Algebraic manipulation: IGCSE Mathematics questions with answers
Free Algebraic manipulation questions for IGCSE Mathematics (0580), part of Algebra & graphs. Each question has the answer and the marking points an examiner would look for.
This question is about writing numbers as the difference of two squares, x2 − y2, where x and y are positive integers.
(a)
Find all the pairs of positive integers (x, y) for which x2 − y2 = 45.
[4]
(b)
Prove that 2026 cannot be written as the difference of two squares of integers.
[4]
(c)
Find how many of the integers from 1 to 100 inclusive cannot be written as the difference of two squares of non-negative integers. Justify your answer fully.
[3]
Show answers and mark scheme
Part
Answer and marking points
Marks
(a)
(23, 22), (9, 6), (7, 2)
(x − y)(x + y) = 45 M1 uses factor pairs 1 × 45, 3 × 15, 5 × 9 M1 two correct pairs A1 all three pairs and no others A1
4
(b)
x − y and x + y have the same parity, so the product is odd or a multiple of 4; 2026 = 2 × odd
x − y and x + y differ by 2y, so they are both odd or both even M1 both odd ⇒ product odd; both even ⇒ product a multiple of 4 A1 2026 is even but 2026 ÷ 2 = 1013 is odd, so not a multiple of 4 A1 so impossible A1
4
(c)
25
every odd number 2k + 1 = (k + 1)2 − k2 and every multiple of 4, 4k = (k + 1)2 − (k − 1)2B1 so exactly the numbers of the form 4k + 2 fail (by (b)) M1 25 A1
3
Question 2Structured · 4 marksA* challenge
Make x the subject of the formula y = 2x + 3x − 1.
[4]
Show answers and mark scheme
Part
Answer and marking points
Marks
x = (y + 3) ÷ (y − 2)
y(x − 1) = 2x + 3 M1 xy − y = 2x + 3 M1 x(y − 2) = y + 3 M1 x = y + 3y − 2A1
4
Question 3Structured · 3 marks
(a)
Expand and simplify (x − 3)(x + 3).
[2]
(b)
Factorise x2 − 25.
[1]
Show answers and mark scheme
Part
Answer and marking points
Marks
(a)
x2 − 9
three terms correct M1 x2 − 9 A1
2
(b)
(x − 5)(x + 5)
(x − 5)(x + 5) B1
1
Question 4Structured · 9 marksA* challenge
(a)
Show that 1 × 2 × 3 × 4 + 1, 2 × 3 × 4 × 5 + 1 and 3 × 4 × 5 × 6 + 1 are all perfect squares.
[2]
(b)
Prove that the product of any four consecutive integers, plus 1, is always a perfect square.
[5]
(c)
Hence find, without a calculator, the value of √(50 × 51 × 52 × 53 + 1).
[2]
Show answers and mark scheme
Part
Answer and marking points
Marks
(a)
25, 121, 361 = 5², 11², 19²
25, 121, 361 B1 identified as 52, 112, 192B1
2
(b)
n(n + 1)(n + 2)(n + 3) + 1 = (n² + 3n + 1)²
pairs the outer and inner factors: n(n + 3) and (n + 1)(n + 2) M1 (n2 + 3n)(n2 + 3n + 2) + 1 A1 let m = n2 + 3n: m(m + 2) + 1 = (m + 1)2M1 = (n2 + 3n + 1)2A1 clear conclusion A1
Riyaz Shaikh has mentored IGCSE, IB and A Level students for 15+ years, one-on-one, online and in Mumbai. The hardest questions on this site are the ones he uses to find a student's weak points.
Original Cambridge-style practice questions written for mentorriyaz.in. These are not Cambridge past paper questions, and this site is not affiliated with or endorsed by Cambridge University Press & Assessment. Updated 2026-11-04. Paper generator · mentorriyaz.in