Number
📌 0580 Topic 1 — Number C E · Types of number · Fractions, decimals, percentages · Indices & standard form · Surds E · Ratio & proportion · Financial maths · Limits of accuracy E
C Core topics are examinable on Papers 1 & 3 (Core tier).
E Extended topics appear on Papers 2 & 4 only.ClipSAT Chapter 0580 Topics covered Tier 1 Number Types of number, fractions, %, ratio & proportion, standard form, surds, indices, financial maths, limits of accuracy C E 2 Algebra & Graphs Expressions, equations, sequences, inequalities, simultaneous equations, graph types C E 3 Coordinate Geometry Gradient, midpoint, distance, equation of a line, perpendicular lines C E 4 Mensuration Perimeter, area, arc, sector, surface area, volume of 3D shapes C E 5 Geometry Angles, polygons, circle theorems, similarity & congruence, construction & loci C E 6 Trigonometry SOH-CAH-TOA, sine/cosine rules, area of triangle, bearings, 3D trig C E 7 Transformations & Vectors Translation, reflection, rotation, enlargement, column vectors, vector paths, magnitude C E 8 Probability & Statistics Probability, tree diagrams, conditional probability, averages, spread, frequency tables, histograms, cumulative frequency, box plots C E 9 Functions Function notation, domain, composite, inverse E 10 Sets & Venn Diagrams Set notation, union, intersection, complement, two- and three-set Venn diagrams C E 11 Matrices E Matrix operations, determinant, inverse, solving simultaneous equations E
The numerical backbone of the whole syllabus: working confidently with fractions, percentages, ratio, standard form, indices and rounding. Most marks here are earned by careful, methodical arithmetic rather than clever tricks.
A number is in standard form when written as \( a\times 10^{n} \) with \( 1\le a<10 \) and \( n \) an integer. Large numbers take a positive \( n \); small numbers (less than 1) take a negative \( n \).
Write \( 0.000452 \) in standard form. Move the decimal point four places right to land between 4 and 5, so the exponent is \( -4 \):
A coat costs $60 after a 20% discount. Find its original price. The sale price is 80% of the original, so divide rather than add back 20%:
A surd is an irrational root left in exact form (e.g. \(\sqrt{3}\)). Key rules: \(\sqrt{ab}=\sqrt{a}\cdot\sqrt{b}\); \(\dfrac{1}{\sqrt{a}}=\dfrac{\sqrt{a}}{a}\) (rationalising a denominator). To rationalise \(\dfrac{p}{a+\sqrt{b}}\), multiply numerator and denominator by the conjugate \(a-\sqrt{b}\).
Simplify \(\sqrt{48}-\sqrt{27}\).
A length is measured as \(12.4\) cm, rounded to 1 d.p. State the upper and lower bounds. Half the smallest unit = 0.05.
Lower bound: \(12.35\) cm. Upper bound: \(12.45\) cm.
Algebra & Graphs
📌 0580 Topic 2 — Algebra & Graphs C E · Expressions & equations · Sequences · Inequalities · Simultaneous equations · Graph recognition & sketching · Kinematics graphs E
Expanding, factorising, solving and rearranging — plus reading and sketching graphs. This is the strand that feeds directly into AS and A-Level, so the habits you build here matter well beyond the exam.
To factorise \( x^{2}+bx+c \), find two numbers that multiply to \( c \) and add to \( b \). They become the constants in \( (x+p)(x+q) \).
Solve the simultaneous equations \( x+y=7 \) and \( 2x-y=2 \). Adding eliminates \( y \): \( 3x=9 \), so \( x=3 \). Substitute back into the first equation:
Find the \( n \)th term of \( 5,\,8,\,11,\,14,\ldots \) The common difference is \( 3 \) and the term before the first would be \( 2 \):
An arithmetic sequence increases/decreases by a fixed common difference \(d\). The \(n\)th term is \(u_n = a+(n-1)d\) where \(a\) is the first term. A geometric sequence multiplies by a common ratio \(r\): \(u_n=ar^{n-1}\).
