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📐 IGCSE Mathematics 0580 (Extended)
✓ Final exam expectations
Based on ticked syllabus areas
May/June 2025 readiness
✅ Use checkboxes to track revision – print or save as PDF to keep a physical record.
📘 Topic area — complete syllabus requirement
☑️ ✔ Tick when confident
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1. Number
Arithmetic · bounds · ratio · percentages
Identify natural, integer, prime, square, cube numbers; find HCF/LCM
Distinguish rational & irrational numbers; use reciprocals
Rounding to significant figures / decimal places; estimate calculations
Upper and lower bounds (limits of accuracy); bound calculations for results
Ratio: simplify, divide quantity in given ratio; proportional reasoning (maps, best value, exchange rates, density, pressure)
Rates: average speed, speed/distance/time; solve problems involving rate (e.g. pressure, flow)
Percentages: % of quantity, express as %, % increase/decrease, simple & compound interest, reverse percentages
2. Algebra & Graphs
Surds · inequalities · sequences · functions · graphs
Efficient calculator use (time, money display); interpret display (4.8 = $4.80)
Time calculations (seconds, minutes, hours, days); 12/24-hour clock; timetables
Money: currency conversion; financial applications
Exponential growth & decay (depreciation, population change)
Surds: simplify expressions, rationalise the denominator
Algebraic substitution; use letters as generalised numbers
Inequalities: number line representation, solve linear inequalities; graphical inequalities in two variables; define region
Sequences: term-to-term rule, linear/quadratic/cubic/exponential sequences; nth term
Direct/inverse proportion: algebraic expressions; find unknown quantities
Graphs in practical situations: travel graphs, conversion graphs; distance-time & speed-time graphs; area under speed-time graph = distance
Graphs of functions: tables, draw/interpret y = axⁿ, exponential; solve equations graphically; sketch linear, quadratic, cubic, reciprocal, exponential (turning points, roots, asymptotes)
3. Coordinate Geometry
Straight lines · gradient · length · perpendicular/parallel
Interpret and use Cartesian coordinates in 2D
Draw straight-line graphs from linear equations
Find gradient of a line from two points
Calculate length of a line segment; midpoint coordinates
Obtain equation of a straight line (y = mx + c)
Equation of line parallel to a given line
Equation of line perpendicular to a given line (negative reciprocal gradients)
4. Geometry
Constructions · similarity · angles · circle theorems
Geometrical vocabulary: point, line, plane, parallel, perpendicular, bearing, acute/obtuse/reflex, similar, congruent, scale factor; polygons, circles
Constructions: measure/draw lines & angles; construct triangle using ruler & compasses; nets
Scale drawings and three-figure bearings
Similarity: length scale factor, area & volume ratios for similar shapes/solids
Symmetry: line symmetry, rotational symmetry; symmetry in prisms, pyramids, cylinders, cones
Angle properties: angles at a point =360°, straight line=180°, vertically opposite, triangle sum, quadrilateral sum; interior/exterior angles of regular polygons
Circle theorems I: angle in semicircle=90°, tangent⊥radius, centre twice circumference, angles in same segment, cyclic quadrilateral (supplementary), alternate segment theorem
Circle theorems II: equal chords equidistant from centre; perpendicular bisector of chord passes through centre; tangents from external point equal length
5. Mensuration
Area · volume · surface area · compound shapes
Use metric units (mass, length, area, volume, capacity); convert between units
Perimeter and area: rectangle, triangle, parallelogram, trapezium
Circles, arcs, sectors: circumference, area, arc length, sector area (fraction of circle)
Surface area and volume: cuboid, prism, cylinder, sphere, pyramid, cone
Compound shapes & parts of shapes: perimeter, area, surface area, volume problems
6. Trigonometry
Pythagoras · SOH CAH TOA · sine/cosine rule · 3D
Pythagoras' theorem in 2D and simple 3D contexts
Sine, cosine, tangent ratios for acute angles; solve right-angled triangle problems
Angles of elevation and depression; shortest distance from point to line
Exact trigonometric values: sin, cos, tan for 0°,30°,45°,60°,90° (tan up to 60°)
Sketch graphs y=sin x, y=cos x, y=tan x (0° to 360°); solve trig equations
Sine rule & cosine rule for any triangle (non-right-angled)
Area of triangle = ½ ab sin C
3D trigonometry: angle between line and plane; Pythagoras in 3D
7. Probability
Single event · combined events · tree diagrams · conditional
Probability scale 0–1; use notation; P(not A) = 1 – P(A)
Relative frequency as estimate of probability; expected frequencies
Combined events: sample space diagrams, tree diagrams
Conditional probability using tree diagrams, tables
8. Statistics
Data classification · interpretation · comparison
Classify and tabulate statistical data (discrete, continuous)
Read, interpret and draw inferences from tables and statistical diagrams
Compare data sets using graphs, tables, measures (mean, median, mode, range)
Recognise restrictions/cautions when drawing conclusions from data
Pick Your Question
Cambridge IGCSE Mathematics 0580/21 · May/June 2025 · Choose a card, solve it, and be ready to explain your method!
