Find the greatest and the least values of the function \[ \sin x - \frac{\sin 2x}{2} + \frac{\sin 3x}{3} \] (i) for all real values of \(x\), and (ii) for \(0 \le x \le \pi\).
Prove that the common chord of the ellipse \[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \] and a circle of curvature envelops the curve \[ \left(\frac{x}{a} + \frac{y}{b}\right)^{\frac{2}{3}} + \left(\frac{x}{a} - \frac{y}{b}\right)^{\frac{2}{3}} = 2. \]
A shell explodes on the ground, and fragments fly from it in all directions with all velocities up to 80 feet a second. Show that a man 160 feet away can be hit by fragments with angles of projection between \(\tan^{-1}2\) and \(\tan^{-1}\frac{1}{2}\), and that he is in danger for \(\sqrt{5}\) seconds.
If \(m\) and \(n\) are unequal integers, prove that \[ \int_X^Y \frac{\sin^2\pi x}{x(x-m)(x-n)}dx = \frac{1}{m-n}\left( \int_{Y-n}^{Y-m} \frac{\sin^2\pi u}{u}du - \int_{X-n}^{X-m} \frac{\sin^2\pi u}{u}du \right) \] and hence, or otherwise, find the value of \[ \int_{-\infty}^\infty \frac{\sin^2\pi x}{x(x-m)(x-n)}dx. \]
Show that the equation \[ \frac{d^n}{dx^n}\left(\frac{1}{x^2+1}\right) = 0 \] has just \(n\) roots (all real), and determine them.
Obtain conditions that the lines \(lx+my+n=0\) and \(l'x+m'y+n'=0\) may be conjugate diameters of the conic \[ ax^2+2hxy+by^2+2gx+2fy+c=0. \] Prove that the equation of the two diameters which are conjugate with respect to each of the conics \[ ax^2+2hxy+by^2=1, \quad a'x^2+2h'xy+b'y^2=1 \] is \[ \begin{vmatrix} y^2 & -xy & x^2 \\ a & h & b \\ a' & h' & b' \end{vmatrix} = 0. \]
State and prove the principle of conservation of momentum for a system of interacting particles. \par A particle of mass \(2m\) is attached by a light inextensible string of length \(l\) to a small ring of mass \(m\) that can slide without friction along a straight horizontal wire. The system is released from rest with the string taut and horizontal. Find the angular velocity and the tension of the string when it is inclined at an angle \(\theta\) to the horizontal.
Find the Cartesian equation of the curve assumed by a uniform string hanging freely under gravity. \par If a uniform string is in a vertical plane and is in contact with a smooth horizontal cylinder of any form of cross-section, so that the plane of the string is perpendicular to the generators, show that the difference in the tensions of the string at any two points is proportional to the vertical distance between these points. \par If the string lies across a number of such cylinders in a vertical plane perpendicular to their generators, show that the catenaries in which the free portions of the string lie all have the same directrix. Show also that the free ends must be at the same level.
By considering the points where the curve \[ x^3 + y^3 = axy \] is met by the line \(y=mx\), obtain a parametric representation \(x=\phi(m), y=\psi(m)\) of the curve. \par Show that the three points with parameters \(m_1, m_2, m_3\) are collinear if and only if \[ m_1 m_2 m_3 = -1. \] The tangents at three collinear points \(P_1, P_2, P_3\) of the curve meet the curve again in \(Q_1, Q_2, Q_3\), respectively. Prove that \(Q_1, Q_2, Q_3\) are collinear.
If \(\Sigma = 0, \alpha = 0, \beta = 0\) are the tangential equations of a conic and two points, interpret geometrically the equations \[ \Sigma + \alpha\beta = 0, \quad \Sigma + \alpha^2 = 0. \] Two conics \(\Sigma_1\) and \(\Sigma_2\) each have double contact with a third conic \(\Sigma\). Show that two of the points of intersection of the common tangents of \(\Sigma_1\) and \(\Sigma_2\) lie on the line joining the poles of the chords of contact of \(\Sigma_1\) and \(\Sigma_2\) with \(\Sigma\) and form with them a harmonic range.