Problems

A plane quadrilateral is formed by the four straight lines \(l_i\) (\(i=1,2,3,4\)), and the point of intersection of \(l_i, l_j\) is denoted by \(A_{ij}\). Prove that the circumcircles of the four triangles formed by sets of three of the lines have a common point. Prove also that the orthocentre \(O_1\) of the triangle with vertices at \(A_{23}, A_{34}, A_{42}\) is on the radical axis of the circles whose diameters are \(A_{14}A_{23}\) and \(A_{12}A_{34}\). Deduce that

1. the orthocentres of the four triangles formed by threes of the lines are collinear;
2. the middle points of the three diagonals of the quadrilateral are collinear.

The equation of a conic, in general homogeneous coordinates, is \[ ax^2+by^2+cz^2+2fyz+2gzx+2hxy=0. \quad (1) \] A variable line through a fixed point \(O\), whose coordinates are \((x',y',z')\), meets the conic in \(P, P'\). Prove that the locus of the harmonic conjugate of \(O\) with respect to \(P, P'\) is a line \(\Lambda\) whose equation is \[ (ax'+hy'+gz')x + (hx'+by'+fz')y + (gx'+fy'+cz')z = 0. \] Shew that, if \(O\) lies on the conic, then \(\Lambda\) passes through \(O\), and conversely. Deduce that the condition that the line \(lx+my+nz=0\) should touch the conic is \[ \begin{vmatrix} a & h & g & l \\ h & b & f & m \\ g & f & c & n \\ l & m & n & 0 \end{vmatrix} = 0. \] Interpret this equation geometrically when the left-hand side of equation (1) is (i) the product of two linear factors, (ii) a perfect square.

If \(\alpha_1, \alpha_2, \dots, \alpha_n\) are the roots of the equation \[ f(x) = x^n + a_1 x^{n-1} + \dots + a_n = 0, \] prove that \[ \frac{f(x)}{x-\alpha_1} + \frac{f(x)}{x-\alpha_2} + \dots + \frac{f(x)}{x-\alpha_n} = nx^{n-1} + (n-1)a_1x^{n-2} + \dots + a_{n-1}. \] If \(s_k\) denotes the sum of the \(k\)th powers of the roots, prove the relation \[ s_k + a_1 s_{k-1} + a_2 s_{k-2} + \dots + a_{k-1}s_1 + ka_k = 0 \quad (k=1, \dots, n). \] Obtain a similar relation for \(k>n\) and shew that, if \(k>n\), \[ \begin{vmatrix} s_k & s_{k-1} & \dots & s_{k-n} \\ s_{k+1} & s_k & \dots & s_{k-n+1} \\ \vdots & \vdots & \ddots & \vdots \\ s_{k+n} & s_{k+n-1} & \dots & s_k \end{vmatrix} = 0. \] Find the value of \(s_k\) for the equation \[ x^n + x^{n-1} + \dots + x + 1 = 0. \]

If \(x > 1\) and \(m\) is a positive integer greater than 1, prove that \[ \frac{x^m-1}{m} - \frac{x^{m-1}-1}{m-1} > 0, \] and hence that \(x^m - 1 > m(x-1)\). Generalise this result to the case in which \(m\) is a rational number greater than 1. Prove that, if \(\alpha, \beta\) are positive rational numbers whose sum is 1, and \(a,b\) are positive, then \[ a^\alpha b^\beta < \alpha a + \beta b, \] unless \(a=b\).

Explain what is meant by the statement \[ \phi(n) \to a \text{ as } n \to \infty, \] where \(n\) is a positive integer. Prove that, if \(\phi(n) \to a\) and \(\psi(n) \to b\) as \(n \to \infty\), then \[ \phi(n)\psi(n) \to ab. \] Determine the behaviour, as \(n \to \infty\), of two of the following:

1. \(n^r x^n\), where \(r\) is an integer, positive, negative or zero;
2. \(\sqrt[n]{x}\), where \(x\) is positive;
3. \(\sqrt[n]{n}\).

If \(Q_n(x) = (1+x^2)^{\frac{n}{2}+1} \frac{d^n y}{dx^n}\), where \(y=\frac{1}{\sqrt{(1+x^2)}}\), prove that \(Q_n(x)\) is a polynomial of degree \(n\) satisfying the following relations:

1. \(Q_{n+1} = (1+x^2)Q_n' - (2n+1)xQ_n\);
2. \(Q_{n+1} + (2n+1)xQ_n + n^2(1+x^2)Q_{n-1} = 0\);
3. \(Q_n' + n^2 Q_{n-1} = 0\);
4. \((1+x^2)Q_n'' - (2n-1)xQ_n' + n^2 Q_n = 0\).

Shew from first principles that necessary and sufficient conditions of equilibrium of a system of co-planar forces are that the sums of the resolved parts of the forces in any two distinct directions should separately vanish, and that the sum of the moments of the forces about any point in the plane should vanish. If the moments of a coplanar system about two points \(A(x_1, y_1)\) and \(B(x_2, y_2)\) are respectively \(G_1\) and \(G_2\), and if the resolved part along \(AB\) of the resultant of the system is \(F\), shew that the ordinate of the point in which the resultant cuts the line \(x=0\) is \[ \begin{vmatrix} y_1 & x_1 & G_1 \\ y_2 & x_2 & G_2 \\ (x_2-x_1) & y_2-y_1 & -Fl \end{vmatrix} \div \begin{vmatrix} x_2-x_1 & G_2-G_1 \\ y_2-y_1 & -Fl \end{vmatrix} \] where \(l=AB\).

State carefully the principle of virtual work. Illustrate the applications of its converse by solving the following problems:

1. Two uniform rods \(AB, BC\), of weights \(W_1\) and \(W_2\) respectively, are freely hinged together at \(B\), and to a wall at \(A, C\), so that \(A\) is vertically above \(C\). The rod \(AB\) is horizontal and the angle \(ABC\) is equal to \(\beta\). Find the horizontal and vertical components of the actions at \(A\) and \(B\).
2. A smooth paraboloid of revolution, the latus rectum of whose generating parabola is \(4a\), is fixed with its axis vertical and its vertex uppermost. An inextensible chain, of length \(2\pi r\) and of mass \(M\), in the form of a horizontal circle rotates with angular velocity \(\omega\), in contact with the paraboloid. Shew that the tension in the chain is \(Mr(g+2a\omega^2)/4\pi a\).

A light inextensible thread is wound on a reel, which may be considered as a uniform circular cylinder of mass \(M\); to the free end of the thread is fastened a mass \(M\), which lies on a smooth horizontal table. The portion of thread on the table is taut, and is perpendicular to an edge of the table. The reel is held below this edge with the portion of the string between the reel and the table vertical, and the system released from rest. Shew that the reel descends vertically. Shew also that the tension in the thread throughout the motion is \(\frac{1}{4}Mg\).

One end of a light inextensible string \(OAB\), in which \(OA=a, AB=b\), is fixed at \(O\), and masses \(m, M\) are carried at \(A, B\) respectively. If the masses move in a vertical plane with the strings taut, write down the equations of motion of the particles in terms of the angles \(\theta, \phi\) which \(OA\) and \(AB\) make with the vertical. If \(m=M\) and the system is released from rest with \(\theta=\alpha, \phi=\alpha+30^\circ\), where \(0 < \alpha < 60^\circ\), shew that the initial tensions in \(OA\) and \(AB\) are \(\frac{8}{5}mg \cos\alpha\) and \(\frac{2\sqrt{3}}{5} mg\cos\alpha\) respectively. Find also the initial angular acceleration of \(OA\).