ISI Entrance Paper 2006 – from Indian Statistical Institute’s B.Stat Entrance

Also see: ISI and CMI Entrance Course at Cheenta

- If the normal to the curve at some point makes an angle with the X-axis, show that the equation of the normal is
- Suppose that a is an irrational number.

(a) If there is a real number b such that both (a+b) and ab are rational numbers, show that a is a quadratic surd. (a is a quadratic surd if it is of the form or for some rationals r and s, where s 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 a is a quadratic surd and a is not a quadratic surd, separately). - Prove that is composite for all values of n greater than 1.

Discussion - In the figure below, E is the midpoint of the arc ABEC and the segment ED is perpendicular to the chord BC at D. If the length of the chord AB is , and that of the segment BD is , determine the length of DC in terms of .

Discussion - Let A,B and C be three points on a circle of radius 1.

(a) Show that the area of the triangle ABC equals

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

(c) Hence or otherwise, show that the area of the triangle ABC is maximum when the triangle is equilateral. - (a) Let . Show that f(x) is an increasing function on , and .

(b) Using part (a) or otherwise, draw graphs of for using the same X and Y axes. - For any positive integer n greater than 1, show that
- Show that there exists a positive real number such that . Hence obtain the set of real numbers c such that has only one real solution.
- Find a four digit number M such that the number has the following properties.

(a) N is also a four digit number

(b) N has the same digits as in M but in reverse order. - Consider a function f on nonnegative integers such that f(0)=1, f(1)=0 and f(n)+f(n-1)=nf(n-1)+(n-1)f(n-2) for . Show that