How 9 Cheenta students ranked in top 100 in ISI and CMI Entrances?

Learn MoreHere, you will find all the questions of ISI Entrance Paper 2014 from Indian Statistical Institute's B.Stat Entrance. You will also get the solutions soon of all the previous year problems.

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**Problem 1:**

Let $ y = x^2 + ax + b $ be a parabola that cuts the coordinate axes at three distinct points. Show that the circle passing through these three points also passes through $(0,1)$. Discussion

**Problem 2: **

Find all such Natural number $n$ such that $7$ divides $5^n + 1$.

**Problem 3: **

Let $\mathbb{R}$ denote the set of real numbers. Find all functions $f: \mathbb{R} \rightarrow \mathbb{R},$ satisfying

$$

|f(x)-f(y)|=2|x-y|

$$

for all $x, y \in \mathbb{R} .$ Justify your answer.

**Problem 4: **

Say $0 < a_1 < a_2 < ... < a_n $ be $n$ real numbers. Show that the equation $\frac{a_1}{a_1 - x } + \frac{a_2}{a_2 - x} + \cdot + \frac{a_n}{a_n - x} = 2015 $ has $n$ real solutions.

**Problem 5: **

Consider the set $S = \{1, 2, 3, ..., j\}$. In a subset $P$ of $S$, Max $P$ be the maximum element of that subset. Show that the sum of all Max $P$ (over all subsets of the set) is $(j-1)2^j + 1 $

**Problem 6: **

There are three unit circles each of which is tangential to the other two. A triangle is drawn such that each side of the triangle is tangential to exactly two of the circle. Find the length of sides of this triangle.

**Problem 7: **

Let $m_1< m_2 < \cdots <m_{k-1}< m_k$ be $k$ distinct positive integers such that their reciprocals are in arithmetic progression.

$1$. Show that $k< m_1 + 2$.

$2$. For any integer $k>0$, give an example of a sequence of $k$ positive integers whose reciprocals are in arithmetic progression.

**Problem 8: **

Let $P(x) = x^7 + x^6 + b_5 x^5 + b_4 x^4 + \cdots + b_0$ and $Q(x) = x^5 + c_4 x^4 + c_3 x^3 + \cdots + c_0$ be polynomial with integer coefficients , Assume that $P(i) = Q(i)$, for integers $i= 1,2,3 \cdot 6$ . Show that there exists a negative integer $r$ such that $P(r) = Q(r)$ .

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