Find a polynomial of the ninth degree \(f(x)\), such that \((x-1)^5\) divides \(f(x)-1\) and \((x+1)^5\) divides \(f(x)+1\). Prove that the quotients do not vanish for any real value of \(x\).
State exactly what the statement "\(y^n e^{-y}\) tends to the limit 0 as \(y\) tends to \(+\infty\)" means. (It may be assumed true without proof.) A function \(f(x)\) of the real variable \(x\) is defined as follows: \begin{align*} f(x) &= e^{-1/x^2} \quad \text{if } x \neq 0, \\ f(x) &= 0 \quad \text{if } x = 0. \end{align*} When \(x \neq 0\), show that its \(n\)th derivative can be written \[ f^{(n)}(x) = G_n(1/x) \cdot e^{-1/x^2}, \] where \(G_n\) is a polynomial of degree \(3n\), and that the coefficients in \(G_n\) are all smaller than \(n!3^n\) in absolute magnitude. (Use mathematical induction.) Hence show that \[ |f^{(n)}(x)| < \frac{(n+1)!3^{n+1}}{|x|^{3n}} e^{-1/x^2} \] if \(0<|x|<1\). Prove that \(f^{(n)}(0)=0\) for all values of \(n\). Is there anything remarkable about this conclusion?
If \(\omega\) is one of the complex cube roots of unity, describe the position in the Argand diagram of the point \(-\omega^2z_1 - \omega z_2\). On the sides of any convex plane hexagon, equilateral triangles are constructed external to it. Their outer vertices are joined to form another hexagon. If \(PQRSTU\) are the mid-points of its sides, show that \(PS, QT\) and \(RU\) are equal and inclined at 60 degrees to one another.
A map of the world is drawn with the parallels of latitude horizontal and the meridians of longitude vertical. The parts near the equator are represented on a scale of 1 cm. to 1000 km. In the neighbourhood of every point, in order to avoid distortion, the north-and-south scale is adjusted to be equal to the east-and-west scale for that latitude. Compare the true distance from the equator of a point in latitude \(\eta\) with the distance measured on the map. [The earth may be assumed to be spherical.] A region is bounded by the meridian \(\xi=0\), the equator \(\eta=0\), and the curve \[ \xi = C(\cos\eta - \cos\beta), \] where \(\xi\) is the longitude, and \(C\) and \(\beta\) are constants less than \(\frac{1}{2}\pi\). Compare its actual area with its area on the map.
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 \(\mu>1\).