Q1
Answer
Solve $\tan x + \mathrm{cot}\ x = 8\cos 2x$ in the range $0 \leq x < \pi$.
$\dfrac{\pi}{24}, \dfrac{5\pi}{24}, \dfrac{13\pi}{24}, \dfrac{17\pi}{24}$
Q2
Answer
The polynomials $\mathrm{f}(x)$ and $\mathrm{g}(x)$ are defined in terms of the constants $a$ and $b$: $$f(x) = a(x^3+1)-bx(x+1)$$ $$g(x) = bx^3-5x^2-2a(x-1)$$
  1. Given that $(x-2)$ is a factor of both $f(x)$ and $g(x)$, determine the value of $a$ and $b$.
  2. By first factorising $f(x)$ and $g(x)$, find another common factor.
  1. $a = 2$, $b = 3$
  2. $(x+1)$
Q3
Answer
Solve $5\times5^{\log x} + 5^{2-\log x} = 30$
$x = 1, 10$
Q4
Answer
Find the first three terms, in ascending powers of $x$, in the expansion of $\sqrt{4+x}$.
$2 + \dfrac{1}{4}x - \dfrac{1}{32}x^2$