1. 1
   Let $x$ be a non-zero real number, and let $P = (2x^{3})^{-2}$ and $Q = \dfrac{6x^{-2}}{3x^{-5}}$.

   (a)

   Which of the following is equal to $P$?

   [1 mark]

   • A$\dfrac{1}{2x^{6}}$
   • B$\dfrac{1}{4x^{6}}$
   • C$\dfrac{1}{4x^{5}}$
   • D$-4x^{6}$

   (b)

   Which of the following is equal to $Q$?

   [1 mark]

   • A$2x^{10}$
   • B$2x^{-7}$
   • C$3x^{3}$
   • D$2x^{3}$

   (c)

   Hence write $PQ$ in the form $ax^{k}$, where $a \in \mathbb{Q}$ and $k \in \mathbb{Z}$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The pH of a solution is given by $\text{pH} = -\log_{10} c$, where $c$ is the concentration of hydrogen ions in mol dm$^{-3}$.

   (a)

   Find the pH of a solution with $c = 10^{-4}$.

   [1 mark]

   • A4
   • B-4
   • C0.0001
   • D$10^{4}$

   (b)

   A solution has pH 2.5. Which of the following gives $c$?

   [1 mark]

   • A$\log_{10}2.5$
   • B$10^{2.5}$
   • C$10^{-2.5}$
   • D$-\log_{10}2.5$

   (c)

   A sample of rainwater has $c = 3.2\times10^{-5}$. Use technology to find its pH, correct to 3 significant figures.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The number of bacteria in culture X, $t$ hours after it is set up, is modelled by $N = 500e^{0.3t}$. The number of bacteria in culture Y is modelled by $M = 1500e^{0.2t}$. A calculator may be used in this question.

   (a)

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

   [3 marks]

   (b)

   Find the exact time at which the two cultures contain the same number of bacteria, and give this time correct to 3 significant figures.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   The loudness $L$ decibels (dB) of a sound of intensity $I$ W m$^{-2}$ is given by $L = 10\log_{10}\left(\dfrac{I}{I_0}\right)$, where $I_0 = 10^{-12}$ W m$^{-2}$.

   (a)

   (i) Find the loudness of a sound of intensity $10^{-5}$ W m$^{-2}$.
   (ii) A rock concert has a loudness of 110 dB. Find the intensity of the sound at the concert.
   (iii) Hence show that the sound at the concert is $10^{4}$ times as intense as the sound in part (i).

   [6 marks]

   (b)

   (i) A speaker produces sound of intensity $3.5\times10^{-4}$ W m$^{-2}$. Use technology to find the loudness of this sound, correct to 3 significant figures.
   (ii) When $n$ identical speakers play together, the total intensity is $n$ times the intensity of one speaker. Safety guidance states that the loudness must not exceed 100 dB. Find the greatest number of these speakers that can play together.

   [6 marks]

   Total for question 4: 12 marks

End of questions