IGCSE · Mathematics 0580 · Topic

Algebra & Graphs: IGCSE Mathematics questions and answers

10 free IGCSE Mathematics Algebra & Graphs questions written in the style of Cambridge Mathematics papers, each with its answer and mark scheme. 4 of them are A* challenge questions, built to catch the mistakes strong students still make.

This topic covers Algebraic manipulation, Algebraic fractions, Indices II, Linear equations, Simultaneous equations, Quadratic equations, Inequalities, Sequences, Proportion, Differentiation, Functions. Work through the questions below, then generate a full Algebra & Graphs paper with a fresh set every time.

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Algebra & graphs sub-topics

Algebraic manipulationAlgebraic fractionsIndices IILinear equationsSimultaneous equationsQuadratic equationsInequalitiesSequencesProportionDifferentiationFunctions

Algebra & graphs practice questions

Question 1Structured · 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
PartAnswer and marking pointsMarks
(a)25, 121, 361 = 5², 11², 19²
25, 121, 361 B1
identified as 52, 112, 192 B1
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)2 M1
= (n2 + 3n + 1)2 A1
clear conclusion A1
5
(c)2651
n = 50 in n2 + 3n + 1 M1
2651 A1
2
Question 2Structured · 10 marksA* challenge

A curve has equation y = x3 − 6x2 + 9x − 3.

(a)

Find dydx.

[2]
(b)

Find the coordinates of the two turning points and determine whether each is a maximum or a minimum. Show your working.

[5]
(c)

Find the equation of the tangent to the curve at the point where x = 0. Give your answer in the form y = mx + c.

[3]
Show answers and mark scheme
PartAnswer and marking pointsMarks
(a)3x2 − 12x + 9
two terms correct B1
all correct B1
2
(b)(1, 1) maximum, (3, −3) minimum
their dydx = 0 M1
x = 1, x = 3 A1
y = 1, y = −3 A1
valid method to determine nature (e.g. sign of gradient either side, or shape of cubic) M1
(1, 1) maximum and (3, −3) minimum A1
5
(c)y = 9x − 3
gradient = 9 B1
point (0, −3) B1
y = 9x − 3 B1
3
Question 3Structured · 3 marksA* challenge

Find all the integer values of n that satisfy
−11 < 6 − 3n ≤ 4.

[3]
Show answers and mark scheme
PartAnswer and marking pointsMarks
1, 2, 3, 4, 5
23 ≤ n or n < 173 (sign reversed correctly) M1
23 ≤ n < 173 A1
1, 2, 3, 4, 5 A1
3
Question 4Structured · 3 marksA* challenge

The nth term of sequence P is 4n − 1.
The nth term of sequence Q is 142 − 7n.
The kth term of P is equal to the kth term of Q.
Find k and the value of this term.

[3]
Show answers and mark scheme
PartAnswer and marking pointsMarks
k = 13, term = 51
4k − 1 = 142 − 7k M1
k = 13 A1
51 A1
3
Question 5Structured · 3 marks

Solve the inequality -5 < 2x + 1 ≤ 9 and list the integer values of x.

[3]
Show answers and mark scheme
PartAnswer and marking pointsMarks
-3 < x ≤ 4; −2, −1, 0, 1, 2, 3, 4
subtract 1 and divide by 2 M1
-3 < x ≤ 4 A1
integers listed B1
3
Question 6Structured · 3 marks

y is inversely proportional to the square root of x. When x = 4, y = 6.

Find y when x = 16.

[3]
Show answers and mark scheme
PartAnswer and marking pointsMarks
3
y = k√x M1
k = 12 A1
3 A1
3
Question 7Structured · 5 marks
(a)

Simplify x5 × x6.

[1]
(b)

Simplify (3y5)2.

[2]
(c)

Find the value of 64−1/3.

[2]
Show answers and mark scheme
PartAnswer and marking pointsMarks
(a)x11
x11 B1
1
(b)9y10
9y10 B2
B1 for one part correct
2
(c)14
∛64 = 4 seen M1
14 A1
2
Question 8Structured · 3 marks
(a)

y is inversely proportional to x. When x = 3, y = 12. Find a formula for y in terms of x.

[2]
(b)

Find y when x = 8.

[1]
Show answers and mark scheme
PartAnswer and marking pointsMarks
(a)y = 36x
k = 36 M1
y = 36x A1
2
(b)4.5
4.5 B1
1
Question 9Structured · 4 marks
(a)

Solve 4x + 9 < −15.

[2]
(b)

List all the integers n that satisfy −1 < 2n + 1 ≤ 9.

[2]
Show answers and mark scheme
PartAnswer and marking pointsMarks
(a)x < −6
4x < −24 M1
x < −6 A1
2
(b)0, 1, 2, 3, 4
-1 < n ≤ 4 seen M1
0, 1, 2, 3, 4 A1
2
Question 10Structured · 3 marks
(a)

Solve 2x + 4 < 8.

[2]
(b)

List the positive integers that satisfy the inequality.

[1]
Show answers and mark scheme
PartAnswer and marking pointsMarks
(a)x < 2
2x < 4 M1
x < 2 A1
2
(b)1
1 B1
1

Other IGCSE Mathematics topics

NumberCoordinate geometryGeometryMensurationTrigonometryTransformations & vectorsProbabilityStatistics

Frequently asked questions

What is in IGCSE Mathematics Algebra & Graphs?

In the 0580 syllabus, Algebra & Graphs covers Algebraic manipulation, Algebraic fractions, Indices II, Linear equations, Simultaneous equations, Quadratic equations, Inequalities, Sequences, Proportion, Differentiation, Functions.

Where can I get Algebra & Graphs questions with mark schemes?

This page has 10 Algebra & Graphs questions with answers. For more, the free generator on mentorriyaz.in builds a new Algebra & Graphs paper with a full mark scheme each time you click.

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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.

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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