Find the \(n\)th term of: 5, 9, 13, 17, … \(d=4\), first term \(a=5\). \(u_n=5+(n-1)\times4=4n+1\).
Solve like an equation but reverse the inequality sign when multiplying or dividing by a negative number. Graph types you must recognise: linear \(y=mx+c\) (straight line), quadratic \(y=ax^2+bx+c\) (parabola), cubic \(y=ax^3+\ldots\) (S-shape), reciprocal \(y=\dfrac{k}{x}\) (two branches), exponential \(y=a^x\) (crosses \(y\)-axis at 1).
Solve \(3-2x<11\) and represent on a number line. Open circle at \(-4\), arrow pointing right.
Coordinate Geometry
📌 0580 Topic 3 — Coordinate Geometry C E · Gradient · Midpoint & distance · Equation of a line · Parallel & perpendicular lines
Everything about straight lines on the \( xy \)-plane: gradient, length, midpoint and the equation \( y=mx+c \). Parallel lines share a gradient; perpendicular gradients multiply to \( -1 \).
For points \((x_1,y_1)\) and \((x_2,y_2)\): gradient \(m=\dfrac{y_2-y_1}{x_2-x_1}\); midpoint \(=\!\left(\dfrac{x_1+x_2}{2},\dfrac{y_1+y_2}{2}\right)\); distance \(=\!\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}\). Two lines are parallel if they have equal gradients. Two lines are perpendicular if the product of their gradients is \(-1\): \(m_1\times m_2=-1\).
Line \(\ell\) has equation \(y=3x-2\). Find the equation of the line perpendicular to \(\ell\) passing through \((6,1)\). Perpendicular gradient: \(m_\perp=-\tfrac{1}{3}\).
Find the equation of the line through \( (1,2) \) and \( (4,11) \). Gradient \( m=\dfrac{11-2}{4-1}=3 \). Use \( (1,2) \): \( 2=3(1)+c \Rightarrow c=-1 \):
Mensuration
📌 0580 Topic 4 — Mensuration C E · Perimeter & area · Arc length & sector area · Volume & surface area of prisms, cylinders, cones, spheres
Perimeter, area, surface area and volume — from circles and trapezia up to cylinders, cones and spheres. Read each question carefully for the value of \( \pi \) it expects (often \( 3.14 \) or \( \tfrac{22}{7} \)).
Circumference \( C=2\pi r=\pi d \); area \( A=\pi r^{2} \). An arc and sector are simply fractions of these, scaled by \( \dfrac{\theta}{360} \).
Find the volume of a cylinder with \( r=3 \) cm and \( h=10 \) cm. Take \( \pi=3.14 \).
Find the volume of a sphere of radius \( 3 \) cm. Take \( \pi=3.14 \).
Geometry
📌 0580 Topic 5 — Geometry C E · Angle properties · Polygons · Circle theorems E · Similarity & congruence · Construction & loci
Angle reasoning is the heart of IGCSE geometry: angles on lines and in polygons, and the family of circle theorems. Every answer should be justified with a reason, not just a number.
Interior angles of an \( n \)-sided polygon sum to \( (n-2)\times 180^{\circ} \). The exterior angles always sum to \( 360^{\circ} \), so a regular polygon has each exterior angle \( \dfrac{360^{\circ}}{n} \).
① The angle at the centre is twice the angle at the circumference on the same arc. ② The angle in a semicircle is \( 90^{\circ} \). ③ Angles in the same segment are equal. ④ Opposite angles of a cyclic quadrilateral sum to \( 180^{\circ} \). ⑤ A tangent meets the radius at \( 90^{\circ} \).
The first theorem is the one students doubt most — so watch it directly. Point \( P \) slides around the major arc while \( A \) and \( B \) stay fixed; the inscribed angle never changes, and it is always exactly half the central angle.
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Wherever \(P\) sits on the arc, \( \angle APB=\tfrac12\,\angle AOB \). The inscribed angle holds steady.
