Two particles are placed at the points A and B on a rough plane inclined at 45\(^\circ\) to the horizontal; AB is a line of greatest slope, and A is above B. The coefficient of friction \(\mu\) between either particle and the plane is greater than unity. A light string ACB joins the particles, the part AC being horizontal and the part CB being vertical. Prove that, if a gradually increasing force perpendicular to and away from the plane is applied at C, equilibrium is broken in one of two ways according as the ratio of the weights of the particles is less than or greater than \(\dfrac{\mu+1}{\mu-1}\).
Points L, M are taken on the sides AB, AC, respectively, of a triangle ABC so that \(BL = \lambda.BA\) and \(CM=\mu.CA\), and the lines BM, CL meet in P. Prove that the ratio of the area of the triangle PBC to the area of ABC is \[ \frac{\lambda\mu}{\lambda+\mu-\lambda\mu}. \] Points X, Y, Z are taken on the sides BC, CA, AB, respectively, of a triangle ABC so that \(BX = \frac{1}{3}BC\), \(CY=\frac{1}{3}CA\), \(AZ=\frac{1}{3}AB\). Find the ratio of the area of the triangle formed by the lines AX, BY, CZ to the area of the triangle ABC.
A vertical tower of height \(h\) stands on the top of a hill and the angles of elevation of the top of the tower above the horizontal, as seen from two points on the hillside in a straight line with the base of the tower and at distances \(a\) and \(b\) from the base, are \(\alpha\) and \(\beta\), respectively. Prove that \[ \frac{\sin^2 (\alpha - \beta)}{h^2} = \frac{\cos^2 \alpha}{a^2} + \frac{\cos^2 \beta}{b^2} - \frac{2 \cos \alpha \cos \beta \cos (\alpha - \beta)}{ab}. \]
Points F, G, H, K are taken on a conic such that FG, GH, HK pass through fixed points A, B, C respectively. Prove that, in general, KF envelops a conic, but that, if A, B, C are collinear, then KF passes through a fixed point.
A force F acts in a given plane at a point P. Define the work done by F when P is displaced from A to B along a given curve in the plane. If two such forces \(F_1\) and \(F_2\) acting at P have a resultant F, prove that the work done by F in the displacement is the sum of the work done by \(F_1\) in the displacement and that done by \(F_2\). \par If the coordinates of P referred to rectangular axes are \((x,y)\), and a force F acting at P has components \((Ay, 0)\) along these axes, prove that the work done by F when P is displaced along a straight line from \((x_1, y_1)\) to \((x_2, y_2)\) is \(\frac{1}{2}A(x_2-x_1)(y_1+y_2)\). \par Prove also that the work done when P is displaced round a closed path is proportional to the area enclosed by the path.
Show that the equation \[ r\frac{\partial}{\partial r}\left(r\frac{\partial u}{\partial r}\right) + \frac{\partial^2 u}{\partial \theta^2} = 0 \] can be satisfied (identically) by taking \(u = r^n P\), where \(n\) is any positive integer and P is a certain polynomial of degree \(n\) in \(\cos\theta\), and that P must be a constant multiple of \[ 1 - \frac{n^2}{2!}\cos^2\theta + \frac{n^2(n^2-2^2)}{4!}\cos^4\theta - \frac{n^2(n^2-2^2)(n^2-4^2)}{6!}\cos^6\theta + \dots \] or of \[ \cos\theta - \frac{n^2-1^2}{3!}\cos^3\theta + \frac{(n^2-1^2)(n^2-3^2)}{5!}\cos^5\theta - \dots, \] according as \(n\) is even or odd, the summation continuing as far as the term in \(\cos^n\theta\). \par Verify that \(u=r^n \cos n\theta\) satisfies the above equation, and hence, or otherwise, express \(\cos n\theta\) as a polynomial in \(\cos\theta\).
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 \(0 \le \theta < \pi\).
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 \(d_2\).
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.
(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 \(2^{10}/3^{15}\).