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Explore the Back-StoryHere, 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 be positive reals with . Then show that

**Problem 2: **

Consider three positive real numbers and . Show that there cannot exist two distinct positive integers and such that both and hold.

**Problem 3: **

Let denote the set of real numbers. Suppose a function satisfies for all . Show that

(i) is one-one,

(ii) cannot be strictly decreasing, and

(iii) if is strictly increasing, then for all .

**Problem 4: **

Let be a twice differentiable function on the open interval such that . Suppose also satisfies and , for all . Show that .

**Problem 5: **

is a trapezium such that and >1. Suppose and are points on and respectively, such that

Prove that is a parallelogram.

**Problem 6: **

Let be a complex number such that both and have modulus . If for a positive integer , is an -th root of unity, then show that is also an -th root of unity and is a multiple of .

**Problem 7:**

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

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

**Problem 8: **

Let . Arrange in increasing order of magnitude. Justify your answer.

**Problem 9: **

Consider all non-empty subsets of the set . For every such subset, we find the product of the reciprocals of each of its elements. Denote the sum of all these products as . For example,

1. Show that .

2. Hence or otherwise, deduce that .

**Problem 10: **

Show that the triangle whose angles satisfy the equality is right angled.

Here, 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 be positive reals with . Then show that

**Problem 2: **

Consider three positive real numbers and . Show that there cannot exist two distinct positive integers and such that both and hold.

**Problem 3: **

Let denote the set of real numbers. Suppose a function satisfies for all . Show that

(i) is one-one,

(ii) cannot be strictly decreasing, and

(iii) if is strictly increasing, then for all .

**Problem 4: **

Let be a twice differentiable function on the open interval such that . Suppose also satisfies and , for all . Show that .

**Problem 5: **

is a trapezium such that and >1. Suppose and are points on and respectively, such that

Prove that is a parallelogram.

**Problem 6: **

Let be a complex number such that both and have modulus . If for a positive integer , is an -th root of unity, then show that is also an -th root of unity and is a multiple of .

**Problem 7:**

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

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

**Problem 8: **

Let . Arrange in increasing order of magnitude. Justify your answer.

**Problem 9: **

Consider all non-empty subsets of the set . For every such subset, we find the product of the reciprocals of each of its elements. Denote the sum of all these products as . For example,

1. Show that .

2. Hence or otherwise, deduce that .

**Problem 10: **

Show that the triangle whose angles satisfy the equality is right angled.

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problem 8 is based on basic inequalities

Let f be a twice differentiable function on the open interval (-1,1) such that f(0)=1. Suppose f also satisfies f(x)≥0,f′(x)≤0 and f”(x)≤f(x), for all x\ge 0. Show that f′(0)≥−2–√.

How to proceed with no. 4?

Can you help me solving the 19 number question ??...

Consider the L-shaped diagram below ..that's one