Tuesday, 22 July 2014

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

If the normal to the curve $\mathbf{ x^{\frac{2}{3}}+y^{\frac23}=a^{\frac23} }$ at some point makes an angle $\mathbf{\theta}$ with the X-axis, show that the equation of the normal is $\mathbf{y\cos\theta-x\sin\theta=a\cos 2\theta}$

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 $\mathbf{r+\sqrt{s}}$ or $\mathbf{r-\sqrt{s}}$ for some rationals r and s, where s is not the square of a rational number).
(b) Show that there are two real numbers $\mathbf{b_1}$ and $\mathbf{b_2}$ such that
i) $\mathbf{a+b_1}$ is rational but $\mathbf{ab_1}$ is irrational.
ii) $\mathbf{a+b_2}$ is irrational but $\mathbf{ab_2}$ is rational.
(Hint: Consider the two cases, where a is a quadratic surd and a is not a quadratic surd, separately).

Prove that $\mathbf{n^4 + 4^{n}}$ is composite for all values of n greater than 1.

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 $\mathbf{l_1}$ , and that of the segment BD is $\mathbf{l_2}$ , determine the length of DC in terms of $\mathbf{l_1, l_2}$ .

Let A,B and C be three points on a circle of radius 1.
(a) Show that the area of the triangle ABC equals $\mathbf{\frac12(\sin(2\angle ABC)+\sin(2\angle BCA)+\sin(2\angle CAB))}$
(b) Suppose that the magnitude of $\mathbf{\angle ABC}$ is fixed. Then show that the area of the triangle ABC is maximized when $\mathbf{\angle BCA=\angle CAB}$
(c) Hence or otherwise, show that the area of the triangle ABC is maximum when the triangle is equilateral.

(a) Let $\mathbf{f(x)=x-xe^{-\frac1x}, \ \ x>0 }$ . Show that f(x) is an increasing function on $\mathbf{(0,\infty)}$ , and $\mathbf{\lim_{x\to\infty} f(x)=1}$ .
(b) Using part (a) or otherwise, draw graphs of $\mathbf{y=x-1, y=x, y=x+1, \text{and} y=xe^{-\frac{1}{|x|}}}$ for $\mathbf{-\infty < x < \infty}$ using the same X and Y axes.

for any positive integer n greater than 1, show that $\mathbf{2^n < \binom{2n}{n} <\frac{2^n}{\prod\limits_{i=0}^{n-1} \left(1-\frac{i}{n}\right)}}$

Show that there exists a positive real number $\mathbf{x\neq 2}$ such that $\mathbf{\log_2x=\frac{x}{2}}$ . Hence obtain the set of real numbers c such that $\mathbf{\frac{\log_2x}{x}=c}$ has only one real solution.

Find a four digit number M such that the number $\mathbf{N=4\times M}$ 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.

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 $\mathbf{n \ge 2}$ . Show that $\mathbf{\frac{f(n)}{n!}=\sum_{k=0}^n \frac{(-1)^k}{k!}}$

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