If the angular momentum of all rotating parts may be assumed to be in the same direction as that of the wheels, and proportional to the car's speed, determine whether the maximum attainable retardation is greater or less than if this angular momentum were negligible.
A lamina is moving in any manner in a plane. The coordinates of a point \(P\) fixed in the lamina are \((X,Y)\) with respect to axes with origin \(O\) fixed in space, and \((x,y)\) with respect to axes, with origin \(O'\), fixed in the lamina. The velocity of \(O'\) has components \((u,v)\) parallel to the \((X,Y)\) axes and the \(x\)-axis makes an angle \(\theta\) with the \(X\)-axis, \(u,v\) and \(\theta\) being given functions of time. Determine the components parallel to the \((X,Y)\) axes of the velocity and acceleration of \(P\) in terms of \(x,y\) and \(u,v\) and \(\theta\) and their time derivatives. Show that in general the points where the acceleration is perpendicular to the velocity at any given instant lie on a circle. \(A\) and \(B\) are two points fixed in the lamina distant \(l\) apart, and they are constrained to move along \(OX\) and \(OY\) respectively, their displacements from \(O\) at time \(t\) being \(l\sin nt\) and \(l\cos nt\). Show that at any instant the points where the velocity is perpendicular to the acceleration lie on the straight line joining \(O\) to the instantaneous centre of rotation.