| Exam Board | AQA |
|---|---|
| Module | Paper 2 (Paper 2) |
| Year | 2024 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Exponential Equations & Modelling |
| Type | y vs ln(x) linear graph |
| Difficulty | Moderate -0.3 This is a straightforward logarithmic modeling question requiring substitution into a formula, simultaneous equations, and basic logarithm manipulation. The algebraic steps are clearly signposted (including 'show that'), and the final part tests understanding of domain restrictions. Slightly easier than average due to the structured guidance and routine techniques, though the logarithm manipulation and model validity check elevate it slightly above pure recall. |
| Spec | 1.06f Laws of logarithms: addition, subtraction, power rules1.06h Logarithmic graphs: reduce y=ax^n and y=kb^x to linear form |
| Age in months (\(x\)) | 3 | 24 |
| Median mass (\(y\)) | 6.4 | 12 |
| Answer | Marks |
|---|---|
| 8(a)(i) | Obtains 12=a+blog 24 |
| Answer | Marks | Guidance |
|---|---|---|
| ISW | 3.4 | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| Subtotal | 1 | |
| Q | Marking instructions | AO |
| Answer | Marks | Guidance |
|---|---|---|
| 8(a)(ii) | Eliminates a to obtain an | |
| equation in b | 3.1a | M1 |
| Answer | Marks | Guidance |
|---|---|---|
| where 3h = their 24 | 1.1a | M1 |
| Answer | Marks | Guidance |
|---|---|---|
| AG | 2.1 | R1 |
| Subtotal | 3 | |
| Q | Marking instructions | AO |
| Answer | Marks | Guidance |
|---|---|---|
| 8(a)(iii) | Obtains AWRT 3.44 | 1.1b |
| Subtotal | 1 | |
| Q | Marking instructions | AO |
| Answer | Marks |
|---|---|
| 8(b) | Substitutes a value for |
| Answer | Marks | Guidance |
|---|---|---|
| 0.28 | 3.4 | M1 |
| Answer | Marks | Guidance |
|---|---|---|
| or use of weight throughout | 3.5a | R1 |
| Subtotal | 2 | |
| Question 8 Total | 7 | |
| Q | Marking instructions | AO |
Question 8:
--- 8(a)(i) ---
8(a)(i) | Obtains 12=a+blog 24
10
ISW | 3.4 | B1 | 12=a+blog 24
10
Subtotal | 1
Q | Marking instructions | AO | Marks | Typical solution
--- 8(a)(ii) ---
8(a)(ii) | Eliminates a to obtain an
equation in b | 3.1a | M1 | 12=a+blog 24
10
−(6.4=a+blog 3)
10
5.6=blog 24−blog 3
10 10
24
=blog
10
3
=blog 8
10
5.6
b=
log 8
10
Obtains blog h from
10
blog their24−blog 3 or
10 10
their24
blog
10
3
where 3h = their 24 | 1.1a | M1
Completes a reasoned
5.6
argument to show b=
log 8
10
24
Must include log OE or
10
3
log 24=log 8×3
10 10
AG | 2.1 | R1
Subtotal | 3
Q | Marking instructions | AO | Marks | Typical solution
--- 8(a)(iii) ---
8(a)(iii) | Obtains AWRT 3.44 | 1.1b | B1 | a = 3.44
Subtotal | 1
Q | Marking instructions | AO | Marks | Typical solution
--- 8(b) ---
8(b) | Substitutes a value for
0< x≤0.25into the model with
their a and b = AWRT 6.2
PI by correct negative y-value
Or
Substitutes x = 0 into the model
with their a and b = AWRT 6.2
and states that the value for y is
undefined
Or
Substitutes y = 0 into the correct
model with and gets x = AWRT
0.28 | 3.4 | M1 | When x =0.25
y =3.44+6.2log 0.25
10
=−0.29
The model predicts a negative median
mass for a monkey that is one week
old, therefore it is unsuitable for use
with monkeys 1 week old or less.
Completes reasoned argument
to find a correct median mass
for their value of x and
concludes that the model cannot
be used to predict the median
mass of monkeys less than one
week old.
Condone that the model cannot
be used to predict the median
mass of monkeys for their value
of x where 0< x≤0.25
Condone omittance of median
or use of weight throughout | 3.5a | R1
Subtotal | 2
Question 8 Total | 7
Q | Marking instructions | AO | Marks | Typical solutionA zookeeper models the median mass of infant monkeys born at their zoo, up to the age of 2 years, by the formula
$$y = a + b \log_{10} x$$
where $y$ is the median mass in kilograms, $x$ is age in months and $a$ and $b$ are constants.
The zookeeper uses the data shown below to determine the values of $a$ and $b$.
\begin{tabular}{|l|c|c|}
\hline
Age in months ($x$) & 3 & 24 \\
\hline
Median mass ($y$) & 6.4 & 12 \\
\hline
\end{tabular}
\begin{enumerate}[label=(\alph*)]
\item The zookeeper uses the data for monkeys aged 3 months to write the correct equation
$$6.4 = a + b \log_{10} 3$$
\begin{enumerate}[label=(\roman*)]
\item Use the data for monkeys aged 24 months to write a second equation.
[1 mark]
\item Show that
$$b = \frac{5.6}{\log_{10} 8}$$
[3 marks]
\item Find the value of $a$.
Give your answer to two decimal places.
[1 mark]
\end{enumerate}
\item Use a suitable value for $x$ to determine whether the model can be used to predict the median mass of monkeys less than one week old.
[2 marks]
\end{enumerate}
\hfill \mbox{\textit{AQA Paper 2 2024 Q8 [7]}}