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Explore the Back-StoryHere, you will find all the questions of ISI Entrance Paper 2008 from Indian Statistical Institute's B.Stat Entrance. You will also get the solutions soon of all the previous year problems.

**Problem 1:**

** **Of all triangles with given perimeter, find the triangle with the maximum area. Justify your answer

**Problem 2:**

A $40$ feet high screen is put on a vertical wall $10$ feet above your eye-level. How far should you stand to maximize the angle subtended by the screen (from top to bottom) at your eye?

**Problem 3:**

Study the derivatives of the function

$\mathbf{y=\sqrt[3]{x^3-4x}}$

and sketch its graph on the real line.

**Problem 4:**

Suppose $P$ and $Q$ are the centres of two disjoint circles $\mathbf{C_1}$ and $\mathbf{C_2}$ respectively, such that $P$ lies outside $\mathbf{C_2}$ and $Q$ lies outside $\mathbf{C_1}$. Two tangents are drawn from the point $P$ to the circle $\mathbf{C_2}$, which intersect the circle $\mathbf{C_1}$ at point $A$ and $B$. Similarly, two tangents are drawn from the point $Q$ to the circle $ \mathbf{C_1}$, which intersect the circle $\mathbf{C_2}$ at points $M$ and $N$. Show that $AB=MN$

**Problem 5:**

Suppose $ABC$ is a triangle with inradius $r$. The incircle touches the sides $BC$, $CA$, and $AB$ at $D,E$ and $F$ respectively. If $BD=x$, $CE=y$ and $AF=z$, then show that $\mathbf{r^2=\frac{xyz}{x+y+z}}$

**Problem 6:**

Evaluate: $\lim_{n \to\infty} \frac{1}{2n} \ln {{2n} \choose{n}}$

**Problem 7:**

Consider the equation $\mathbf{x^5+x=10}$. Show that

(a) the equation has only one real root;

(b) this root lies between $1$ and $2$;

(c) this root must be irrational.

**Problem 8:**

In how many ways can you divide the set of eight numbers $ \mathbf{{2,3,\cdots,9}}$ into $4$ pairs such that no pair of numbers has $ \mathbf{\text{gcd} }$ equal to $2$?

**Problem 9:**

Suppose $S$ is the set of all positive integers. For $\mathbf{a,b \in S}$, define

$\mathbf{a * b=\frac{\text{lcm}[a,b]}{\text{gcd}(a,b)}}$

For example $8 * 12=6$.

Show that exactly two of the following three properties are satisfied:

(i) If $\mathbf{a,b \in S}$, then $\mathbf{a * b \in S}$.

(ii) $\mathbf{(a*b)*c=a*(b*c)}$ for all $\mathbf{a,b,c \in S}$.

(iii) There exists an element $ \mathbf{i \in S}$ such that $\mathbf{a *i =a}$ for all $\mathbf{a \in S}$

**Problem 10:**

Two subsets $A$ and $B$ of the ($x,y$)-plane are said to be equivalent if there exists a function $f: A$ to $B$ which is both one-to-one and onto.

(i) Show that any two line segments in the plane are equivalent.

(ii) Show that any two circles in the plane are equivalent.

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How to solve problem number 5 and 6?

Solution to 6: Let y be the given limit. Then e^y = (2nCn)^(1/2n). Now 2nCn = (2n)!/(n!)^2 . We divide both Nr and Dr by (2n)^2n. Hence e^y = [ {(2n)!/(2n)^2n}^(1/2n)]/[{n!/n^n}^2]^(1/2n)*2. Now limit n tends to infinity (n!/n^n)^(1/2n) is e^-1. Hence on simplification limit n tends to infinity e^y =2 and finally result is log 2. Do it yourself to understand the solution.

you can expand the factorial and use definite integral as a sum of infinite terms. the things multiplied in the log can be split and then take as a summation which can be used as an integral by putting r/n as x and 1/n as dx