Problems

(i) If \(x\) is positive and not equal to 1 and \(p\) is rational and not equal to 0 or 1, prove that \(x^p-1\) is less than or greater than \(p(x-1)\) according as \(p\) is between 0 and 1 or is outside these limits. \par (ii) If \(a_1, a_2, \dots, a_n\) are positive, show that \[ \frac{a_1+a_2+\dots+a_n}{n} \ge (a_1 a_2 \dots a_n)^{1/n}. \] Prove that, if \(x,y,z\) are positive and \(x+y+z=1\), the greatest value of \(x^2 y^7 z^6\) is \(2^{10}/3^{15}\).

The mass of an electron is found to vary with the velocity according to the law \[ m = \frac{\lambda}{\sqrt{(1-v^2/\mu)}}, \] where \(m\) is the mass in grams, \(v\) is the velocity in centimetres per second, \(\lambda = 9 \times 10^{-28}\) and \(\mu=9 \times 10^{20}\). Write down the law relating the mass M in kilograms with the velocity V in kilometres per minute. \par Explain and justify the statement that the constants \(\lambda\) and \(\mu\) have the dimensions of mass and (velocity)\(^2\) respectively.

Two parabolas have a common focus S and a common tangent \(t\), and their directrices \(d_1, d_2\) intersect at D. P is a variable point of \(t\), and PQ is the harmonic conjugate of PS with respect to the remaining tangents from P to the parabolas. By reciprocation or otherwise, prove that the envelope of PQ is a parabola touching \(t\), whose focus is S and whose directrix is the harmonic conjugate of DS with respect to \(d_1\) and \(d_2\).

Show how to find the sum of the sines of \(n\) angles in arithmetical progression. \par Simplify the expression \[ \sum_{r=1}^{n} \sin (r-\tfrac{1}{2})\alpha \cos (r-\tfrac{3}{4})\theta, \] and show that (i) if \(\alpha = \pi\) the expression is never negative in the range \(0 \le \theta \le \pi\), and (ii) if \(\alpha = \pi/n~(n>1)\) the expression changes sign just once in \(0 \le \theta < \pi\).

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 \[ \int_{-\infty}^\infty \frac{\sin^2\pi x}{x(x-m)(x-n)}dx. \]

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.

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 \[ \left(\frac{x}{a} + \frac{y}{b}\right)^{\frac{2}{3}} + \left(\frac{x}{a} - \frac{y}{b}\right)^{\frac{2}{3}} = 2. \]

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 \(0 \le x \le \pi\).

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.

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.