Select Page

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

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 $\displaystyle{l_1, l_2}$.

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 $\displaystyle{\frac{f(n)}{n!}=\sum_{k=0}^n \frac{(-1)^k}{k!}}$