Click here to do these on a digital whiteboard. Note that these are a repeat of the holiday questions.
Q1
Answer
Given that $(x+1)$ and $(x-3)$ are both factors of $$ax^3 - 3x^2 - 8x + d$$ fully factorise it.
$(x+1)(x-3)(2x+1)$
Q2
Answer
Solve
  1. $3^m = 81$
  2. $(36p^4)^{\frac{1}{2}} = 24$
  3. $5^n \times 5^{n+4} = 25$
  1. $4$
  2. $2$ or $-2$
  3. $-1$
Q3
Answer
The temperature, $T$, of a cup of tea is modelled by $$T = 20 + b\mathrm{e}^{-kt}$$ where $t$ is the time in minutes after it was brewed. Initially, the temperature is $100^{\circ}$C, and after $5$ minutes the temperature drops to $60^{\circ}$C. Find the time since it was brewed when the tea reaches $50^{\circ}$C.
$7.075$ minutes
Q4
Answer
Given $\mathrm{f}(x) = \dfrac{4}{x} - 3x + 2$, find
  1. $\mathrm{f}'(x)$
  2. $\mathrm{f}''(0.5)$
  1. $-4x^{-2} - 3$
  2. $64$
Q5
Answer
Solve $(3x-2)^4 - 5(3x-2)^2 + 4 = 0$
$0, \dfrac{1}{3}, 1, \dfrac{4}{3}$