Morgan Stanley organized a contest Q.E.D in 2012 to promote Mathematics in Finance among the students of IITB. It kicked off with a talk on Quantitative Finance by Ashwin Rao (Managing Director, Morgan Stanley India). For the contest, there were 3 elimination rounds. Round #1 had the following questions.

**Objective Section**

**1)**Let P be a projection, so P*P = P. Compute inverse of (I - cP) given c = 1/2, where I is the identity matrix.**2)**Two creepers, C1 and C2, are both climbing up and round a cylinder. C1 twists clockwise and C2 anticlockwise, both start at the same point at the bottom. Before they reach the top of the cylinder C1 had made 5 complete twists and C2 had made 3 complete twists. Not counting the bottom and the top, how many times do their paths intersect?**3)**Compute Σ ( i^5 / 1000^6 ) over i = 1 to 1000, approximately.**4)**For strings of length m + n, with m 0's and n 1's. Find the expected number of switches from 0 to 1 (a switch can be thought of as presence of '01' in the given string).**5)**Compute ∫ (*dt*)^2.**6)**Let*I*be an n*n identity matrix and*J*be an n*n matrix of all ones. Find the rank of*I -*(1/n)*J*.

**Subjective Section**

**1)**You have N cars that are all travelling the same direction on an infinitely long one-way highway. Unfortunately, they are all going different speeds, and cannot pass each other. Eventually the cars will clump up in one or more traffic jams. In terms of N, what is the expected number of clumps of cars?**2)**Say a[1], a[2], a[3], ... a[N] be a permutation of first N natural numbers. a[i] is a maxima if a[i-1] < a[i] > a[i+1]. Find*E*[ number of maximas in a random permutation ].**3)**Prove that*E*[x] = ∫ (1 -*F(x)) dx*over the range [0, ∞], where F is the cumulative distribution function.