Click here to do these on a digital whiteboard. Note that these are a repeat of the holiday questions.
Q1
Answer
Solve $x^{\frac{2}{3}} + 3x^{\frac{1}{3}} - 10 = 0$
$x = 8$, $x = -125$
Q2
Answer
  1. Write $2x^2 - 24x + 80$ in the form $a(x-b)^2 + c$
  2. State the equation of the line of symmetry of the curve $y = 2x^2 - 24x + 80$
  3. State the equation of the tangent to the curve $y = 2x^2 - 24x + 80$ at its minimum point
  1. $2(x-6)^2 + 8$
  2. $x = 6$
  3. $y = 8$
Q3
Answer
Solve the simultaneous equations $$y = x^2 + x + 1$$ $$x + 2y = 4$$
$\left(\frac{1}{2},\frac{7}{4}\right)$, $(-2, 3)$
Q4
Answer
For the shape below, the perimeter must be at least $39$ and the area must be less than $99$. Determine the set of possible values for $x$.
$\dfrac{5}{2} \leq x < 3$
Q5
Answer
Prove, by contradiction, that there is no largest value of $x$ such that $3 < x < 4$
Assume there is a largest value $k$, but $\dfrac{k+4}{2}$ is always larger and in the interval.
Q6
Answer
Given $\mathrm{f}(x) = 8x^3 + \dfrac{1}{x^3}$, find $\mathrm{f}''(x)$
$48x + 12x^{-5}$