Show how to reduce the equation (in homogeneous coordinates) of any non-degenerate conic, \(ax^2 + by^2 + cz^2 + 2fyz + 2gzx + 2hxy = 0\), to the form \(\eta^2 = \xi\zeta\) where \(\xi\), \(\eta\), \(\zeta\) are certain linear expressions in \(x\), \(y\) and \(z\). In what circumstances can the equations of two given conics be simultaneously reduced to the forms \(\eta^2 = \xi\zeta\) and \(\eta^2 = k\xi\zeta\), where \(k\) is a constant? Two parabolas \(S_1\), \(S_2\) touch at a point \(P_0\) and have parallel axes. From any point \(P\) of \(S_1\) two tangents are drawn to \(S_2\), cutting \(S_1\) again at \(Q\), \(R\) respectively. Prove that, as \(P\) varies, the line \(QR\) envelopes a third parabola touching \(S_1\) and \(S_2\) at \(P_0\) and with axis parallel to those of \(S_1\) and \(S_2\).
(a) A light inextensible string is pulled against a rough curve in a plane. Given that at a point \(P\) on the string, which lies at a distance \(s\) measured along the string from a fixed point of it, the radius of curvature is \(\rho\), the tension \(T\), normal and tangential reactions per unit length \(N\) and \(R\) respectively, set up the equations of equilibrium. (b) A weightless string of length \(2l\) is fixed at the endpoints \(A\) and \(B\) which are at distance \(2a\) apart. A wind perpendicular to \(AB\) blows the string into a curve, exerting on it a normal force \(k \cos \psi\) per unit length, where \(\psi\) is the angle between the normal and the direction of the wind and \(k\) is constant. Find the intrinsic equation of the curve. Show that the tension in the string must exceed \(2kl/\pi\).
Two boys, \(A\) and \(B\), each of mass \(m\), hang at rest at the ends of a light inextensible rope which runs without slipping over a smoothly mounted pulley of radius \(a\) and moment of inertia \(I\). At a signal the boys begin to race, \(A\) and \(B\) climbing with constant speeds \(u\) and \(v\) respectively \((u > v)\) relative to the rope. Show that, in a race through height \(h\), \(A\) can give \(B\) any start less than \(h\{(I + ma^2) u + ma^2 v\}\) and win.
A light inextensible string, carrying equal masses \(m\) at the two ends, hangs over two smooth pegs \(A\), \(B\) at the same level and at distance \(2a\) apart. A mass \(2m\) is attached at the point \(C\) of the string which lies midway between \(A\) and \(B\), and the system is then released from rest. In the subsequent motion the angle between \(AC\) and the vertical is \(\theta\). Find the velocity of the mass \(2m\) as a function of \(\theta\) as long as neither mass \(m\) has reached the corresponding peg. Find also the tension in the string when \(\theta = \frac{1}{4}\pi\).
A string \(ABCD\), whose elasticity can be neglected, is stretched at tension \(T\) the fixed points \(A\) and \(D\) on a smooth horizontal table. Equal masses \(m\) are attached along the string, with small velocity \(v\). Assuming that the tension in the string off at right angles, and \(C\), Hence find the subsequent displacements of \(B\) and \(C\) as functions of time.
The displacement \(x\) of the indicator in a seismograph is related to the displacement \(s\) of the ground by the equation \(\ddot{x} + 2\lambda\omega\dot{x} + \omega^2x = -M\ddot{s}\) where \(\lambda\), \(\omega\), \(M\) are (positive) constants of the instrument and \((\,)\) denotes differentiation with respect to the time \(t\). (i) Show that the free motion of the seismograph (given by the solution of (1) identically zero) takes different forms according as \(\lambda \gtrless 1\). (ii) Show that if \(\lambda < 1\) the ratio \(\epsilon\) of the amplitudes of two successive half-swings is given by \(\log \epsilon = \pi\lambda/\sqrt{1-\lambda^2}\). (iii) Given \(\lambda = 1\), find a formula for the response \((x)\) of the instrument to an earth movement in which \(s\) is a given function \(f(t)\). How should the constants of the instrument be adjusted so that \(x\) is approximately proportional to \((a)\) the displacement \(s\), or \((b)\) the acceleration \(\ddot{s}\)?
A circular table has radius 1 ft. Five equal circular discs are symmetrically placed so as to cover the table completely. What is their minimum radius? Without detailed calculation, show that it is possible to cover the table by means of five slightly smaller discs.
A table is laid with \(2n\) places in a row. A party of \(2k\) dons, where \(k \leq n\), sit down in such a way that the number of empty spaces between any two of them is even. (It may be zero.) The number of empty spaces at the ends of the row need not be even. In how many ways can this be done?
The sum \(s(m,n)\) is defined by \[ s(m,n) = \sum_{r=1}^n \frac{1}{r}, \] where \(n \geq m \geq 2\). Show that \(s(m,n)\) is never an integer, by proving the following two propositions or otherwise.
The inscribed circle \(\Gamma\) of a triangle \(ABC\) touches the sides of the triangle at \(D\), \(E\), \(F\). Prove that the circumcircle of the triangle \(ABC\) and the nine-point circle of the triangle \(DEF\) are inverse with respect to \(\Gamma\). Show further that the orthocentre and centroid of the triangle \(DEF\) lie on the line joining the circumcentre and incentre of the triangle \(ABC\).