Click here to do these on a digital whiteboard. Note that these are a repeat of the holiday questions.
Q1
Answer
Solve $5 - 8x - x^2 \leq 0$.
$x \leq -4 - \sqrt{21}$, $x \geq -4 + \sqrt{21}$
Q2
Answer
Evaluate $2^{-1} \times 32^{\frac{4}{5}}$
$8$
Q3
Answer
  1. Simplify $(3x+1)^2 - 2(2x-3)^2$.
  2. Find the coefficient of $x^3$ in the expansion of $$(2x^3 - 3x^2 + 4x - 3)(x^2 - 2x + 1)$$
  1. $x^2 + 30x - 17$
  2. $12$
Q4
Answer
Express the following in terms of $\log_{10} x$ and $\log_{10} y$
  1. Express $\log_{10}\dfrac{x}{y}$ and $\log_{10} (10x^2y)$ in terms of $\log_{10} x$ and $\log_{10} y$.
  2. Given that $$2\log_{10}\dfrac{x}{y} = 1 + \log_{10} (10x^2y)$$ find $y$ correct to 3 decimal places.
  1. $\log_{10} x - \log_{10} y$ and $1 + 2\log_{10} x + \log_{10} y$
  2. $0.215$
Q5
Answer
Use proof by contradition to prove that the sum of any positive rational number with its reciprocal is always greater than $2$.
Assume $\dfrac{a}{b} + \dfrac{b}{a} \leq 2 \Rightarrow (a-b)^2 \leq 0$
Q6
Answer
Simplify $(3+2x)^3 - (3-2x)^3$
$108x + 16x^3$