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Edexcel Pure Paper 1 2024 Question 15
Edexcel Pure Paper 1 2024 Question 15 – Algebra and Proof
❓ The Question
🧠 Before you start
This question feels quite different from the rest of the paper.
Part (i) is familiar. You’ve probably done that many times.
Part (ii) is where people hesitate a bit — mainly because it’s not obvious how to start.
The key thing to remember is: you’re not trying to find values. You’re trying to show none exist.
✏️ Working
Part (i)
Start with:
k^2 – 4k + 5
k^2 – 4k + 5 = (k – 2)^2 + 1
Now pause for a second.
(k – 2)^2 is always zero or positive.
So adding 1 means it can’t ever reach zero.
So:
(k – 2)^2 + 1 > 0
That’s really all they want.
Part (ii)
Now the proof.
We assume the opposite of what we’re trying to show.
So assume there are positive integers x and y such that:
(3x + 2y)(2x – 5y) = 28
At this point, don’t overthink it.
28 is small, so just look at its factor pairs.
Positive ones:
1 \times 28, \quad 2 \times 14, \quad 4 \times 7
We match these with:
3x + 2y \quad \text{and} \quad 2x – 5y
Try a case.
Case 1
3x + 2y = 14
2x – 5y = 2
Multiply first by 2:
6x + 4y = 28
Multiply second by 3:
6x – 15y = 6
Subtract:
19y = 22
y = \frac{22}{19}
Not an integer. So this case doesn’t work.
Case 2
3x + 2y = 7
2x – 5y = 4
Same idea.
6x + 4y = 14
6x – 15y = 12
Subtract:
19y = 2
y = \frac{2}{19}
Again, not an integer.
Case 3
3x + 2y = 28
2x – 5y = 1
6x + 4y = 56
6x – 15y = 3
Subtract:
19y = 53
y = \frac{53}{19}
Still not an integer.
At this point, you’ve tested all the realistic positive factor pairs.
Nothing works.
So the assumption that integer solutions exist leads nowhere.
Final line
Hence, there are no positive integers x and y satisfying:
(3x + 2y)(2x – 5y) = 28
🎯 Where the Marks Are
Part (i) is quick:
- complete the square
- state it’s always positive
Part (ii) is more about process.
- try valid factor pairs
- solve correctly
- reject non-integers
- finish with a proper conclusion
Miss that last sentence and it’s not really a proof.
⚠️ What Went Wrong
A lot of issues came from misunderstanding what was being asked.
Some students tried to expand everything and solve it algebraically. That didn’t really lead anywhere.
Others checked just one case and stopped.
That’s not enough — you need to rule out all possibilities.
There were also answers where the working was fine, but no conclusion was written.
That costs marks, even if everything else is correct.
In part (i), a common slip was completing the square but not actually explaining why the result is always positive.
Just writing the expression isn’t quite enough.
💡 The takeaway
This question is more about logic than algebra.
You’re not solving — you’re ruling things out.
Once you see that, the structure becomes clearer.
🚀 If This Felt Difficult
If this one felt a bit unfamiliar, that’s normal.
Proof questions don’t come up as often, so it’s easy to feel unsure about how much to write or where to start.
Going through a few more examples like this can really help — especially seeing how full solutions are structured step by step. That kind of support is exactly what you get with help with maths when you focus on exam-style questions.
Over time, getting comfortable with this type of reasoning is one of the things that really helps you improve your A level maths grade, particularly on the longer questions at the end of the paper.
🔗 Next Steps
👨🏫Author Bio
Written by S Mahandru a maths tutor who focuses on helping students make sense of exam questions without overcomplicating them.
The aim is always to keep things clear, step-by-step, and realistic to how you’d actually work in an exam.
❓ Frequently Asked Questions
📌 Why complete the square in part (i)?
It shows the expression can’t be zero or negative.
📌 Do I need every factor pair?
Yes — otherwise you haven’t ruled everything out.
📌 What if I only test one case?
You won’t get full marks.
📌 Why is the final sentence important?
Because it completes the contradiction.