Click here to do these on a digital whiteboard. Note that these are a repeat of the holiday questions.
Q1
Answer
  1. Solve $x^2 - 8x + 11 = 0$ giving your answers in exact form
  2. Sketch the curve $y = x^2 - 8x + 11$ labelling the coordinates of the points where it crosses the axes
  3. Solve $y - 8\sqrt{y} + 11 = 0$, giving your answers in the form $a \pm b\sqrt{5}$
  1. $4\pm\sqrt{5}$
  2. Correct sketch
  3. $y = 21 \pm 8\sqrt{5}$
Q2
Answer
Solve the inequality $x^2 + 8x + 10 \geq 0$
$x \leq -4 - \sqrt{6}$, $x\geq -4 + \sqrt{6}$
Q3
Answer
Solve the equations
  1. $x^{\frac{1}{3}} = 2$
  2. $10^y = 1$
  3. $\left(z^{-2}\right)^2 = \dfrac{1}{81}$
  1. $x = 8$
  2. $y = 0$
  3. $z = \pm 3$
Q4
Answer
Express $\dfrac{5}{2-\sqrt{3}}$ in the form $a + b\sqrt{3}$ where $a$ and $b$ are integers
$10+5\sqrt{3}$
Q5
Answer
  1. Sketch on the same axes the curves wtih equations $y = a^x$ and $y = 2b^x$ where $a > 1$ and $0 < b < 1$. Label any intersections with the coordinate axes.
  2. The curves intersect. Prove that the $x$ coordinate of the point of intersection is $\dfrac{1}{\log_2 a - \log_2 b}$
  1. Exponential growth for $a > 1$, intersect at $(0,1)$ and exponential decay for $0 < b < 1$, intersect at $(0,2)$
  2. Use of log laws