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.
  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 \(\displaystyle{l_1, l_2}\).
  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 \(\displaystyle{\frac{f(n)}{n!}=\sum_{k=0}^n \frac{(-1)^k}{k!}}\)

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