Q1
Answer
$$\mathrm{f}(x) = x^2-(k+8)x+(8k+1)$$
  1. Find the discriminant of $\mathrm{f}(x)$ in terms of $k$. Give your answer as a simplified quadratic.
  2. If the equation $\mathrm{f}(x)=0$ has two equal roots, find the possible values of $k$.
  3. Show that when $k = 8$, $\mathrm{f}(x) > 0$ for all values of $x$.
  1. $k^2 - 16k + 60$
  2. $6, 10$
  3. Discriminant: $-4$ no roots and e.g. $\mathrm{f}(0) > 0$
Q2
Answer
Sketch the graph of $y = \dfrac{4}{x-6}+5$, labelling any asymptotes and any points of intersection with the axes.
Compared to $y = \dfrac{1}{x}$ - translate 6 right, stretch parallel to $y$ axis scale factor 4, translate up 5
Q3
Answer
Prove that, for all values of $x$, $x^2+6x+18 > 2 - \frac{1}{2}x$.
Prove by completing the square on the quadratic
Q4
Answer
A fish tank in the shape of a cuboid is to be made from 1600 cm$^2$ of glass. The fish tank will hav a square base of side length $x$ cm, and no lid. No glass is wasted, and the glass can be assumed to be very thin.
  1. Show that the volume $V$ cm$^3$ of the fish tank is given by $V=400x-\dfrac{x^3}{4}$.
  2. Given that $x$ can vary, use differentiation to find the maximum or minimum value of $V$.
  3. Justify that the value of $V$ you found in part b is a maximum.
  1. Surface: $1600 = x^2 + 4xy$, Volume: $x^2y$ and substitute
  2. $400 - \dfrac{3}{4}x^2 = 0$ and $x = \dfrac{40\sqrt{3}}{3}$
  3. $-\dfrac{6}{4}x < 0$ when $x > 0$