Prove that the equation of the chord joining the points \(P(ap^2, 2ap), Q(aq^2, 2aq)\) of the parabola \(y^2=4ax\) is \[ 2x-(p+q)y+2pqa=0. \] Prove also that, if a circle through P, Q cuts the parabola again in \(U(au^2, 2au), V(av^2, 2av)\), then \[ u+v = -(p+q), \] and that, if this circle passes through the focus \((a,0)\), then \[ uv = \frac{p^2+pq+q^2+3}{1+pq}. \] Hence, or otherwise, prove that, if the circle through the focus of a parabola and two points P, Q on it (both lying "above" the axis, so that \(p,q\) are positive) cuts the parabola in two further real points, then the chord PQ meets the axis of the parabola at a distance from the focus exceeding the length \(4a\) of the latus rectum.
Prove that the conics through four distinct points in general position cut an involution on an arbitrary line. A system of conics is known to be such that each conic passes through three given distinct (non-collinear) points. It is also known that there exists a certain straight line (not through any of the given points) on which the conics cut pairs of points in involution. Determine whether the conics have a fourth common point. \subsubsection*{SECTION B}
A uniform beam of thickness \(2c\) rests horizontally upon a fixed perfectly rough circular cylinder of radius \(a\) whose axis is horizontal and perpendicular to the direction of the length of the beam. The beam is then rolled on the cylinder keeping its length perpendicular to the axis of the cylinder. Show that there is a position of equilibrium with the beam inclined to the horizontal if \(a>c\) and that this position is unstable.
A bola consists of two particles each of mass \(m\) joined by a light string of length \(2\pi a\). The bola travels in a horizontal plane with velocity \(V\) with the string taut and in a direction perpendicular to the string. The mid-point of the string hits a fixed vertical right circular cylindrical post of radius \(a\). Describe the ensuing motion in the case when the particles and the post are perfectly elastic, and show that the time \(T\) between the instant when the string hits the post and the instant when the particles meet is \(a\pi^2/2V\).
A long plank of length \(2l\) and mass \(m\) is supported horizontally at its two ends by vertical ropes, the weaker of which can only stand a tension \(\frac{5}{4}mg\). A man of mass \(m\) walks across the plank starting at the stronger rope. When the weaker rope breaks the man clings to the plank at the position he has reached. Show that when the weaker rope breaks the tension in the stronger rope suddenly becomes \(1\frac{3}{16}mg\).
A particle lies on a horizontal plank at a distance \(a\) to the right of a point \(O\) of the plank. The coefficient of friction between the particle and the plank is \(\mu\). The plank is then rotated with constant positive angular velocity \(\omega\) about a horizontal axis through the point \(O\) of the plank and perpendicular to its length. Show that when the particle begins to move \[ \ddot{x} \pm 2\mu\omega\dot{x} - x\omega^2 = -g\sec\alpha\sin(\omega t\pm\alpha), \] where \(x\) is the distance of the particle from \(O\) and \(\tan\alpha=\mu\), the positive or negative sign being taken according as the particle moves away from or towards \(O\). Also show that, if \(\mu< 1\) and \(a > g\mu\omega^{-2}\), the particle begins to move outwards at once, but that, if \(\mu< 1\) and \(0 < a< g\mu\omega^{-2}\), the particle begins to move inwards at time \(t\) given by \[ \omega t = \alpha + \sin^{-1}(\omega^2 a \cos\alpha \cdot g^{-1}). \] Discuss the cases that arise when \(\mu>1\).
Two beads each of mass \(m\) are threaded on to a smooth straight rod one end of which is freely hinged to a fixed point. They are connected by an elastic string of natural length \(l\) and modulus \(\lambda\). The rod is set in uniform rotation in a horizontal plane with angular velocity \(\omega\). Show that, if \(2\lambda< ml\omega^2\), the string must in general ultimately break.
Show that \[ \sum_{r=0}^n r(r+1)\dots(r+k-1) = \frac{1}{k+1}n(n+1)\dots(n+k). \] Deduce that, if \(a_{r+1} = a_r/(1+ra_r)\), then \[ \sum_{r=0}^n \frac{1}{a_ra_{r+2}} = \frac{n+1}{a_0} + \frac{n+1}{3a_0^2}(n^2+2n+3) + \frac{1}{20}(n-1)n(n+1)(n+2)(n+3). \]
The sequence \(a_1, a_2, a_3, \dots\) is defined by means of the relations \[ a_1=3, \quad a_{p+1} = \frac{a_p^2+5}{2a_p} \quad (p>0). \] Prove that \[ 0 < a_{p+1} - \sqrt{5} \le \frac{(3-\sqrt{5})^{2^p}}{(2\sqrt{5})^{2^p-1}} < 6 \times \left(\frac{2}{11}\right)^{2^p}. \] Hence show that \(a_p \to \sqrt{5}\) as \(p\to\infty\). Use this process to calculate \(\sqrt{5}\) correct to four places of decimals.
A regimental dinner is attended by \(n\) officers who leave their caps in an ante-room before going in to dine. At the conclusion of the dinner there is a certain amount of confusion in the ante-room with the result that each officer emerges wearing a cap which is not his own. If \(p_r\) is the chance of this occurring for a dinner of \(r\) officers show that \[ p_n + \frac{p_{n-1}}{1!} + \frac{p_{n-2}}{2!} + \dots + \frac{p_0}{n!} = 1, \] where \(p_0=1\). By forming the series \(\displaystyle\sum_{n=0}^\infty p_n x^n\) and multiplying by \(e^x\), or otherwise, deduce that \[ p_n = 1 - \frac{1}{1!} + \frac{1}{2!} - \dots + (-1)^n \frac{1}{n!}. \]