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]