**IIT Patna (Mtech in CS, Research Fellowship)**

- What is the probability of getting 2 consecutive heads in a fair coin toss?
- What is the probability of getting a prime number when 1–10 is randomly sampled?
- What is the probability of getting a prime number given that the number chosen is greater than 5?
- You are given 2 unbounded numbers to add (assume that these 2 numbers can exceed the max capacity size_of_int of the given machine). Write a program to store and add these 2 numbers. (Hint. Use linked list)

**IIT Jodhpur (Mtech in CS, Self Sponsored)**

- Basic questions of Joint and Conditional Probability.
- You are given an array of size n from with consecutive integers but one number is missing. E.g. {1,2,3,5,6,7}. Write an algorithm to find this missing number. What is the running time of the best algorithm for this? Can you use the formula for the sum of natural numbers till n to design this?
- Define inverse of a matrix, mathematically and intuitively.
- When does a matrix not have an inverse?

**IITH (Mtech in CS, Self Sponsored)**

- F(n) = 100n; G(n) = n². F(n) = O(g(N)), is this true or false? In what cases is this true or false?
- You are given an array ‘a’ of size ‘n’, where all the elements are 0 except for 1 element which is 1. You have a function Sum(a, j) which return the sum of the elements of the array from i → j. E.g Sum(a,5) will return the sum of elements from a[1] to a[5]. Using Sum(), what is the least number of calls needed to determine the index r S.T a[r] = 1. (Hint: use divide and conquer)
- Given a polynomial P(x) = x⁵ + 4x³ + 6x + 5. Assuming you are given only one temporary variable t and the value of x, what is the minimum number of operations needed to compute P(x).
- A word of variable size is generated with random permutations of the alphabet a-z. What is the probability that the letter ‘a’ appears before ‘b’ (assuming that they appear only once). (Ans: 50%)

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