Problem 1

Painting a room has a linear “time complexity” because it will take double the time to paint an area double the size of the room. Based on this definition of time complexity the most optimum algorithm to solve a problem will be one that has:

1. exponential time complexity;
2. logarithmic time complexity;
3. parabolic time complexity;
4. None of the above;

Problem 2

The following group of objects demonstrates degeneracy

1. A point and a circle
2. A point and an ellipse
3. A point and a square
4. both (a) and (b)
5. A point and a triangle

( Hint: In math a degenerate case is a limiting case when a class of object changes its nature so as to belong to another class due to the limiting case of a variable. What happens to a circle when the radius approaches zero? Similarly, what happens to an ellipse when the eccentricity approaches zero?)

Problem 3

Which of the following polynomials have degenerate roots?

( Hint: Roots (“zeros” of a polynomial) which coincide, i.e. are equal, are said to be degenerate.)

Problem 4

If  σ  is the standard deviation of a time series S_i (of asset prices) and  Max (S_i)  is the maximum value of the series and is the minimum value of  Min (S_i)  the series which has  n  observations. Range is defined as the difference of the maximum and the minimum of the series. Then which of the following holds true:

1. 
2. 
3. 
4. None of the above;

( Hint : This is perhaps one of the most profound areas of mathematical finance, i.e. the application of probability and distribution to make trading decisions. What should matter is the expected payoff from a trade and not the respective probabilities of the up move and the down move

Problem 5

Of the following, one pair of numbers is “taxicab” numbers (Hardy-Ramanujam number), i.e. the number is a sum of two cubes?

1. 81 and 459
2. 2 and 1729
3. 3 and 1165
4. 195 and 1956

Answers:

1. (b)
2. (d)
3. (a)
4. (a)
5. (b)

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