Q1
Answer
  1. Expand $(1+3x)^8$ in ascending powers of $x$ up to and including the term in $x^3$.
  2. Use this expression to find an approximation for $1.03^8$ to 5 significant figures.
  1. $1 + 24x + 252x^2 + 1512x^3$
  2. $1.2667$
Q2
Answer
Show that $x+2$ is a factor of $$2x^3-x^2-13x-6$$ and hence fully factorise it.
Factor theorem with $x = -2$, $(x+2)(x-3)(2x+1)$
Q3
Answer
The population $P$ of bacteria in an experiment can be modelled by the formula $P=100e^{0.4t}$ where $t$ is the time in hours after the experiment began.
  1. Use the model to estimate the population of bacteria 7 hours after the experiment began.
  2. Interpret the meaning of the constant $100$ in the model.
  3. How many whole hours after the experiment began does the population of bacteria first exceed 1 million according to the model?
  1. $1644$
  2. Amount of bacteria at the start
  3. $24$
Q4
Answer
Solve algebraically, showing each step of your working, the equation $$(8^{x-1})^2-18(8^{x-1})+32=0$$
Use substitution $y = 8^{x-1}$ to get $x = \dfrac{7}{3}$ or $x = \dfrac{4}{3}$