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# I.S.I. B.STAT ENTRANCE SUBJECTIVE 2006

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 $$\displaystyle{\frac{f(n)}{n!}=\sum_{k=0}^n \frac{(-1)^k}{k!}}$$
May 4, 2014