Mathematics_Today_-_October_2016

DERANGEMENTS Derangement means destroy the arrangement i.e., rearranging the objects in such a way that no object remains at its ...

The total number of flags with three horizontal strips, in order, that can be formed using 2 identical red, 2 identical green a ...

Two players P 1 and P 2 play a series of ‘2n’ games. Each game can result in either a win or loss for P 1. Total number of ways ...

We must have xW + xR + xB = 10. Required number of ways = Number of non-negative integral solutions of xW + xR + xB = 10 = 3 + 1 ...

(a) : Let the number be n = x 1 x 2 x 3 Since x 1 + x 2 + x 3 is even. That means there are following cases : (i) x 1 , x 2 , x ...

 CLASS XI Series 6 SEQUENCE A sequence is an arrangement of numbers in a definite order according to a certain rule. Note : ...

z Let a be the first term, r be the common ratio and n be the number of terms such that a, ar, ar^2 , ar^3 , ..... is a G.P., th ...

Find the relation between x and y such that the rth mean between x and 2y may be same as the rth mean between 2x and y, if n me ...

⇒ −− −−− −− −−− ()() ()()() , ()( ) ()()() abac , abbcca bcba abbcca ()() ()()() cacb abbcca −− −−−^ are in A.P. ⇒ − − − − − − 1 ...

Either (a – c)^2 = 0 ⇒ a = c and from (1), b =aa+ =aabc⇒ == 2 Or, (a + c)^2 + 2ac = 0 ⇒ a^2 + c^2 + 4ac = 0 or ac acor ac b (^22 ...

the expansion of (a + b)n is 4096, then greatest coefficient in the expansion is (a) 1594 (b) 792 (c) 924 (d) 2924 In the binom ...

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The number of seven digit integers, with sum of the digits equal to 10 and formed by using any of the digits 1, 2 and 3 only, i ...

* Alok Kumar is a winner of INDIAN NATIONAL MATHEMATICS OLYMPIAD (INMO-91). He trains IIT and Olympiad aspirants. Integration of ...

INTEGRATION BY SUBSTITUTION When integrand is a function i.e., ∫fx xdx[()] () :φφ′ Here, we put φ(x) = t, so that φ ′(x)dx = dt ...

(ii) xA xkxA dx 2 422 − ∫ ++ Divide numerator and denominator by x^2 and substitute x A x +=t, A being positive constant. (iii) ...

LEIBNITZ’S RULE z If f(x) is continuous and u(x), v(x) are differentiable functions in the interval [a, b], then, d dx ftdt fvx ...

(a) logsin3x – logsin5x + c (b)^1 3 3 1 5 log sin xxc++log sin 5 (c)^1 3 3 1 5 log sin xxc− log sin 5 + (d) 3logsin3x – 5logsin5 ...

(log ) (log ) x x − dx + ⎧ ⎨ ⎪ ⎩⎪ ⎫ ⎬ ⎪ ⎭⎪ ∫ 1 1 2 2 is equal to (a) xe x c x 1 +^2 + (b) x x c (log )^2 + 1 + (c) log (log ) ...

(c)^1 1 1 (^21) 2 e ex ex c − − log sin + cos , (e is greater than 1) (d)^2 1 1 (^21) 2 e exe x ex c − ++− log cos sin + cos , ...