CMI Entrance 2013 Question Paper

C.M.I. ENTRANCE 2012

CHENNAI MATHEMATICAL INSTITUTE
B.SC. MATH ENTRANCE 2012

ANSWER FIVE 6 MARK QUESTIONS AND SEVEN OUT 10 MARK QUESTIONS.

6 mark questions

1. Find the number of real solutions of x = 99 sin \(\pi \) x
2. Find \(\displaystyle\lim_{x\to\infty}\dfrac{x^{100} ln(x)}{e^x tan^{-1}(\frac{\pi}{3} + sin x)}\)
3. (part A)Suppose there are k students and n identical chocolates. The chocolates are to be distributed one by one to the students (with each student having equal probability of receiving each chocolate). Find the probability of a particular student getting at least one chocolate.
   (part B) Suppose the number of ways of distributing the k chocolates to n students be \(\dbinom{n+k-1}{k}\). Find the probability of a particular student getting at least one chocolate.
4. Show that \(\dfrac{ln 12}{ln 18}\) is an irrational number.
5. Give an example of a polynomial with real coefficients such that \(P(\sqrt{2} + i)=0\). Further given an example of a polynomial with rational coefficients such that \(P(\sqrt{2} + i)=0\).
6. Say f(1) = 2; f(2) = 3, f(3) = 1; then show that f'(x) = 0 for some x (given that f is a continuously differentiable function defined on all real numbers).

10 mark questions

1. (part A) Suppose a plane has 2n points; n red points and n blue points. One blue point and one red point is joined by a line segment. Like this n line segments are drawn by pairing a red and a blue point. Prove that each such scheme of pairing segments will have two segments which do not intersect each other.
   (part B) Suppose the position of the n red points are given. Prove that we can put n blue points in such a way that there are two segments (produced in the manner described in part A) which do not intersect each other.
2. (part A) Let ABCD be any quadrilateral. E, F, G and H be the mid points of the sides AB, BC, CD and DA respectively. Prove that EFGH is a parallelogram whose area is half of the quadrilateral ABCD.
   (part B) Suppose the coordinates of E, F, G, H are given:  E (0,0) , F(0, -1), G (1, -1) , H (1, 0). Find all points A in the first quadrant such that E, F, G, H be the midpoints of quadrilateral ABCD.
3. Let f be a function whose domain and co-domain be non negative natural numbers such that f(f(f(n))) < f(n+1). Prove that:
   (a) If f(n) = 0 then n = 0.
   (b) f(n) < n+1
   (c) If f(x) < n then x<n
   Using the above prove that f is an identity function, that is f(n) = n.
4. Consider a sequence \(c_{n+2} = a c_{n+1} + b c_n\) for \(n \ge 0\) where \(c_0 = 0\). If gcd(b, k) = 1 then show that k divides n for infinitely many n.
5. Find out the value of \(x^{2012} + \dfrac{1}{x^{2012}}\) when \(x + \dfrac{1}{x} = \dfrac{\sqrt{5} + 1}{2}\).
   Hint
   (a) Show that \(|{r + \dfrac{1}{r}}|\ge 2\) for all real r.
   (b) Prove that \(\sin \dfrac{\pi}{5} < \cos \dfrac{2\pi}{5} < \sin \dfrac{3\pi}{5}\).
6. A polynomial P(x) takes values \(prime^{positive number}\) for every positive integer n,then show that p(x) is a constant polynomial.
   If such a polynomial exist then show that there also exists a polynomial g(x)= \(prime^l\) where l is a fixed number.
7. Consider a set A = {1, 2, ... , n}. Suppose \(A_1 , A_2 , ... , A_k \) be subsets of set A such that any two of them consists at least one common element. Show that the greatest value of k is \(2^{n-1}\). Further, show that if they any two of them have a common element but intersection of all of them is a null set then the greatest value of k is \(2^{n-1}\).
8. Suppose \(\displaystyle x = \sum_{i=1}^{10} \dfrac{1}{10 \sqrt 3} . \dfrac{1}{1+ (\dfrac{i}{10 \sqrt 3})^2}\) and \(\displaystyle y = \sum_{i=0}^9 \dfrac{1}{10 \sqrt 3} . \dfrac{1}{1+ (\dfrac{i}{10 \sqrt 3})^2}\)
1.  Show that \(x < \dfrac{\pi}{6} < y \) 
2. \(\dfrac{x+y}{2} < \dfrac{\pi}{6} \)

[b]OTHERS PLEASE CONTRIBUTE THE REST OF THE QUESTIONS (AND SOLUTIONS). WE ARE TRYING ON OUR END TO DO THE SAME[/b]

I.S.I. Entrance Model Test @ Cheenta Schedule

The model test are based on the entrances of Indian Statistical Institute B.Stat and B.Math Entrance which are scheduled on 13th May, 2012.

All Cheenta Model Test will be held on Saturdays. The tests will start at 10 AM in the morning in our Kolkata center.

External Students may also appear in the tests. For registration call 09804005499, 07381944396 or mail us at helpdesk at cheenta dot com.

Model Test	Date
1	17th December, 2011
2	31st December, 2011
3	14th January, 2012
4	28th January, 2012
5	11th February, 2012
6	25th February, 2012
7	10th March, 2012
8	24th March, 2012
9	7th April, 2012
10	21st April, 2012

CLASS PREVIEW: WEEK AHEAD

INSIDE CLASSROOM: WEEK AHEAD

INDIAN MATH IVY

Indian Statistical Institute (I.S.I.), Chennai Mathematical Institute (C.M.I.) and Institute of Mathematics and Application (I.M.A.) can be regarded as three Indian institutions that provided world class mathematics course at undergraduate level. The B.Stat Course at I.S.I. is also world famous. The courses at C.M.I. and I.M.A. have computer science as second major.

Each of these three institutes conduct entrance test at the month of May. The prospective students are expected to know more rigorous mathematics than the IIT's. Apart from High School math, the entrance test syllabus includes Math Olympiad curriculum- plane geometry, combinatorics, number theory, inequalities, functional equation, polynomials and more.

Indian National Math Olympiad qualified students are not required to take the Entrance test. They will directly appear for the interview. Yes, there is an interview after the entrance test where students are asked math related questions.

The entrance tests are on Mathematics only. There is no age bar and non science students may also sit in the test provided he/she had pure maths in High School. The alumni of these three institutes are working in the most prestigious labs, universities and corporate sectors of the world. India hopes to regenerate it's mathematical fervor through state-of-art institutes like these.

Preparatory guidance for I.S.I., C.M.I. and I.M.A. entrances (and Math Olympiads) are difficult to find. Check out our website for some help.