PG Admissions

Admission season to UG is over. Now, let’s focus on Admissions to PG.

Our examinations of interest are mainly:

  • M.Math Admission Test at ISI
  • M.Sc in Mathematics Admission Test at CMI
  • JAM 2017, organised by IIT D (Official Site) (MA & MS)
  • NBHM 2017 (Ph.D and M.Sc)
  • TIFR Entrance Examination (Mathematics)

Stay tuned for more details.

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ISI Admission Test 2016 B.Math/B.Stat

  1. In a sports tournament of n players, each pair of players plays exactly one match against each other. There are no draws. Prove that the players can be arranged in an order P_1, P_2, ..., P_n, such that P_i defeats P_{i+1} \forall i=1, 2, ..., n-1.
  2. Consider the polynomial ax^3 + bx^2 + cx + d, where ad is odd and bc is even. Prove that all roots of the polynomial cannot be rational.
  3. P(x) = x^n + a_1x^{n-1} + ... + a_{n} is a polynomial with real coefficients. a_1^2<a_2. Prove that all roots of P(x) cannot be real.
  4. Let ABCD be a square. Let A lie on the positive x-axis and B on the positive y-axis. Suppose the vertex C lies in the first quadrant and has co-ordinates (u,v). Then find the area of the square in terms of u and v.
  5. Prove that there exists a right-angled triangle with rational sides and area d iff there exist rational numbers x, y, z such that x^2, y^2, z^2 are in arithmetic progression with common difference d.
  6. Suppose in a \triangle ABC, A, B, C denote the three angles and a, b, c denote the three sides opposite to the corresponding angles. Prove that, if sin\ (A-B) = \frac{a}{a+b}\sin{A} \cos{B}-\frac{b}{a+b}\sin{B}\cos{A}, then \triangle ABC is isosceles.
  7. f is a differentiable function, such that f(f(x))=x, where x\in [0,1]. Also, f(0)=1. Find the value of \int_0^1(x-f(x))^{2016}dx
  8. (a_n)_{n\geq 1} is a sequence of real numbers satisfying a_{n+1}=\frac{3a_n}{2+a_n}. (i) If 0<a_1<1, then prove that the sequence a_n is increasing and hence, \lim_{n \to \infty} a_n = 1. (ii) If a_1>1, then prove that the sequence a_n is decreasing and hence, \lim_{n \to \infty} a_n = 1.

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