SPECIAL CHALLENGE MIOTP WEEK 4

1)Find the value of

Based on Srinivas Ramanujans problem 

2)In ∆ABC the sides of the triangle  are integers angle B equals to 90 degree BD is perpendicular to AC  CD equals to 31³ let the value of Cos A be m/n.  m and n are relatively Prime integers find the value of m + n

3)If, for a positive integer n, the quadratic equation,

x(x + 1) + (x + 1)(x + 2) +.....+ (x + n – 1)(x + n) = 10n 

has two consecutive integral solutions, then n is equal to(JEE MAINS 2017 INDIA)

4)A coin comes up head with probability p greater than 0 and till with probability 1 - p greater than 0 suppose the probability of 4 heads and 5 tail is equal to 1/36 the probability of getting 5 head and 4 tail . Two person plays a game with two such coins the person who will first get two head will win the match what is the probability for the first person to win the match?

5)Let

 be the number of permutations of the set which have exactly k fixed points. Prove that

.

(Remark: A permutation f of a set S is a one-to-one mapping of S onto itself. An element i in S
 is called a fixed point of the permutation f if f(i)=i.)