Edexcel Pure Paper 1 2024 Question 10

Edexcel Pure Paper 1 2024 Question 10 – Tangents, Integration and Area

❓ The Question

🧠 Before you start

This is a longer question, but it’s built from standard ideas.

Nothing new — just:

differentiation
straight lines
integration

What matters here is how you link them together.

If the structure is clear, the marks come quite naturally. If it isn’t, it gets messy fast.

✏️ Working

🔹 Part (a)

You are asked to verify that $x = 4$ is where the curve crosses the x-axis.

The curve is:

$y = 8x – x^{5/2}$

Substitute $x = 4$ :

$y = 8(4) – 4^{5/2}$

$= 32 – (\sqrt{4})^5$

$= 32 - (2^5)$

$= 32 - 32 = 0$

So the point lies on the x-axis.

🔹 Part (b)

Differentiate:

$y = 8x – x^{5/2}$

$\frac{dy}{dx} = 8 – \frac{5}{2}x^{3/2}$

Now substitute $x = 4$ :

$\frac{dy}{dx} = 8 – \frac{5}{2}(4^{3/2})$

$= 8 – \frac{5}{2}(8)$

$= 8 - 20 = -12$

So the gradient at A is -12.

Use point-slope form with point (4, 0):

$y - 0 = -12(x - 4)$

$y = -12x + 48$

Rearrange:

$12x + y = 48$

🔹 Part (c)

You need the area between:

curve
line $y = 8x$
line $12x + y = 48$

First find where the two lines meet:

$y = 8x$
$12x + y = 48$

Substitute:

$12x + 8x = 48$

$20x = 48$

$x = 2.4$

So the triangle runs from 0 to 2.4 to 4.

Step 1: Area of triangle

Height at intersection:

$y = 8(2.4) = 19.2$

Triangle area:

$\frac{1}{2} \times 4 \times 19.2 = 38.4$

$= \frac{192}{5}$

Step 2: Area under curve

$\int_0^4 (8x – x^{5/2}) dx$

Integrate:

$= \left[4x^2 – \frac{2}{7}x^{7/2} \right]_0^4$

Substitute:

$= 4(16) – \frac{2}{7}(4^{7/2})$

$= 64 – \frac{2}{7}(128)$

$= 64 – \frac{256}{7}$

Step 3: Final area

$\frac{192}{5} – \left(64 – \frac{256}{7}\right)$

Convert:

$= \frac{192}{5} – 64 + \frac{256}{7}$

Final:

$= \frac{384}{35}$

🎯 Where the Marks Are

Part (a):

B1 for correct substitution

Part (b):

B1 derivative
M1 substitution + tangent
A1 correct equation form

Part (c):

M1 finding intersection
M1 valid method (triangle or split integral)
B1 correct integration
M1 combining areas correctly
A1 final exact answer

The key here is that marks are spread across method.
You don’t need perfection everywhere — but you do need a clear plan.

⚠️ What Went Wrong

This is where things really separated stronger answers.

A lot of students could do the individual steps — but didn’t join them together well.

Part (a) caused unnecessary problems.

It said “verify”, but many treated it as “solve”. That led to longer working and more chance of errors. A quick substitution was all that was needed.

Part (b) was generally fine, but marks were lost on presentation.

Some students:

jumped straight to the answer
didn’t show differentiation clearly
or left the equation in the wrong form

That final rearrangement mattered more than people expected.

Part (c) was the main issue.

Not because integration was hard — but because the structure wasn’t clear.

Common problems:

not finding the intersection first
trying to integrate everything in one go
mixing up which function was on top
using decimals instead of exact values

Once decimals appeared, accuracy was usually lost.

Some also made the question harder than needed.

Using trigonometry or splitting shapes awkwardly just introduced more risk.

The cleaner approach was always:
triangle minus curve.

💡 The takeaway

This question is really about control.

If you:

slow down
set it up clearly
keep everything exact

then it works.

🚀 If This Felt Difficult

If this felt messy, it’s usually not because the maths is hard — it’s because the structure isn’t secure yet.

Working through similar problems in a structured maths revision course helps build that step-by-step thinking.

And if you want to improve how you approach longer questions, regular practice with A level maths tutoring can make a noticeable difference.

🔗 Next Steps

← Question 9
→ Question 11

👨‍🏫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

📌Do I have to find the intersection first?

Yes — without it, the rest of the area is unclear.

📌Why not use decimals?

Because the question asks for an exact answer.

📌What’s the best method for area?

Triangle minus curve is usually the cleanest.

📌Where are most marks lost?

In part (c), due to unclear setup rather than difficult maths.

Edexcel Pure Paper 1 2024 Question 10