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# ISI Entrance Paper 2016 – B.Stat, B.Math Subjective

ISI Entrance Paper 2016 – from Indian Statistical Institute’s B.Stat, B.Math Entrance

1. In a sports tournament of $n$ players, each pair of players plays against each other exactly one match and there are no draws. Show that the players can be arranged in an order $P_1,P_2, …. , P_n$ such that $P_i$ defeats $P_{i+1}$ for all $1 \le i \le n-1$.
2. Consider the polynomial $ax^3+bx^2+cx+d$ where $a,b,c,d$ are integers such that $ad$ is odd and $bc$ is even. Prove that not all of its roots are rational.
3. If $P(x)=x^n+a_1x^{n-1}+…+a_{n-1}$ be a polynomial with real coefficients and $a_1^2<a_2$ then prove that not all roots of $P(x)$ are real.
4. Given a square $ABCD$ with two consecutive vertices, say $A$ and $B$ on the positive $x$-axis and positive $y$-axis respectively. Suppose the other vertice $C$ lying in the first quadrant has coordinates $(u , v)$. Then find the area of the square $ABCD$ in terms of $u$ and $v$.
5. Prove that there exists a right angle triangle with rational sides and area $d$ if and only if $x^2,y^2$ and $z^2$ are squares of rational numbers and are in Arithmetic Progression
Here $d$ is an integer.
6. Suppose in a triangle $\triangle ABC$, $A$ , $B$ , $C$ are the three angles and $a$ , $b$ , $c$ are the lengths of the sides opposite to the angles respectively. Then prove that if $sin(A-B)= \frac{a}{a+b}\sin A \cos B – \frac{b}{a+b}\sin B \cos A$ then the triangle $\triangle ABC$ is isoscelos.
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. Suppose that $(a_n)_{n\geq 1}$ is a sequence of real numbers satisfying $a_{n+1} = \frac{3a_n}{2+a_n}$.
1. Suppose $0 < a_1 <1$, then prove that the sequence $a_n$ is increasing and hence show that $\lim_{n \to \infty} a_n =1$.
2.  Suppose $a_1 >1$, then prove that the sequence $a_n$ is decreasing and hence show that $\lim_{n \to \infty} a_n =1$. ## By Dr. Ashani Dasgupta

Ph.D. in Mathematics, University of Wisconsin, Milwaukee, United States.

Research Interest: Geometric Group Theory, Relatively Hyperbolic Groups.

Founder, Cheenta

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