Find each interior angle of a regular pentagon. Interior sum \( =(5-2)\times 180^{\circ}=540^{\circ} \); divide equally among five angles:
Points \( A \) and \( B \) lie on a circle, centre \( O \), with \( \angle AOB=80^{\circ} \). Find the angle subtended by \( AB \) at the circumference. By the centre–circumference theorem the inscribed angle is half the central angle:
Congruent shapes are identical in shape and size (SSS, SAS, ASA, RHS). Similar shapes have the same angles; corresponding sides are in the same ratio \(k\). Area scales as \(k^2\); volume scales as \(k^3\).
Two similar triangles have corresponding sides in the ratio \(3:5\). If the smaller triangle has area \(18\text{ cm}^2\), find the area of the larger.
A locus is the set of all points satisfying a given condition. Key loci: (1) equidistant from a fixed point → circle; (2) equidistant from two fixed points → perpendicular bisector; (3) equidistant from two lines → angle bisector; (4) fixed distance from a line → pair of parallel lines.
Trigonometry
📌 0580 Topic 6 — Trigonometry C E · SOH-CAH-TOA & Pythagoras · Sine & cosine rules · Area \(\tfrac{1}{2}ab\sin C\) · Bearings · 3D trigonometry E
Right-angled triangles via Pythagoras and SOH-CAH-TOA, then any triangle through the sine and cosine rules. Bearings questions are trigonometry in disguise — always start with a clear diagram.
For an acute angle \( \theta \) in a right triangle, \( \sin\theta=\dfrac{\text{opp}}{\text{hyp}} \), \( \cos\theta=\dfrac{\text{adj}}{\text{hyp}} \), \( \tan\theta=\dfrac{\text{opp}}{\text{adj}} \) — the rule remembered as SOH-CAH-TOA.
Drag the angle and watch the three ratios respond. The hypotenuse is fixed in length; the opposite and adjacent sides — and the ratios built from them — change together.
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Opposite & adjacent are drawn in indigo, the hypotenuse in amber. As \( \theta\to 90^{\circ} \), \( \tan\theta \) grows without bound.
A right triangle has hypotenuse \( 10 \) and an angle of \( 30^{\circ} \). Find the side opposite that angle.
In a triangle, \( a=7 \), \( b=8 \) and the included angle \( C=60^{\circ} \). Find \( c \). Apply the cosine rule with \( \cos 60^{\circ}=0.5 \):
A bearing is a clockwise angle measured from North, always written as a three-digit number (e.g. 045°, 270°). Draw a North line at the starting point, then measure clockwise. The bearing from B back to A is the back-bearing = bearing ± 180°.
A ship sails 40 km on a bearing of 130° from port P. How far east and south of P is it? East component: \(40\sin50°\approx30.6\) km. South component: \(40\cos50°\approx25.7\) km. (Angle from South = 180°−130° = 50°.)
Identify a right-angled triangle within the 3D figure: find a face diagonal using Pythagoras, then use that as one side of a second right triangle. The angle of elevation is measured upward from horizontal; the angle of depression downward from horizontal.
A cuboid has length 8 cm, width 6 cm, height 3 cm. Find the angle the space diagonal makes with the base. Base diagonal \(=\sqrt{8^2+6^2}=10\) cm.
Transformations & Vectors
📌 0580 Topic 7 — Transformations & Vectors C E · Translation · Reflection · Rotation · Enlargement · Combined transformations E · Vectors & vector paths · Magnitude E
Four transformations move shapes around the plane, and column vectors describe the moves precisely. A full description must name the transformation and its defining data.
Translation — give the column vector. Reflection — give the mirror line. Rotation — give the centre, angle and direction. Enlargement — give the centre and scale factor (a fraction shrinks; a negative factor flips through the centre).
📊 Cambridge IGCSE Mathematics 0580 — Topic Weighting
Source: Cambridge Assessment International Education — Syllabus 0580 (2025–2027)
Pick a transformation and compare the faint original triangle with its solid image. The mapping rule \( (x,y)\to(\ldots) \) updates underneath.
