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Explore the Back-StoryHere, you will find all the questions of ISI Entrance Paper 2006 from Indian Statistical Institute's B. Stat Entrance. You will also get the solutions soon of all the previous year problems.

**Problem1 : **

If the normal to the curve at some point makes an angle with the -axis, show that the equation of the normal is

**Problem 2 :**

Suppose that is an irrational number.

(a) If there is a real number such that both and are rational numbers, show that is a quadratic surd. ( is a quadratic surd if it is of the form or for some rationals and , where is not the square of a rational number).

(b) Show that there are two real numbers and such that

i) is rational but is irrational.

ii) is irrational but is rational.

(Hint: Consider the two cases, where is a quadratic surd and is not a quadratic surd, separately).

**Problem3 :**

Prove that is composite for all values of greater than .

Discussion

**Problem4 :**

In the figure below, is the midpoint of the arc and the segment is perpendicular to the chord at . If the length of the chord is , and that of the segment is , determine the length of in terms of .

Discussion

**Problem 5 :**

Let and be three points on a circle of radius .

(a) Show that the area of the triangle equals

(b) Suppose that the magnitude of is fixed. Then show that the area of the triangle is maximized when

(c) Hence or otherwise, show that the area of the triangle is maximum when the triangle is equilateral.

**Problem 6 :**

(a) Let . Show that f(x) is an increasing function on , and .

(b) Using part (a) or otherwise, draw graphs of , and for using the same and axes.

**Problem 7 :**

For any positive integer greater than , show that

**Problem 8:**

Show that there exists a positive real number such that . Hence obtain the set of real numbers such that has only one real solution.

**Problem 9 :**

Find a four digit number such that the number has the following properties.

(a) is also a four digit number

(b) has the same digits as in but in reverse order.

**Problem 10 :**

Consider a function on nonnegative integers such that , and +=+ for . Show that =

Here, you will find all the questions of ISI Entrance Paper 2006 from Indian Statistical Institute's B. Stat Entrance. You will also get the solutions soon of all the previous year problems.

**Problem1 : **

If the normal to the curve at some point makes an angle with the -axis, show that the equation of the normal is

**Problem 2 :**

Suppose that is an irrational number.

(a) If there is a real number such that both and are rational numbers, show that is a quadratic surd. ( is a quadratic surd if it is of the form or for some rationals and , where is not the square of a rational number).

(b) Show that there are two real numbers and such that

i) is rational but is irrational.

ii) is irrational but is rational.

(Hint: Consider the two cases, where is a quadratic surd and is not a quadratic surd, separately).

**Problem3 :**

Prove that is composite for all values of greater than .

Discussion

**Problem4 :**

In the figure below, is the midpoint of the arc and the segment is perpendicular to the chord at . If the length of the chord is , and that of the segment is , determine the length of in terms of .

Discussion

**Problem 5 :**

Let and be three points on a circle of radius .

(a) Show that the area of the triangle equals

(b) Suppose that the magnitude of is fixed. Then show that the area of the triangle is maximized when

(c) Hence or otherwise, show that the area of the triangle is maximum when the triangle is equilateral.

**Problem 6 :**

(a) Let . Show that f(x) is an increasing function on , and .

(b) Using part (a) or otherwise, draw graphs of , and for using the same and axes.

**Problem 7 :**

For any positive integer greater than , show that

**Problem 8:**

Show that there exists a positive real number such that . Hence obtain the set of real numbers such that has only one real solution.

**Problem 9 :**

Find a four digit number such that the number has the following properties.

(a) is also a four digit number

(b) has the same digits as in but in reverse order.

**Problem 10 :**

Consider a function on nonnegative integers such that , and +=+ for . Show that =

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