Q1
Answer
A curve has equation $y=\dfrac{2(x-1)}{x^2+3}$ and crosses the $x$ axis at the point $A$.
  1. Show that the normal to the curve at the point $A$ has equation $y=2-2x$.
  2. Find the coordinates of any stationary points on the curve.
  1. $y = 2 - 2x$
  2. $\left(3, \frac{1}{3}\right)$, $(-1,-1)$
Q2
Answer
Given that $f(x) = \dfrac{\sqrt{x-1}}{\sqrt{x+1}}$,
  1. Find $f'(x)$.
  2. Hence evaluate $\displaystyle\int \dfrac{1}{(x+1)\sqrt{x^2-1}}\ \mathrm{d}x$.
  1. $\dfrac{1}{(x+1)\sqrt{x^2-1}}$
  2. $\sqrt{\dfrac{x-1}{x+1}} + c$
Q3
Answer
  1. Sketch the graph of $y=8^x$, stating the points of any intersections with the axes
  2. Describe fully the transformation that transforms the graph $y=8^x$ to $y=8^{x-1}+5$
  1. Standard graph
  2. Translation by $1$ right and $5$ up