Edexcel Pure Paper 1 2024 Question 7

Edexcel Pure Paper 1 2024 Question 7 – Differential Equation Modelling

❓ The Question

🧠 First Thoughts

This is a standard modelling question.

Nothing too unusual, but it mixes a few ideas:

solving a differential equation
using a condition
then actually interpreting the result

The maths itself is manageable — it’s just keeping everything organised.

✏️ Working

Part (a)

You’re given:

$\frac{dH}{dt} = -0.12e^{-0.2t}$

To get $H$ , you integrate.

So:

$H = \int -0.12e^{-0.2t} , dt$

Now, integrating $e^{-0.2t}$ brings in a factor.

$\int e^{-0.2t} dt = \frac{1}{-0.2}e^{-0.2t}$

So:

$H = -0.12 \cdot \frac{1}{-0.2} e^{-0.2t} + C$

That simplifies to:

$H = 0.6e^{-0.2t} + C$

This matches the given form:

$H = Ae^{-0.2t} + B$

Part (b)

Now use the initial condition.

At $t = 0$ , the tank is full, so:

$H = 1.5$

Substitute:

$1.5 = 0.6(1) + C$

So:

$C = 0.9$

So the model becomes:

$H = 0.6e^{-0.2t} + 0.9$

Now solve for when:

$H = 1.2$

Substitute:

$1.2 = 0.6e^{-0.2t} + 0.9$

Rearrange:

$0.3 = 0.6e^{-0.2t}$

$0.5 = e^{-0.2t}$

Take logs:

$\ln(0.5) = -0.2t$

So:

$t = \frac{\ln(0.5)}{-0.2}$

This gives:

$t \approx 3.47 \text{ hours}$

Convert to hours and minutes:

$0.47 \times 60 \approx 28$

Part (c)

As $t \to \infty$ , the exponential term goes to zero.

So:

$H \to 0.9$

That represents the height of the hole from the bottom.

🎯 Where the Marks Are

This is an 8-mark question, so the marks aren’t all at the end — they’re spread through the working.

You pick them up as you go.

The first couple of marks are usually for getting started properly.
If the setup is right, you’re already on the board.

Then most of the marks sit in the middle.

That’s where you’re doing the actual maths — solving something, rearranging, maybe substituting a value back in.
If the method is clear here, you’ll still get credit even if the final number isn’t perfect.

There’s usually a mark towards the end for finishing it properly.

That might be:

a final value
a correct interval

or a short conclusion

⚠️ Where it tends to go wrong

A lot of answers start well… and then just stop.

Or the method is there, but it’s not written clearly enough for marks to be given.

Another common one — getting most of the way through, then losing accuracy with a small slip.

💡 The main thing

On a question like this, it’s not about one big step.

It’s about building it up.

If each line makes sense, the marks follow quite naturally.

⚠️ What Went Wrong

A few things showed up:

forgetting the constant after integration
mistakes with the exponential when rearranging
not converting time properly into minutes

And in part (c), some answers gave a number but didn’t explain what it represented.

💡 One Thing That Helps

Once you’ve got your model, pause and check it makes sense.

At $t = 0$ , does it give 1.5?

If not, something’s already gone wrong.

🚀 If This Felt Tricky

If the integration step felt fine but the rest didn’t, that’s normal — it’s the modelling that usually causes problems.

Working through a few of these with online maths tuition can help make the steps feel more predictable.

If you want a more structured approach overall, an A level maths study course will help link the maths to the interpretation, which is what these questions are really testing.

🔗 Next Steps

← Question 6
→ Question 8

👨‍🏫Author Bio

S. Mahandru is an experienced A Level Maths teacher and founder of Exam.Tips, specialising in exam-focused revision techniques and helping students achieve top grades.

❓ Frequently Asked Questions

📌 Why do we add a constant when integrating?

Because there are infinitely many solutions — the constant fixes the specific one.

📌 What happens as time increases?

The exponential term shrinks towards zero.

📌 Why take logs in part (b)?

To solve an equation involving an exponential.

📌 What’s the main idea here?

Linking calculus with a real situation, not just doing the maths.

Edexcel Pure Paper 1 2024 Question 7