Q1
Algebra
[2 marks]
Simplify.
7c − 5d + c + 3d
Think about…Collect like terms — group the c terms together and the d terms together.
💬 Can you explain what "like terms" means to the class?
Q2
Geometry
[4 marks]
Two parallel lines are cut by two straight lines. The angles marked are 158° and 76°.
Find the values of w, x and y.
Think about…Alternate angles, co-interior angles, and angles on a straight line. Which angle rules apply where?
💬 Which angle rule did you use first, and why?
Q3
Number
[3 marks]
Sally invests $1500 at 3% per year simple interest.
Work out the total value of her investment at the end of 6 years.
Think about…Simple interest formula: I = PRT ÷ 100. Then add the interest to the principal.
💬 How would the answer change if it were compound interest?
Q4
Number – Fractions
[3 marks]
Work out.
5/6 − 2/3 × 3/8
Think about…Order of operations — multiplication before subtraction. Simplify the product first, then find a common denominator.
💬 Why do we multiply before we subtract?
Q5
Geometry
[2 marks]
The interior angle of a regular polygon is 150°.
Find the number of sides of this polygon.
Think about…Exterior angle = 180° − 150°. The exterior angles of any polygon sum to 360°.
💬 What type of polygon is this? Can you name it?
Q6a
Graphs
[2 marks]
The line x + y = 7 is drawn on a grid.
Draw the line y = 2x + 1 on the same grid.
Think about…Find at least two points on y = 2x + 1 by substituting values of x. Plot and join them.
💬 What is the gradient and y-intercept of your line?
Q6b
Simultaneous Equations
[1 mark]
Use the graph to solve the simultaneous equations:
x + y = 7
y = 2x + 1
Think about…The solution is the point where the two lines intersect. Read the coordinates carefully.
💬 How could you check your answer algebraically?
Q7
Number – Decimals
[3 marks]
Write the recurring decimal 0.26 as a fraction in its simplest form.
Think about…Let x = 0.2666… Multiply by 10 and by 100, then subtract to eliminate the recurring part.
💬 How do you know when to multiply by 10, 100, or 1000?
Q9
Statistics
[3 marks]
Test marks of 4, 5 and 8 have frequencies 2, 4 and n.
The mean mark is 6.
Work out the value of n.
Think about…Mean = (sum of all values) ÷ (total frequency). Write an equation and solve for n.
💬 How does changing n affect the mean?
Q10a
Circle Theorems
[1 mark]
A, B and C are on a circle, centre O. Angle ACO = 35°.
Find angle AOC.
Think about…OA = OC (radii), so triangle AOC is isosceles. Use this to find angle AOC.
💬 Why are OA and OC equal?
Q10b
Circle Theorems
[1 mark]
A, B and C are on a circle, centre O. Angle ACO = 35° and angle BCO = 40°.
Find angle ABC.
Think about…The angle at the centre is twice the angle at the circumference. What is angle ACB first?
💬 State the circle theorem you used.
Q10c
Circle Theorems
[1 mark]
DE is a tangent to the circle at A. Angle ABC = 55°.
Find angle DAC.
Think about…The tangent-chord angle equals the inscribed angle in the alternate segment (Alternate Segment Theorem).
💬 In your own words, what does the Alternate Segment Theorem say?
Q10d
Circle Theorems
[1 mark]
Angle OAB = ?
Using the circle (centre O, with angle ABC = 55°), find angle OAB.
Think about…OA = OB (radii), so triangle OAB is isosceles. Use angle ABC and the isosceles property.
💬 How many circle theorem properties did Q10 use in total?
Q13a
Surface Area
[5 marks]
Solid A is a hemisphere + cone, both with radius 6 cm; cone slant edge = 10 cm.