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Mapping: (x, y) → (x, −y)
Reflect the point \( (3,2) \) in the \( x \)-axis. Reflection in the \( x \)-axis sends \( (x,y)\to(x,-y) \):
Find the magnitude of the vector \( \begin{pmatrix} 3 \\ 4 \end{pmatrix} \).
The magnitude of \(\mathbf{a}=\binom{x}{y}\) is \(|\mathbf{a}|=\sqrt{x^2+y^2}\). Vectors add tip-to-tail: \(\overrightarrow{AC}=\overrightarrow{AB}+\overrightarrow{BC}\). To find a vector path, combine known vectors (you may need to reverse direction, which negates the vector).
\(O\) is the origin; \(\overrightarrow{OA}=\mathbf{a}\), \(\overrightarrow{OB}=\mathbf{b}\). \(M\) is the midpoint of \(AB\). Find \(\overrightarrow{OM}\).
Probability & Statistics
📌 0580 Topic 8 — Probability & Statistics C E · Probability & tree diagrams · Conditional probability E · Mean, median, mode · Frequency diagrams & histograms E · Cumulative frequency & box plots E
Measuring chance and summarising data: the probability scale, combined events with tree diagrams, the three averages, and reading frequency and cumulative-frequency diagrams.
Probability runs from \( 0 \) (impossible) to \( 1 \) (certain). For equally likely outcomes, \( P(E)=\dfrac{\text{number of favourable outcomes}}{\text{total number of outcomes}} \).
A bag holds 3 red and 5 blue counters. One is drawn at random. Find \( P(\text{red}) \). There are \( 3+5=8 \) equally likely counters, 3 of them red:
Find the mean of \( 4,\,8,\,10,\,6,\,2 \).
For grouped data with unequal class widths, plot frequency density on the \(y\)-axis, not frequency. \(\text{Frequency density}=\dfrac{\text{frequency}}{\text{class width}}\). Area of each bar equals the frequency for that class.
A class interval 20–30 has frequency 15; class width = 10. Conversely, a bar of height 2.4 over interval 30–45 (width 15) represents frequency \(2.4\times15=36\).
Without replacement: after removing one item, the total decreases, so probabilities on the second branch change. Conditional probability: \(P(A\mid B)=\dfrac{P(A\cap B)}{P(B)}\). Multiply along branches for "and"; add separate branches for "or".
A bag has 5 red and 3 blue balls. Two balls are drawn without replacement. Find \(P(\text{both red})\).
Functions
📌 0580 Topic 2 (Extended) — Functions E · Function notation · Domain & range · Composite functions · Inverse functions
Function notation, composite and inverse functions form a core Extended-tier topic.
\( f(x) \) gives the output for an input \( x \). The composite \( fg(x)=f(g(x)) \) is applied inside-out; the inverse \( f^{-1} \) reverses \( f \).
If \( f(x)=2x+3 \), find \( f^{-1}(x) \).
Given \(f(x)=2x+3\) and \(g(x)=x^{2}\), find (a) \(fg(3)\), (b) \(f^{-1}(x)\). (a) \(fg(3)=f(9)=2(9)+3=21\). (b) Let \(y=2x+3\). Swap and solve: \(x=\dfrac{y-3}{2}\). So \(f^{-1}(x)=\dfrac{x-3}{2}\).
Sets & Venn Diagrams
📌 0580 Topic 1/2 — Sets & Venn Diagrams C E · Set notation · Union, intersection, complement · Two- and three-set Venn diagrams · Solving problems with \(n(A)\)
Sets group elements; Venn diagrams display unions, intersections and complements.
Union \( A\cup B \) (in either set), intersection \( A\cap B \) (in both), complement \( A' \) (not in \( A \)); \( n(A) \) counts the elements of \( A \).
If \( A=\{1,2,3\} \) and \( B=\{2,3,4\} \), find \( A\cap B \).
In a class of 30: 14 play football (F), 11 play basketball (B), 8 play tennis (T); 4 play F and B, 3 play F and T, 2 play B and T, 1 plays all three. Use the formula: \(n(F\cup B\cup T)=n(F)+n(B)+n(T)-n(F\cap B)-n(F\cap T)-n(B\cap T)+n(F\cap B\cap T)\) So \(30-25=5\) students play none of the three sports.
