This cheatsheet condenses the O Level E Math syllabus into seven pages of rules, methods and calculator steps. It runs from numbers and algebra, through graphs, geometry and trigonometry, to mensuration, matrices, finance, vectors, sets, probability and statistics.
Many topics are graded from Level 1 to Level 3 questions and come with short rules to remember, such as when to flip an inequality sign, how to handle an obtuse angle from the sine rule, and which property to look for first in a circle question. This makes it useful as a final check before prelims and the O Level papers.
What these notes cover
- •Numbers, HCF and LCM
- •Number Patterns and Proportion
- •Expansion, Factorisation and Algebraic Fractions
- •Changing the Subject, Equations and Inequalities
- •Indices and Standard Form
- •Quadratic Graphs and Coordinate Geometry
- •Speed–Time Graphs
- •Angles, Polygons, Congruency and Similarity
- •Trigonometry and Bearings
- •Mensuration and Circle Properties
- •Matrices, Finance and Vectors
- •Sets, Probability and Statistics
Key definitions
- Rational number
- A number that can be expressed as a fraction. Recurring decimals are included.
- Irrational number
- A number that cannot be expressed as a fraction, such as π. Key it into the calculator to check.
- Prime number
- A number divisible only by 1 and itself (2, 3, 5 …). A composite number has more than 2 factors (4, 6, 8, 9 …).
- Proper subset (A ⊂ B)
- Every element of A is in B and B has at least one element that is not in A. A subset (A ⊆ B) only requires every element of A to be in B.
- Retardation
- Another word for deceleration. It already carries a negative meaning, so no negative sign is needed in front of the value.
Key formulas
General term of a linear sequence
Tₙ = dn + (a − d)
a = first term, d = common difference. For 2, 5, 8, 11, 14: Tₙ = 3n + (2 − 3) = 3n − 1. To find the (n + 1)th term, replace n with n + 1.
Angles in a polygon
Sum of interior angles = (n − 2) × 180°; sum of exterior angles = 360°
n = number of sides. Interior angle + exterior angle = 180°. For a regular polygon, divide by n to find each angle.
Turning point of a quadratic graph
Found by completing the square. The coefficient of x² must be positive and 1 first, so factorise it out if it is not.
Similar figures
l = corresponding lengths, A = areas, V = volumes. Area to length: square root; length to volume, mass or cost: cube. Area ratio 4 : 25 gives length ratio 2 : 5 and volume ratio 8 : 125.
Compound interest periods
Rate per period = annual rate ÷ periods per year; number of periods = years × periods per year
Periods per year: 1 (yearly), 2 (semi-annually), 4 (quarterly), 12 (monthly). 12% per annum for 3 years compounded quarterly gives 3% over 12 periods.
Range and interquartile range
Range = highest value − lowest value; IQR = UQ − LQ
UQ = 75th percentile, LQ = 25th percentile, median = 50th percentile.
Worked example
Solve 4x − 40 < −2x + 8 ≤ 3x − 12.
- 1Split the double inequality into two: 4x − 40 < −2x + 8 and −2x + 8 ≤ 3x − 12.
- 2First inequality: 4x + 2x < 8 + 40, so 6x < 48 and x < 8.
- 3Second inequality: 8 + 12 ≤ 3x + 2x, so 20 ≤ 5x and x ≥ 4.
- 4Combine both results: ∴ 4 ≤ x < 8.
Watch out: Rewriting 4 ≤ x as x ≥ 4 swaps the sides, so the sign reverses. The sign also flips whenever you multiply or divide by a negative number.
Common mistakes
Taking the acute angle from the sine rule when the angle is obtuse
Wrong: Using sin⁻¹ 0.5 = 30° from the calculator even though the diagram shows the angle is more than 90°.
Right: sin 30° = sin 150° = 0.5, so check the diagram. If the angle is obtuse, take 180° − 30° = 150°.
Proving congruency with ASS or AAA
Wrong: Concluding two triangles are congruent using ASS or AAA.
Right: Use SSS, SAS, AAS or RHS, which need at least one side, and name the vertices in corresponding order.
Dropping the ± when square-rooting
Wrong: Square-rooting both sides and writing only x − 4 = 2, so x = 6.
Right: Include ±: x − 4 = ±2, so x = 2 or x = 6.
Questions students ask
How do you find HCF and LCM using prime factorisation?
Write each number as a product of prime factors, for example 40 = 2³ × 5 and 125 = 5³. The HCF takes the common primes with the lower power, so HCF = 5. The LCM takes every prime with the higher power, so LCM = 2³ × 5³ = 1000.
How do you find the general term of a number pattern?
For a sequence with first term a and common difference d, the general term is dn + (a − d), so 2, 5, 8, 11, 14 gives 3n − 1. To test whether 28 is in the sequence, solve 3n − 1 = 28. Since n is not an integer, 28 does not lie in the sequence.
What does the area under a speed-time graph represent?
The area under a speed–time graph gives the total distance travelled, and its gradient gives the acceleration. On a distance–time graph, the gradient gives the speed. Always check the axes and units before you start.
Should I use the median or interquartile range to compare two sets of data?
Use the median to compare performance and the interquartile range to compare consistency. Shifting a cumulative frequency curve changes the median but not the IQR. Making the curve steeper or flatter changes the IQR but may not change the median.
In the full PDF
- •Mensuration table with surface area and volume for the cone, sphere, hemisphere, cylinder, prism and pyramid, plus arcs, sectors and segments
- •All circle properties, the Radius–Tangent–Diameter check and special questions such as whether a point lies inside, on or outside a circle
- •Labelled diagrams for quadratic graph sketching, curve–line intersection and speed–time graphs
- •Further worked examples on perfect squares and cubes, finding a number from its HCF and LCM, proportion, indices and the 'hence solve' method
- •Sets and Venn diagram notation table, matrices, and the income tax rate table
- •Calculator steps for mean and standard deviation on Casio and Sharp models, and model answers for misleading statistical graphs