Credits @gaurav_sharma There is a frog who could climb either 1 stair or 3 stairs in one shot. In how many ways he could reach at 10th stair ? Fibonacci with a gap of 1 : 1, 1, 2, 3, 4, 6, 9, 13, 19, 28 Answer : 28 Method 2 :1 step in 1 way2 steps in 1 way3 steps in 2 ways4 steps in 3 ways5 steps in 4 ways6 steps in 6 ways7 steps in 9 ways8 steps in 13 ways9 steps in 19 ways10 steps in 28 waysSo 28 should be the answer Method 3 : x + 3y = 10(1 , 3 ) -> 4(4 , 2 ) -> 6!/4!2! = 15(7 , 1) -> 8!/7! = 8(10 , 0) -> 1TOTAL = 15 + 4 + 8 + 1 = 28

Mass point Geometry lecture - @gaurav_sharma

Kaprekar's constant 6174 is known as Kaprekar's constant. Take any four-digit number, using at least two different digits. (Leading zeros are allowed.)Arrange the digits in descending and then in ascending order to get two four-digit numbers, adding leading zeros if necessary.Subtract the smaller number from the bigger number.Go back to step 2 and repeat.The above process will always yield 6174, in at most 7 iterations. The only four-digit numbers for which Kaprekar's routine does not reach 6174 are repdigits such as 1111, which give the result 0000 after a single iteration.

A quadratic function f (x ) = ax^2 + bx + c, can be expressed in the standard form : a(x-h)^2 + kby completing the square. The graph of f(x) is a parabola with vertex (h,k); the parabola opens upward if a > 0 or downward if a < 0. Maximum or Minimum Value of a Quadratic Function Let f be a quadratic function with standard form f (x) = a( x − h )^2 + k.The maximum or minimum value of f occurs at x = hIf a > 0, then the minimum value of f is f(h) = k.If a < 0, then the maximum value of is f (h) = k We now derive a formula for the maximum or minimum of the quadratic functionF(x) = ax^2 + bx + c.For either of the two cases (the quadratic having a maxima or a minima), the maxima or the minima,as the case may be, will occur when x = - b/2athe maximum or minimum value is f(-b/2a) = c - b^2/4aremember that - b/2a = sum of roots/2

If two objects A and B start simultaneously from opposite points and, after meeting, reach their destinations in ‘a’ and ‘b’ hours respectively (i.e. A takes ‘a hrs’ to travel from the meeting point to his destination and B takes ‘b hrs’ to travel from the meeting point to his destination), then the ratio of their speeds is given by:Sa/Sb = √(b/a) i.e. Ratio of speeds is given by the square root of the inverse ratio of time taken. Two trains A and B starting from two points and travelling in opposite directions, reach their destinations 9 hours and 4 hours respectively after meeting each other. If the train A travels at 80kmph, find the rate at which the train B runs. Sa/Sb = √(4/9) = 2/3This gives us that the ratio of the speed of A : speed of B as 2 : 3.Since speed of A is 80 kmph, speed of B must be 80 * (3/2) = 120 kmph A and B start from Opladen and Cologne respectively at the same time and travel towards each other at constant speeds along the same route. After meeting at a point between Opladen and Cologne, A and B proceed to their destinations of Cologne and Opladen respectively. A reaches Cologne 40 minutes after the two meet and B reaches Opladen 90 minutes after their meeting. How long did A take to cover the distance between Opladen and Cologne? Sa/Sb = sqrt(90/40) = 3/2 This gives us that the ratio of the speed of A : speed of B as 3:2. We know that time taken is inversely proportional to speed. If ratio of speed of A and B is 3:2, the time taken to travel the same distance will be in the ratio 2:3. Therefore, since B takes 90 mins to travel from the meeting point to Opladen, A must have taken 60 (= 90*2/3) mins to travel from Opladen to the meeting point(adsbygoogle = window.adsbygoogle || []).push({}); So time taken by A to travel from Opladen to Cologne must be 60 + 40 mins = 1 hr 40 mins [Credits: Veritasprep]