Click here to do these on a digital whiteboard. Note that these are a repeat of the holiday questions.
Q1
Answer
Solve $8x^3 + \dfrac{1}{x^3} = -9$
$x = -\frac{1}{2}, -1$
Q2
Answer
Solve the equations
  1. $10^a = 0.1$
  2. $(25b^2)^{\frac{1}{2}} = 15$
  3. $c^{-\frac{1}{3}} = \dfrac{1}{2}$
  1. $a = -1$
  2. $b = \pm 3$
  3. $c = 8$
Q3
Answer
$$\mathrm{f}(x) = -4x^3 + 9x^2 + 10x - 3$$
  1. Verify that the curve $y = \mathrm{f}(x)$ passes through the point $(3, 0)$ and hence state a factor of $\mathrm{f}(x)$.
  2. Write $\mathrm{f}(x)$ as a product of this linear factor and a quadratic factor.
  3. Hence fully factorise $\mathrm{f}(x)$.
  1. $(x-3)$ is a factor
  2. $(x-3)(-4x^2-3x+1)$
  3. $(x-3)(1-4x)(x+1)$
Q4
Answer
Given that $\log_a x = p$ and $\log_a y = q$, express $$\log_a\left(\dfrac{a^2x^3}{y}\right)$$ in terms of $p$ and $q$.
$2 + 3p - q$
Q5
Answer
For the curve $y = 27 + 9x - 3x^2 - x^3$,
  1. Find the coordinates of the stationary points
  2. Determine whether each is a minimum or a maximum
  3. State the set of values of $x$ for which the curve is increasing
  1. $(-3, 0)$ and $(1,32)$
  2. min, max
  3. $-3 < x < 1$