1. 1
   Let $f(x) = \arctan(2x)$, $g(x) = 3^{x}$ and $h(x) = \log_{2}(x^{2} + 1)$, for $x \in \mathbb{R}$.

   (a)

   Find $f'(x)$.

   [1 mark]

   • A$\dfrac{1}{1 + 4x^{2}}$
   • B$\dfrac{2}{1 + 2x^{2}}$
   • C$2\sec^{2}(2x)$
   • D$\dfrac{2}{1 + 4x^{2}}$

   (b)

   Find $g'(2)$.

   [1 mark]

   • A$9\ln 3$
   • B$6$
   • C$\dfrac{9}{\ln 3}$
   • D$9$

   (c)

   Find the exact value of $h'(1)$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   Let $f(x) = \sec x + \tan x$, for $-\dfrac{\pi}{2} < x < \dfrac{\pi}{2}$.

   (a)

   Find $f'(x)$.

   [1 mark]

   • A$-\sec x\tan x + \sec^{2}x$
   • B$\sec x\tan x + \sec^{2}x$
   • C$\sec x\tan x - \operatorname{cosec}^{2}x$
   • D$\sec x\tan x + \dfrac{1}{1 + x^{2}}$

   (b)

   Find the exact value of $f'\left(\dfrac{\pi}{4}\right)$.

   [1 mark]

   • A$2 - \sqrt{2}$
   • B$2\sqrt{2}$
   • C$2 + \sqrt{2}$
   • D$\sqrt{2}$

   (c)

   Show that $f'(x) = f(x)\sec x$, and hence explain why $f$ is increasing for $-\dfrac{\pi}{2} < x < \dfrac{\pi}{2}$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   Let $q(x) = \dfrac{x + 7}{(x - 1)(x + 3)}$, for $x > 1$.

   (a)

   Express $q(x)$ in partial fractions.

   [3 marks]

   (b)

   Hence find $\displaystyle\int_{2}^{5} q(x)\,dx$, giving your answer in the form $\ln k$, where $k \in \mathbb{Z}$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   The height of a young plant is $h$ cm, $t$ days after it is first measured. A botanist models its rate of growth by $\dfrac{dh}{dt} = \dfrac{20}{t^{2} - 4t + 8}$, for $t \ge 0$. When it is first measured, the plant is $5$ cm tall.

   (a)

   By completing the square, find an expression for $h$ in terms of $t$.

   [6 marks]

   (b)

   (i) Find the greatest rate of growth of the plant according to the model, and the height of the plant at the time when this occurs.
   (ii) Find the height that the plant approaches in the long term, and comment on what the model predicts about its height.

   [6 marks]

   Total for question 4: 12 marks

End of questions