Click here to do these on a digital whiteboard. Note that these are a repeat of the holiday questions.
Q1
Answer
Solve $$\log_3 (4x+7) - \log_3 x = 2$$
$1.4$
Q2
Answer
Prove that the difference of any rational number with an irrational number is irrational.
Use proof by contradiction $\dfrac{a}{b} - n = \dfrac{c}{d}$ and $n$ is rational.
Q3
Answer
The points $A, B, C$ have coordinates $(5,1), (p, 7), (8, 2)$ respectively.
  1. Given the distance $AB$ is twice the distance $AC$, calculate the possible values of $p$.
  2. Given also the line passing through $AB$ has equation $y = 3x - 14$, find the coordinates of the midpoint of $AB$.
  1. $p = 3, 7$
  2. $(6, 4)$
Q4
Answer
  1. Find the first three terms, in ascending powers of $x$, of the expansion of $(1 - 2x)^{12}$
  2. Hence find the coefficient of $x^2$ in the expansion of $(1 + 3x)(1 - 2x)^{12}$
  1. $1 - 24x + 264x^2$
  2. $192$