Q1
Answer
A curve with equation $y = \dfrac{x+2}{\sqrt{x-2}}$ has a minimum at point $P$.
  1. Find and simplify an expression for $\dfrac{\mathrm{d}y}{\mathrm{d}x}$.
  2. Find the coordinates of $P$.
  3. Find the equation of the normal to the curve at the point where $x = 3$.
  1. $\dfrac{x-6}{2(x-2)^{\frac{3}{2}}}$
  2. $(6,4)$
  3. $2x-3y+9=0$
Q2
Answer
$$f(x) = \dfrac{6x}{(x-1)(x+2)} - \dfrac{2}{x-1}$$
  1. Show that $f(x) = \dfrac{4}{x+2}$.
  2. Find an equation for the tangent to the curve at the point with $x$ coordinate 2, giving your answer in the form $ax+by=c$, where $a$, $b$ and $c$ are integers.
  1. Algebra
  2. $x + 4y = 6$
Q3
Answer
For each statement, either prove that it is true or find a counter example to prove that it is false.
  1. If $a$ and $b$ are two different irrational numbers, then $(a+b)$ is irrational.
  2. If $m$ and $n$ are consecutive odd numbers, then $(m + n)$ is divisible by $4$.
  3. For all real values of $x$, $\cos x\leq 1 + \sin x$.
  1. Counterexample e.g. $a = \sqrt{2}$ and $b = -\sqrt{2}$
  2. Use $2n-1$ and $2n+1$ as consecutive odd numbers, true
  3. Counterexample e.g. $x = 300^{\circ}$
Q4
Answer
Find the area of the region bound by the curve $y = \dfrac{12}{(2x+1)^3}$, the coordinate axes, and the line $x=1$.
$\dfrac{8}{3}$