10273 problems found
A, B, C, D, E are five points in space, no four lying in the same plane. From each of the five points, the three lines are drawn which meet the pairs of opposite edges of the tetrahedron formed by the remaining four points. Prove that of this set of fifteen lines, the three lines which meet AB all do so at the same point and are coplanar.
The equation \(x^2 + ax + b = 0\) has real roots \(\alpha, \beta\). Form the quadratic equation with roots \(\alpha^2 - k^2, \beta^2 - k^2\), and show that \(\alpha, \beta\) are outside the interval \((-k, k)\) if \[ (b + k^2)^2 > k^2a^2 > 2k^2 (b + k^2). \] Find the conditions that the roots of the equation \[ x^4 + px^3 + qx^2 + px + 1 = 0 \] may be all real and unequal.
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.
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 \(\dfrac{\mu+1}{\mu-1}\).
A point A lies in the plane of a circle S and outside the circle. Find the loci of the centres of circles which pass through A and
(i) Prove that an approximate solution of the equation \[ xe^{x-1} + x - 2 = \epsilon, \] where \(\epsilon\) is small, is \[ x = 1 + \tfrac{1}{3}\epsilon - \tfrac{1}{18}\epsilon^2. \] (ii) Assuming that \(\sin^{-1} x\) may be expanded in ascending powers of \(x\), find the first three non-zero terms in the expansion. The expansion of \(\sin x\) in powers of \(x\) may be assumed, if required.
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 \(\cos\theta\).
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.
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 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 \[ \frac{\sin^2 (\alpha - \beta)}{h^2} = \frac{\cos^2 \alpha}{a^2} + \frac{\cos^2 \beta}{b^2} - \frac{2 \cos \alpha \cos \beta \cos (\alpha - \beta)}{ab}. \]