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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.

Objective

  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.
    Find

     

    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

Subjective

  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
      Discussion
  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)
      Discussion
  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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