IITPAL

  Rope around The Earth: Wrap a thread tightly around the earth (along the equator). Now add an extra 2 meters of thread to this length and then wrap the new thread again around the earth in the same circular path (i.e. along the equator).                                                             (Image: Binding rope around the Earth) Now, how high do you think will the thread rise above from the ground ?(Or in other words, what will be the new radius corresponding to this new circumference?) Well, practically one might think that an addition of 2 meter won't make any difference since its too small compared to the size of the earth (which is ~6371000 meters in radius!). And even if it does, the difference will be so small (may be in millimeters or so) that it can easily be neglected? Well, le...

Billiard ball problem can be seen as one elegant mathematical problem. In this article, we will analyse it mathematically. Before we begin, let us first see the problem. Given below is the situation. Can you predict, in which pocket will the ball falls in? Let us now connect this problem with mathematical conditioning. We all know reflection property very well. Now what you have thought is that whenever ball strikes any mirror then it enters into a rectangle which is just a mirror image of it. Now just keep on constructing rectangles with taking common sides CD and BC (initially) then their reflections too. Remember one thing that the ball will keep on moving in its direction until it reaches one of the vertices of our apparent rectangles. and hence it is obvious that the no. of such apparent rectangles that you will need in the horizontal direction will be 67 and in the vertical direction is 100 (smallest possible). being 67 odd hence, it is obvious that it will fall eit...

 Well, the asymptotes are fundamentally virtual tangents. So to know everything about an asymptote, we must know what are tangents. Lines that touch the given curve at a point are its tangent. See the diagram below. The line drawn touches the given circle so is a tangent to the circle. Let us now talk about asymptote, the virtual tangent. The tangent that principally exists but practically does not is asymptotes to the curve. See the diagram below. Blue horizontal dotted lines are asymptotes. Let's now see an applet to visualize the exact meaning of an asymptote.  OPEN THIS APPLET

We all know the easy way to solve the quadratic equation,           a x²+bx+c =0. W e use the quadratic formula for roots. We also have access to similar formulae for roots of cubic equations and quartic equations. The formula for roots of a cubic equation  ax 3 +bx 2 +cx+d=0 is  A similar complex looking formulae exist for roots of a quartic equation. But myster i ously we do not have any such formula for roots of 5 or higher degree polynomials. It seems as if, we can not construct the solutions of a degree 5 or higher degree polynomials just by using addition, subtraction, multiplication, division, and radicals in its coefficients.  Why so? what is so special in this 5-degree polynomial? These were questions that haunted the young Frenchman  Evariste Galois  in the 18th century. He developed a new mathematical object called a “group” that solved this issue in a surprisingly cool way. Galois being shot in a duel.  Image from...