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Learn MoreHere, you will find all the questions of ISI Entrance Paper 2011 from Indian Statistical Institute's B.Stat Entrance. You will also get the solutions soon of all the previous year problems.

**Problem 1:**

Let $x_1, x_2, \cdots , x_n $ be positive reals with $x_1+x_2+\cdots+x_n=1 $. Then show that $ \sum_{i=1}^n \frac{x_i}{2-x_i} \ge \frac{n}{2n-1} $

**Problem 2: **

Consider three positive real numbers $a,b$ and $c$. Show that there cannot exist two distinct positive integers $m$ and $n$ such that both $\mathbf{a^m+b^m=c^m}$ and $\mathbf{ a^n+b^n=c^n} $ hold.

**Problem 3: **

Let $\mathbb{R} $ denote the set of real numbers. Suppose a function $f:R \rightarrow R$ satisfies $f(f(f(x)))=x$ for all $x\in \mathbb{R} $. Show that

(i) $f$ is one-one,

(ii) $f$ cannot be strictly decreasing, and

(iii) if $f$ is strictly increasing, then $f(x)=x$ for all $x \in \mathbb{R} $.

**Problem 4: **

Let $f$ be a twice differentiable function on the open interval $(-1,1)$ such that $f(0)=1$. Suppose $f$ also satisfies $f(x) \ge 0, f'(x) \le 0 $ and $f''(x)\le f(x)$, for all $x\ge 0$. Show that $f'(0) \ge -\sqrt2$.

**Problem 5: **

$ABCD$ is a trapezium such that $\mathbf{AB\parallel DC} $ and $ \mathbf{\frac{AB}{DC}=\alpha} $ >1. Suppose $P$ and $Q$ are points on $AC$ and $BD$ respectively, such that $\mathbf{\frac{AP}{AC}=\frac{BQ}{BD}=\frac{\alpha -1}{\alpha+1}} $

Prove that $PQCD$ is a parallelogram.

**Problem 6: **

Let $\mathbf{\alpha } $ be a complex number such that both $\mathbf{ \alpha } $ and $\mathbf{\alpha+1 } $ have modulus $1$. If for a positive integer $n$, $ \mathbf{ 1+\alpha } $ is an $n$-th root of unity, then show that $\mathbf{ \alpha } $ is also an $n$-th root of unity and $n$ is a multiple of $6$.

**Problem 7:**

** **(i) Show that there cannot exists three prime numbers, each greater than $3$, which are in arithmetic progression with a common difference less than $5$.

(ii) Let $k > 3$ be an integer. Show that it is not possible for $k$ prime numbers, each greater than $k$, to be in an arithmetic progression with a common difference less than or equal to $k+1$.

**Problem 8: **

Let $\mathbf{I_n =\int_{0}^{n\pi} \frac{\sin x}{1+x} , dx , n=1,2,3,4} $ . Arrange $\mathbf{I_1, I_2, I_3, I_4 } $ in increasing order of magnitude. Justify your answer.

**Problem 9: **

Consider all non-empty subsets of the set $\mathbf{{1,2\cdots,n}}$. For every such subset, we find the product of the reciprocals of each of its elements. Denote the sum of all these products as $\mathbf{S_n} $. For example, $ \mathbf{S_3=\frac11+\frac12+\frac13+\frac1{1\cdot 2}+\frac1{1\cdot 3}+\frac1{2\cdot 3} +\frac1{1\cdot 2\cdot 3} } $

1. Show that $\mathbf{S_n=\frac1n+\left(1+\frac1n\right)S_{n-1} }$.

2. Hence or otherwise, deduce that $\mathbf{S_n=n} $.

**Problem 10: **

Show that the triangle whose angles satisfy the equality $ \mathbf{\frac{\sin^2A+\sin^2B+\sin^2C}{\cos^2A+\cos^2B+\cos^2C} = 2} $ is right angled.

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