Edexcel Pure Paper 2 2024 Question 13 Solution

Edexcel Pure Paper 2 2024 Question 13 – Exponential Model

❓ The Question

🧠 Before you start

This one is quite typical once you recognise what’s going on.

You’re given a straight line, but it’s not for P — it’s for log P. That’s the key thing. So straight away, that points to an exponential model in the background.

After that, it’s mostly just translating between the two. Nothing especially new, just making sure each step lines up.

✏️ Working

Part (a)

The graph is of $\log_{10} P$ against $t$ , and it’s a straight line.

So it must be something like:

$\log_{10} P = mt + c$

From the information given:

gradient is $0.0054$
intercept is $0.81$

So we can write:

$\log_{10} P = 0.0054t + 0.81$

Now, we also know the model is:

$P = ab^t$

If you take logs of that:

$\log_{10} P = \log_{10} a + t\log_{10} b$

So you just match terms. Nothing complicated here.

$\log_{10} a = 0.81$
$\log_{10} b = 0.0054$

Now convert them.

For $a$ :

$a = 10^{0.81}$

Which gives:

$a \approx 6.457$

For $b$ :

$b = 10^{0.0054}$

That comes out as:

$b \approx 1.013$

So those are the two constants.

Part (b)

This part looks simple, but it’s easy to be too vague.

For $a$ :

When $t = 0$ , the model gives:

$P = a$

So that’s just the starting value.

In context, that’s the population in 2004.

For $b$ :

This one is slightly different. It’s not an amount, it’s a factor.

Each year, the population gets multiplied by $b$ .

So it represents the yearly growth factor — how the population changes from one year to the next.

Part (c)

Now we use the model.

The year is 2030, so:

$t = 2030 - 2004 = 26$

Substitute into:

$P = 6.457(1.013)^{26}$

Now just evaluate it.

(1.013)^{26} \approx 1.40

Multiply:

$P \approx 6.457 \times 1.40 \approx 9.0$

So the population is about 9 billion.

You don’t need it overly precise here — the mark scheme allows a rounded value.

Part (d)

This is where you step back a bit.

The model is based on data from 2004 to 2007. That’s only a few years.

But we’re using it to predict 2030, which is quite a long way ahead.

So it’s not really safe to assume the same pattern continues that far.

So a valid point is:

The model may not be reliable because it is being used outside the range of the original data.

🎯 Final answers

(a)
$a \approx 6.457, \quad b \approx 1.013$

(b)(i)
Population in 2004

(b)(ii)
Yearly growth factor

(d)
Not reliable for long-term prediction

⚠️ What tends to go wrong

This question is usually fine overall, but a few small things show up.

Some people use the wrong base for logs, which causes problems straight away.

Others round too early, especially when finding b.

And in part (b), answers can be too general — saying “growth” without linking it clearly to the model.

💡 One small thing

If the graph is straight in log form, the original relationship is exponential. That link comes up a lot.

🚀 If this felt okay

That’s expected.

This is more about understanding what the model represents than doing anything technically difficult.

Working through more of these helps build confidence, especially if you’re trying to learn with a maths tutor who can point out those links clearly.

And if interpretation questions feel less certain, getting extra help with A level maths can make a noticeable difference.

🔗 Next Steps

← Question 12 Solution
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👨‍🏫Author Bio

S Mahandru focuses on helping students understand how models behave in exam questions, so they can move beyond just calculation and explain their answers clearly.

❓ Frequently Asked Questions

📌Why use log base 10 here?

Because the graph is plotted using base 10.

📌What does a represent?

The starting value of the population.

📌What does b represent?

The yearly growth factor.

📌Is the answer exact?

No — it’s an estimate based on the model.