1. 1
   Let $a = 27^{\frac{2}{3}}$ and $b = 16^{-\frac{3}{4}}$.

   (a)

   Find the value of $a$.

   [1 mark]

   • A$18$
   • B$81$
   • C$9$
   • D$\frac{1}{9}$

   (b)

   Find the value of $b$.

   [1 mark]

   • A$\frac{1}{8}$
   • B$8$
   • C$-8$
   • D$-12$

   (c)

   Find the exact value of $\dfrac{\sqrt{a}}{b}$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   It is given that $\log_2 x = p$ and $\log_2 y = q$, where $x > 0$ and $y > 0$.

   (a)

   Find an expression for $\log_2\left(8x^3\right)$ in terms of $p$.

   [1 mark]

   • A$8+3p$
   • B$3+3p$
   • C$24p$
   • D$3+p^3$

   (b)

   Find an expression for $\log_2\left(\dfrac{\sqrt{x}}{y}\right)$ in terms of $p$ and $q$.

   [1 mark]

   • A$\sqrt{p}-q$
   • B$\frac{p}{2q}$
   • C$\frac{p-q}{2}$
   • D$\frac{p}{2}-q$

   (c)

   Use the change of base formula to express $\log_8 y$ in terms of $q$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The number of bacteria, $N$, in a laboratory culture $t$ hours after the start of an experiment is modelled by $N = 500\times2^{\frac{t}{3}}$, for $t\ge0$.

   (a)

   Find the time taken for the number of bacteria to reach 4000.

   [3 marks]

   (b)

   Show that the exact time taken for the number of bacteria to reach 3000 is $3 + 3\log_2 3$ hours.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   The functions $f$ and $g$ are defined for $x\in\mathbb{R}$ by $f(x) = 2^{x+1}$ and $g(x) = 3^{x-1}$.

   (a)

   Solve the equation $f(x) = g(x)$, giving your answer in the form $x = \dfrac{\ln p}{\ln q}$, where $p, q\in\mathbb{Q}$.

   [6 marks]

   (b)

   (i) Show that $f(x)\,g(x) = \frac{2}{3}\times6^{x}$.
   (ii) Hence solve $f(x)\,g(x) = 12$, giving your answer in the form $x = 1+\log_6 k$, where $k\in\mathbb{Z}$.

   [6 marks]

   Total for question 4: 12 marks

End of questions