1. 1
   The expression $\dfrac{5x + 1}{x^{2} + x - 2}$ is to be written in the form $\dfrac{A}{x - 1} + \dfrac{B}{x + 2}$, where $A$ and $B$ are constants.

   (a)

   Which identity must hold for all values of $x$?

   [1 mark]

   • A$5x + 1 \equiv A(x - 1) + B(x + 2)$
   • B$5x + 1 \equiv A(x + 2) + B(x - 1)$
   • C$5x + 1 \equiv A + B$
   • D$5x + 1 \equiv (A + B)(x^{2} + x - 2)$

   (b)

   Find the value of $A$.

   [1 mark]

   • A$6$
   • B$-2$
   • C$2$
   • D$3$

   (c)

   Find the value of $B$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   For positive integers $r$, let $u_r = \dfrac{2}{r(r+2)}$.

   (a)

   Which of the following is equal to $u_r$?

   [1 mark]

   • A$\dfrac{1}{r} - \dfrac{1}{r+2}$
   • B$\dfrac{2}{r} - \dfrac{2}{r+2}$
   • C$\dfrac{1}{r} + \dfrac{1}{r+2}$
   • D$\dfrac{1}{r+2} - \dfrac{1}{r}$

   (b)

   Find the value of $u_1 + u_2 + u_3 + u_4$.

   [1 mark]

   • A$\dfrac{4}{5}$
   • B$\dfrac{3}{2}$
   • C$\dfrac{34}{15}$
   • D$\dfrac{17}{15}$

   (c)

   Show that $\displaystyle\sum_{r=1}^{n} u_r = \frac{3}{2} - \frac{1}{n+1} - \frac{1}{n+2}$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The function $f$ is defined by $f(x) = \dfrac{3x + 5}{(x + 1)(x + 3)}$ for $x > -1$.

   (a)

   Express $f(x)$ in partial fractions.

   [3 marks]

   (b)

   Hence show that $f$ is a decreasing function.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A drug is given to a patient. The concentration of the drug in the patient's blood, $t$ hours later, is modelled by $C(t) = \dfrac{20t}{(t + 1)(t + 4)}$ milligrams per litre, for $t \ge 0$. Do not use a calculator in this question.

   (a)

   (i) Express $C(t)$ in partial fractions.
   (ii) Hence find $C'(t)$, and find the time at which the concentration is greatest.

   [6 marks]

   (b)

   Find the exact value of $\displaystyle\int_{0}^{8} C(t)\,\mathrm{d}t$, giving your answer in the form $p\ln 3$, where $p \in \mathbb{Q}$.

   [6 marks]

   Total for question 4: 12 marks

End of questions