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.

## CMI Results

CMI Results are out. Check here! Congratulations to all those who have qualified. If you haven’t don’t worry. There are lots of other places to study Math. And if you want, you can always wait a year. For any help, contact us!

## 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. 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, 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, $\lim_{n \to \infty} a_n = 1$.