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CMI 2015 Objective & Subjective | Problems & Solutions

This post contains Chennai Mathematical Institute, CMI, 2015 Objective, and Subjective Problems and Solutions. Please contribute problems and solutions in the comments.


  1. For all finite word strings comprising A and B only, A string is arranged by dictionary order. eg. ABAA
  2.  For any arbitrary string w, with another string y<w, there cannot always exist a string x, w<x<y
  3. There is an infinite set of strings a1,a2… such that ai<a(i+1) for all i.
  4. There are fewer than 50 strings less than AABBABBA
  • Ten people are seated in a circle. One person contributes five hundred rupees. Every person contributes the average of the money contributed by his two neighbors.
    1. What is the sum contributed by all the ten?
      1. >5000
      2. < 5000
      3. .=5000
      4. Cannot say.
    2. 2. What is maximum contribution by an individual?
      1. 500
      2. =500
      3. none
  • There are 4 bins and 4 balls. Let P(E_i) be the probability of first n balls falling into distinct bins.


    1. P(E_4)
    2. P(E_4|E_3)
    3. P(E_4|E_2)
    4. P(E_3|E_4)
  • Let f(x) = \sin^{-1} (\sin (\pi x)) . Find
    1. f(2.7).
    2. f'(2.7)
    3. integral from 0 to 2.5 of f(x)dx
    4. value of x for which f'(x) does not exist
  • In some country number plates are formed by 2 digits and 3 vowels. It is called confusing if it has both digit 0 and vowel o.
    1. How many such number plates exist?
    2. How many are not confusing
  • A number is called magical if a and b are not coprime to n, a+b is also not coprime to n. For example, 2 is magical as all even numbers are not coprime to 2. Find whether the following numbers are magical
    1. 129
    2. 128
    3. 127
    4. 100
  • a) In the expansion of (1+ \sqrt 2)^10 = \sum_0^10 C_i (\sqrt 2)^i , the term with maximum value is
    b) If (1+\sqrt 2)^n = p_n+q_n \sqrt 2 , where p_n and q_n are integers, \lim_{ n to \infty} \frac {p_n}{q_n} ^{10} is


  1. In a circle, AB be the diameter.. X is an external point. Using straight edge construct a perpendicular to AB from X
    1. If X is inside the circle then how can this be done
  2. a be a positive integer from set {2, 3, 4, ... 9999}. Show that there are exactly two positive integers in that set such that 10000 divides a*a-1.
    1. Put n^2 - 1 in place of 9999. How many positive integers a exists such that n^2 divides a(a-1)
  3. P(x) is a polynomial. Show that \displaystyle { \lim_{t to \infty} \frac{P(t)} {e^t} } exists. Also show that the limit does not depend on the polynomial.
  4. We define function \displaystyle { f(x) = \frac {e^{\frac{-1}{x}}}{x}} when x< 0; f(x) = 0 if x=0 and \displaystyle { f(x) = \frac {e^{\frac{-1}{x}}}{x}}  when x > 0 . Show that the function is continuous and differentiable. Find limit at x =0
  5. p,q,r any real number such that p^2 + q^2 + r^2 = 1
    1. Show that 3*(p^2 q + p^2 r) + 2(r^3 +q^3) \le 2
    2. Suppose f(p,q,r) = 3(p^2 q + p^2 r ) + 2(r^3 +q^3) .  At what values (p,q, r) does f(p,q,r) maximizes and minimizes?
  6. Let g(n) is GCD of (2n+9) and 6n^2+11n-2 then then find greatest value of g(n)

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