Q1
Answer
Given that $f(x)=x^{\frac{3}{2}}(x-3)^3$,
  1. Show that $f'(x) = k\sqrt{x}(x-1)(x-3)^2$, where $k$ is a constant to be found.
  2. Hence find the coordinates of the stationary points of the curve $y=f(x)$.
  1. $\dfrac{9}{2}x^{\frac{1}{2}}(x-1)(x-3)^2$
  2. $(0,0)$, $(1,-8)$, $(3,0)$
Q2
Answer
Evaluate:
  1. $\displaystyle\int \dfrac{x^3}{(x^4-2)^2}\ \mathrm{d}x$
  2. $\displaystyle\int_2^3 \dfrac{x}{\sqrt{x^2-3}}\ \mathrm{d}x$
  1. $-\dfrac{1}{4(x^4-2)} + c$
  2. $\sqrt{6}-1$
Q3
Answer
Find the first three terms, in ascending powers of $x$, of the expansion of $\dfrac{1 + x}{1 - 2x}$
$1 + 3x + 6x^2$