Solid B is a cylinder, radius 4 cm, height h cm.
Their total surface areas are equal. Find h.
Think about…Hemisphere SA = 2πr². Cone curved SA = πrl. Cylinder total SA = 2πr² + 2πrh. Set them equal.
💬 Why don't we include a base circle for the hemisphere in Solid A?
Q13b
Pythagoras
[3 marks]
For Solid A (cone with radius 6 cm and slant height 10 cm):
Work out the height of Solid A (hemisphere + cone).
Think about…Use Pythagoras to find the perpendicular height of the cone: h² + r² = l². Then add the hemisphere's height.
💬 What is the height of a hemisphere with radius r?
Q14a/b
Functions
[3 marks]
Given f(x) = 3x − 4:
(a) Find f(−2)
(b) Find f⁻¹(x)
Think about…(a) Substitute x = −2 directly. (b) Write y = 3x − 4, then rearrange to make x the subject, then swap x and y.
💬 What does the inverse function do geometrically?
Q14c/d
Functions
[5 marks]
Given f(x) = 3x − 4 and g(x) = 4x + 1:
(c) Find fg(x) in the form ax + b
(d) Simplify 2/f(x) − 5/g(x) as a single fraction
Think about…(c) Substitute g(x) into f. (d) Find a common denominator and combine.
💬 Does fg(x) equal gf(x)? Show why or why not.
Q15a
Surds
[2 marks]
Expand and simplify.
(2 − √5)(1 − 3√5)
Think about…Use FOIL (or the grid method). Remember √5 × √5 = 5.
💬 What is the value of √5 × √5? Why?
Q15b
Surds
[2 marks]
Rationalise the denominator. Give your answer in simplest form.
6 / √10
Think about…Multiply numerator and denominator by √10. Then simplify the fraction fully.
💬 Why do we "rationalise"? What does rational mean?
Q16
Algebra – Expand
[3 marks]
Expand and simplify.
(x + 4)(x − 3)(3x + 2)
Think about…Expand any two brackets first, then multiply the result by the third. Collect like terms carefully.
💬 Does the order you expand the brackets matter?
Q17a
Probability
[2 marks]
A bag has 6 red, 3 green and 1 blue marble. Two are picked with replacement.
Find the probability that both marbles are green.
Think about…With replacement means the probabilities don't change. Multiply P(green) × P(green).
💬 How would the answer differ without replacement?
Q17b
Probability – Tree
[5 marks]
A bag has 4 red and 2 yellow counters. Two are picked without replacement.
(i) Complete the tree diagram.
(ii) Find P(exactly one yellow counter).
Think about…After picking the first counter, the total drops to 5. Update the probabilities for the second pick accordingly.
💬 Why do the branch probabilities at each stage sum to 1?
Q18
Algebra – Quadratics
[10 marks]
Anya runs 12 km at x km/h and walks 10 km at (x−4) km/h. Walking takes 1 hour more than running.
Show this gives x² − 2x − 48 = 0, solve it, and find Anya's running time.
Think about…Time = distance ÷ speed. Set up the equation, multiply through to clear fractions, then factorise.
💬 Why do we reject one of the two solutions for x?
Q19
Number – Indices
[2 marks]
Find the value of 27−2/3.
Think about…The denominator of the fraction index is the root, the numerator is the power. Negative index means reciprocal.
💬 Can you write a general rule for am/n?
Q20
Trigonometry
[4 marks]
A right-angled triangle has one side = 6 cm and angle = 30°.
The side adjacent to the 30° angle is x cm.
Find the exact value of x.
Think about…Use exact trig values: tan 30° = 1/√3. Write x = 6/tan30° and rationalise.
💬 What are the exact values of sin, cos and tan for 30°, 45° and 60°?
Q22
Calculus
[6 marks]
A curve has equation y = xⁿ + qx² + 9x and dy/dx = 3x² − 12x + 9.
(a) Find n and q.
(b) Find the coordinates of both turning points.
Think about…Match the derivative term-by-term to find n and q. For turning points, set dy/dx = 0 and solve.
💬 How can you tell which turning point is a maximum and which is a minimum?
Q23
Algebra – Fractions
[3 marks]
Simplify fully.
(2x² + 10x) / (x² − 25)
Think about…Factorise numerator and denominator separately. Look for common factors you can cancel.
💬 What factorisation technique applies to the denominator?
Cambridge IGCSE Mathematics 0580/21 M/J 2025 · Cards prepared for classroom use