CMI 2015

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)

62 Replies to “CMI 2015”

  1. soln of q.no.-1
    Let 2 line drawn meeting A to X and B to X in P and Q at the pepheri of circle respectively.Now join A to Q and B to P.As AB is diameter of circle so angleAPB and angleBQA are 90*.So if we extend the linea AQ and BP,Let it meets at point O ie ORTHOCENTRE of triangle ABX.Now join X to O ie automatically perpendicular to AB.
    HENCE SOLVED

  2. Hint q.no.-3
    as exponential function is more increasing then polynomial functin.
    See derivative of both thats why any polynomial fucnyion upon exponential like e^x as x->infinity is 0 so it is independent of any particular polynomial.

  3. Part A
    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?
    A.5000
    C.=5000
    D. Cannot say.
    2. What is maximum contribution by an individual?
    A.500
    C.=500
    D.none

  4. Anyone has solved question 5???…..part 1 is easy. I wanna know about part 2. Maximum value is ‘2’….is the minimum value ‘-2’???….actually, I have proved it, though I have a confusion!. Please reply!

  5. A11. There are 4 bins and 4 balls. Let P(Ei) be the probability of first n balls falling into distinct bins.
    Find a) P(E4) b)P(E4|E3) c)P(E4|E2) d)P(E3|E4)

      1. How? a) 4/4*3/4*2/4/1/4
        c)2/4*1/4
        Or so I think. The first ball can go into any bin. THe next can go into any bin but that particular one. The next in the remaining two and last one in the specific bin.

  6. A9. Let f(x) = arcsin (sin (pi*x)).
    Find a) f(2.7). b) f'(2.7) c)integral from 0 to 2.5 of f(x)dx
    d)value of x for which f'(x) does not exist

      1. You see, range of arcsin(x) is from pi/2 to -pi/2 . Your answer (.7pi > .5pi) so it is obviously wrong. Now since .7pi is range pi/2 to pi, (for example, if have 2pi/3, then it will become , pi-2pi/3. Similarly pi-7pi/10=3pi/10

  7. A8. In some country number plates are formed by 2 digits and 3 wovels. It is called confusing if it has both digit 0 and vowel o. a) How many such number plates exist? b) How many are not confusing

      1. The question was how many are NOT confusing. WIth the NOT being in bold letters. And the answer is 125*81+64*19. As there can be 125*81 combinations with no 0. And there can be 64*19 combinations with no o. Both cannot be there together. Any one of them can exist

  8. A3. 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 a)129 b)128 c)127 d)100

      1. why is 127 magical ? 127 = 1*127. 1 and 127 are both not coprime to 127, but 1+127=128 is coprime to 127. so 127 is not magical, right ?

      2. Tiyacmi, by that logic neither is 2 a prime as neither 1 nor 2 are coprime to 2 but 3 is coprime to 2. But they give in the question itself that 2 is magical. I do not think 1 is counted as it is neither a prime nor a composite. Take all other numbers coprime to 127. You will find that it is magical. (In fact, all prime numbers are magical)

  9. Subjective 2)
    we know that there exist a number ‘a’ in that set, such that a^2=a(mod n^2). Now let a+b=n^2 [There must be an integer ‘b’ in the set]. then a^2=(n^2-b)^2=b^2(mod n^2) and a^2=a=n^2-b=-b(mod n^2). That means, b^2=-b(mod n^2), so b(b+1)=0(mod n^2) That means, our second integer is (b+1)= (n^2+1-a). Now, if ‘a’ and ‘b+1’ are distinct, then we can find two distinct integers. if not, then a=b+1 => a-b=1 and we know that a+b=n^2, Hence a=(n^2+1)/2 and b=(n^2-1)/2. Now, if n^2 is even, then ‘a’ and ‘b’ are not integers. So, we can find only one integer ‘a’ if n^2 is odd and two distinct integers ‘a’ and (n^2+1-a) when n^2 is even.
    [Is there any problem in this solution? I am not sure!]

    1. Not exactly sure. But I think it depends on the number of prime factors of n as well. Example: n=125

    2. There exist (2) elements in set when the number can be expressed in form ab, where a,b neq 1 and gcd(a,b)=1. You have to first prove that , if n is prime or prime power , then there will be no solutions.

  10. A2 for all finite word strings comprising A and B only, A string is arranged by dictionary order. eg. ABAA<ABB Also, A string with same components as a lesser string but with ehere xtra digits is greater than it eg. AB<ABAA. Write true or false
    A) For any arbitrary string w, with another string y<w, there cannot always exist a string x, w<x<y
    B) There is an infinite set of strings a1,a2… such that ai<a(i+1) for all i.
    C) There are fewer than 50 strings less than AABBABBA

      1. Yes but they are not saying that every set of string is like that are they. There is ANY one set. Take the set of 500 billion B’s as the first set. Every B can be replaced by as many number of A’s as we want and we will get an infinite set

  11. A7. a) In the expansion of (1+√2)^10 = sigma(0-10) Ci*(√2)^i, the term with maximum value is
    b)If (1+√2)^n = pn+qn√2, where pn and qn are integers, limit (n –> infinity) (pn/qn)^10 is

      1. a) Self explanatory – Just expand the binomial expansion.
        b) The ratio converges to √2. √2^10 = 32

      2. Chavan, take the case of (1+√2)^2 = 1+2+2√2 = 3+2√2. Here, p(2)=3 and q(2)=2. In case of (1+√2)^3 = 1+3√2+6+2√2 = 7+5√2. Here, p(3) = 7 and q(3) = 5. This is how you get p(n) and q(n)

  12. For A.8 part b) how many are not confusing ?
    first if it does not contain o : 100*64
    second if it does not contain 0 : 81*125
    now, from this we have to subtract the case where it contains neither o nor 0 since we have counted it twice … so the answer will be :
    100*64 + 81*125 – 81*64 = 11341

  13. I could not avoid a non calculus solution to 5:
    Part1:
    We have after simplifying and substituting p^2 by 1-q^2-r^2,
    f(q,r)=(q+r)(3-(q+r)^2)
    Let q+r=t
    f(t)=t(3-t^2)
    f'(t)=3-3t^2=0 implies t=+1 or t=-1
    It is easy to notice that the function attains maxima when t=+1
    Putting t=1 in the ineq. it is indeed true
    For 2nd part,
    Plug t=-1 and get -2 as minima

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