Matrices E
📌 0580 Topic 2 — Algebra (Extended only) E · Matrix notation & order · Addition & scalar multiplication · Matrix multiplication · Determinant & inverse of a 2×2 matrix · Solving simultaneous equations using the inverse matrix
Matrices are rectangular arrays of numbers. The IGCSE Extended syllabus requires you to add, multiply and invert 2×2 matrices, and use the inverse to solve a pair of simultaneous equations.
For \(A=\begin{pmatrix}a&b\\c&d\end{pmatrix}\): det\((A)=ad-bc\). Inverse: \(A^{-1}=\dfrac{1}{ad-bc}\begin{pmatrix}d&-b\\-c&a\end{pmatrix}\) (exists only when det ≠ 0). Multiplication: \((AB)_{ij}=\sum_k A_{ik}B_{kj}\); only defined when columns of \(A\) = rows of \(B\).
Find \(AB\) where \(A=\begin{pmatrix}2&1\\3&0\end{pmatrix}\) and \(B=\begin{pmatrix}1&4\\-1&2\end{pmatrix}\).
Solve: \(3x+2y=8\) and \(x-y=1\) using the inverse matrix method. Write as \(\begin{pmatrix}3&2\\1&-1\end{pmatrix}\begin{pmatrix}x\\y\end{pmatrix}=\begin{pmatrix}8\\1\end{pmatrix}\). \(\det=-3-2=-5\). \(A^{-1}=\dfrac{1}{-5}\begin{pmatrix}-1&-2\\-1&3\end{pmatrix}=\begin{pmatrix}0.2&0.4\\0.2&-0.6\end{pmatrix}\). Solution: \(x=2,\;y=1\).
Practice set
Fifty exam-style questions spanning all eight topic areas — Number through Probability & Statistics — and tagged by difficulty. Click any problem to reveal a full worked solution.
Write \( 0.000452 \) in standard form.
BasicEvaluate \( \dfrac{3}{4}+\dfrac{1}{6} \).
BasicCommon denominator 12:
\[ \frac{9}{12}+\frac{2}{12}=\frac{11}{12}. \]Increase 80 by 15%.
BasicA jacket costs \$60 after a 20% discount. Find its original price.
Intermediate\$60 is 80% of the original:
\[ \frac{60}{0.8}=\$75. \]Share \$120 in the ratio \( 2:3 \).
Intermediate5 parts, each \( 120\div 5=24 \):
\[ \$48 \ \text{and}\ \$72. \]Evaluate \( 2^{5}\times 2^{3} \).
BasicWrite 56 as a product of its prime factors.
IntermediateA length is 4.7 cm, correct to 1 decimal place. Write its lower bound.
IntermediateHalf of \( 0.1 \) below the rounded value:
\[ 4.65\ \text{cm}. \]Expand \( 3(2x-5) \).
BasicSolve \( 4x+7=23 \).
BasicFactorise \( x^{2}+5x+6 \).
IntermediateTwo numbers multiplying to 6, adding to 5:
\[ (x+2)(x+3). \]Solve \( x^{2}-7x+10=0 \).
Intermediate\( (x-2)(x-5)=0 \):
\[ x=2\ \text{or}\ x=5. \]Make \( r \) the subject of \( C=2\pi r \).
IntermediateSolve simultaneously: \( x+y=7 \), \( 2x-y=2 \).
IntermediateAdd: \( 3x=9 \Rightarrow x=3 \), then \( y=4 \):
\[ x=3,\ y=4. \]Find the \( n \)th term of \( 5,\,8,\,11,\,14,\ldots \)
IntermediateSolve the inequality \( 2x-3<9 \).
AdvancedFind the gradient of the line through \( (1,2) \) and \( (4,11) \).
BasicFind the midpoint of \( (2,3) \) and \( (8,7) \).
IntermediateA line has gradient 2 and passes through \( (0,-1) \). Write its equation.
