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

Also see: ISI and CMI Entrance Course at Cheenta

  1. If the normal to the curve \displaystyle{ x^{\frac{2}{3}}+y^{\frac23}=a^{\frac23} } at some point makes an angle \displaystyle{\theta} with the X-axis, show that the equation of the normal is \displaystyle{y\cos\theta-xsin\theta=a\cos 2\theta}
  2. 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 \displaystyle{r+\sqrt{s}} or \displaystyle{r-\sqrt{s}} for some rationals r and s, where s is not the square of a rational number).
    (b) Show that there are two real numbers \displaystyle{b_1} and \displaystyle{b_2} such that
    i) \displaystyle{a+b_1} is rational but \displaystyle{ab_1} is irrational.
    ii) \displaystyle{a+b_2} is irrational but \displaystyle{ab_2} is rational.
    (Hint: Consider the two cases, where a is a quadratic surd and a is not a quadratic surd, separately).
  3. Prove that \displaystyle{n^4 + 4^{n}} is composite for all values of n greater than 1.
    Discussion
  4. 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 \displaystyle{l_1}, and that of the segment BD is \displaystyle{l_2}, determine the length of DC in terms of \displaystyle{l_1, l_2}.
    B.Stat Entrance 2006 Geometry Problem
    Discussion
  5. Let A,B and C be three points on a circle of radius 1.
    (a) Show that the area of the triangle ABC equals \displaystyle{\frac12(sin(2\angle ABC)+sin(2\angle BCA)+sin(2\angle CAB))}
    (b) Suppose that the magnitude of \displaystyle{\angle ABC} is fixed. Then show that the area of the triangle ABC is maximized when \displaystyle{\angle BCA=\angle CAB}
    (c) Hence or otherwise, show that the area of the triangle ABC is maximum when the triangle is equilateral.
  6. (a) Let \displaystyle{f(x)=x-xe^{-\frac1x}, x>0 }. Show that f(x) is an increasing function on \displaystyle{(0,\infty)}, and \displaystyle{\lim_{x\to\infty} f(x)=1}.
    (b) Using part (a) or otherwise, draw graphs of \displaystyle{y=x-1, y=x, y=x+1, \text{and} y=xe^{-\frac{1}{|x|}}} for \displaystyle{-\infty < x < \infty} using the same X and Y axes.
  7. For any positive integer n greater than 1, show that \displaystyle{2^n < \binom{2n}{n} <\frac{2^n}{\prod\limits_{i=0}^{n-1} \left(1-\frac{i}{n}\right)}}
  8. Show that there exists a positive real number \displaystyle{x\neq 2} such that \displaystyle{\log_2x=\frac{x}{2}}. Hence obtain the set of real numbers c such that \displaystyle{\frac{\log_2x}{x}=c} has only one real solution.
  9. Find a four digit number M such that the number \displaystyle{N=4\times M} 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.
  10. 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 \displaystyle{n \ge 2}. Show that \displaystyle{\frac{f(n)}{n!}=\sum_{k=0}^n \frac{(-1)^k}{k!}}