Q1
Answer
Given that $f(x) = \dfrac{4x^2}{(3x-5)^3}$
  1. Find $f'(x)$ in its simplified form.
  2. Hence find $\displaystyle\int - \dfrac{3x^2+10x}{(3x-5)^4}\ \mathrm{d}x$.
  1. $\dfrac{-4(3x^2+10)}{(3x-5)^4}$
  2. $\dfrac{x^2}{(3x-5)^3} + c$
Q2
Answer
Given that $y=x\sqrt{4x+1}$,
  1. Show that $\dfrac{\mathrm{d}y}{\mathrm{d}x} = \dfrac{6x+1}{\sqrt{4x+1}}$.
  2. Solve the equation $\dfrac{\mathrm{d}y}{\mathrm{d}x} - 5y = 0$
  1. Product rule
  2. $x = \dfrac{1}{4}, \dfrac{1}{5}$
Q3
Answer
Use a counter example to disprove the following statements
  1. For all positive real values of $x$, $\sqrt[3]{x} \leq x$
  2. For all positive integer values of $n$, $(n^3-n+7)$ is prime
  1. Example between $0$ and $1$
  2. $n = 7$ gives $7^3$ which is not prime