IntermediateFind the length of the segment from \( (1,2) \) to \( (4,6) \).
AdvancedFind the area of a circle of radius 7. Take \( \pi=\tfrac{22}{7} \).
BasicFind the circumference of a circle of radius 5. Take \( \pi=3.14 \).
BasicFind the area of a triangle with base 10 and height 6.
IntermediateFind the volume of a cuboid measuring \( 4\times 5\times 3 \).
IntermediateFind the volume of a cylinder with \( r=3 \), \( h=10 \). Take \( \pi=3.14 \).
IntermediateFind the volume of a sphere of radius 3. Take \( \pi=3.14 \).
AdvancedFind the area of a trapezium with parallel sides 6 and 10 and height 4.
IntermediateFind the total surface area of a cube of side 5.
AdvancedSix square faces:
\[ 6\times 5^{2}=150. \]Two angles of a triangle are \( 50^{\circ} \) and \( 60^{\circ} \). Find the third.
BasicFind the sum of the interior angles of a hexagon.
IntermediateFind each interior angle of a regular pentagon.
IntermediateA regular polygon has an exterior angle of \( 30^{\circ} \). How many sides has it?
IntermediateState the size of the angle in a semicircle.
Intermediate\( A \) and \( B \) lie on a circle, centre \( O \), with \( \angle AOB=80^{\circ} \). Find the angle subtended by \( AB \) at the circumference.
AdvancedInscribed angle is half the central angle:
\[ \tfrac12\times 80^{\circ}=40^{\circ}. \]A right triangle has the two shorter sides 3 and 4. Find the hypotenuse.
BasicIn a right triangle the side opposite \( \theta \) is 5 and the hypotenuse is 10. Find \( \theta \).
Intermediate\( \sin\theta=\tfrac{5}{10}=0.5 \):
\[ \theta=30^{\circ}. \]A right triangle has hypotenuse 10 and an angle of \( 30^{\circ} \). Find the opposite side.
IntermediateA triangle has \( a=7 \), \( b=8 \) and included angle \( C=60^{\circ} \). Find \( c \).
AdvancedCosine rule, \( \cos 60^{\circ}=0.5 \):
\[ c^{2}=49+64-56=57,\quad c=\sqrt{57}. \]Find the hypotenuse of a right triangle with legs 6 and 8.
IntermediateIn triangle \( ABC \), \( a=10 \), \( A=30^{\circ} \), \( B=45^{\circ} \). Find \( b \) (sine rule).
AdvancedReflect the point \( (3,2) \) in the \( x \)-axis.
BasicRotate the point \( (1,0) \) by \( 90^{\circ} \) anticlockwise about the origin.
Intermediate\( (x,y)\to(-y,x) \):
\[ (0,1). \]Given \( \mathbf{a}=\begin{pmatrix}2\\3\end{pmatrix} \) and \( \mathbf{b}=\begin{pmatrix}1\\-1\end{pmatrix} \), find \( \mathbf{a}+\mathbf{b} \).
IntermediateFind the magnitude of \( \begin{pmatrix}3\\4\end{pmatrix} \).
AdvancedA fair die is rolled. Find \( P(\text{even number}) \).
BasicThree of six outcomes are even:
\[ \frac{3}{6}=\frac12. \]Find the mean of \( 4,\,8,\,10,\,6,\,2 \).
IntermediateFind the median of \( 3,\,7,\,2,\,9,\,5 \).
IntermediateOrdered: \( 2,3,5,7,9 \); middle value:
\[ 5. \]A bag holds 3 red and 5 blue counters. Find \( P(\text{red}) \).
IntermediateFind the mode of \( 2,\,3,\,3,\,5,\,7,\,3,\,8 \).
IntermediateThe most frequent value:
\[ 3. \]Two fair coins are tossed. Find \( P(\text{two heads}) \).
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Full official exam paper
Mirrors Cambridge IGCSE 0580 Extended tier: structured questions requiring full working. Paper 2 (no calc) and Paper 4 (calc) format.
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