O-Level Math Study Guide | Academic Impact Education

Complete O-Level
Math Study Guide

15 chapters of key formulas, worked examples, and exam tips.

15
Chapters
60+
Formulas
30+
Examples
2
Papers
Start with Chapter 1
Chapter 1

Quadratic & Fractional Equations

IntermediateTextbook p. 1

Three solution methods: factorisation, completing the square, and the quadratic formula. Extends to fractional equations that reduce to quadratics.

Key Concepts
Factorisation
\[ ax^2+bx+c=0 \Rightarrow (px+q)(rx+s)=0 \]
Set each factor to zero. Works when the quadratic factorises neatly.
Quadratic Formula
\[ x=\frac{-b\pm\sqrt{b^2-4ac}}{2a} \]
Works for all quadratics. Discriminant \(\Delta=b^2-4ac\) gives number of real roots.
Completing the Square
\[ x^2+bx=\left(x+\frac{b}{2}\right)^2-\frac{b^2}{4} \]
Finds vertex of parabola; solves equations without nice factors.
Discriminant
\[ \Delta=b^2-4ac \]
\(\Delta>0\): two roots  |  \(\Delta=0\): one root  |  \(\Delta<0\): no real roots
Worked Example
Example 1.1Solving by Factorisation
Solve \(2x^2-5x-3=0\).
Exam Tip

Check for excluded values — if a solution makes any denominator zero, reject it. State this check in your working.

Common Mistakes
MistakeCorrect Approach
✗Dividing both sides by \(x\) (loses a root)Bring all terms to one side and factorise
✗Forgetting \(\pm\) in the quadratic formulaAlways write both \(+\) and \(-\) cases
✗Not checking excluded values in fractional equationsSubstitute back and verify denominators are non-zero
Chapter 2

Simultaneous Linear Inequalities

FoundationTextbook p. 9

Solving pairs of linear inequalities and representing solutions on a number line.

Key Rules
Inequality Operations
\[ \text{If }a>b\text{ and }c>0:\;ac>bc \]
Multiplying or dividing by a negative number reverses the inequality sign.
Combined Inequalities
\[ a\leq x\leq b \]
The solution is the intersection of both inequalities — values satisfying both simultaneously.
Worked Example
Example 2.1Simultaneous Inequalities
Find values of \(x\) satisfying both \(3x-2>7\) and \(2x+1\leq11\).
Exam Tip

When asked for integer solutions, list them explicitly after finding the range.

Chapter 3

Indices & Standard Form

FoundationTextbook p. 17

Laws of indices are foundational across algebra and geometry. Standard form handles very large or very small numbers.

Laws of Indices
Multiplication
\[a^m\times a^n=a^{m+n}\]
Division
\[a^m\div a^n=a^{m-n}\]
Power of a Power
\[(a^m)^n=a^{mn}\]
Negative & Fractional
\[a^{-n}=\frac{1}{a^n},\quad a^{1/n}=\sqrt[n]{a}\]
Zero Index
\[a^0=1\quad(a\neq0)\]
Standard Form
\[A\times10^n,\quad1\leq A<10\]
e.g. \(0.00045=4.5\times10^{-4}\)
Exam Tip

Convert roots to fractional indices first, then apply the laws. Prevents sign errors.

Chapter 4

Coordinate Geometry

IntermediateTextbook p. 25

Key formulas for lines and their relationships on the coordinate plane.

Essential Formulas
Gradient
\[m=\frac{y_2-y_1}{x_2-x_1}\]
Distance
\[d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}\]
Midpoint
\[M=\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)\]
Equation of Line
\[y-y_1=m(x-x_1)\]
Parallel Lines
\[m_1=m_2\]
Perpendicular Lines
\[m_1\times m_2=-1\]
Exam Tip

The perpendicular gradient is the negative reciprocal: if \(m=\frac{2}{3}\), the perpendicular gradient is \(-\frac{3}{2}\).

Chapter 5

Graphs of Functions & Graphical Solution

IntermediateTextbook p. 33

Shapes and transformations of key function families for graphical problem-solving.

Key Function Families
Quadratic \(y=ax^2+bx+c\)
Parabola. Opens up if \(a>0\), down if \(a<0\). Vertex at \(x=-\frac{b}{2a}\).
Reciprocal \(y=\frac{a}{x}\)
Hyperbola with asymptotes at \(x=0\) and \(y=0\).
Exponential \(y=a^x\)
Passes through \((0,1)\). Always positive. Asymptote at \(y=0\).
Cubic \(y=ax^3+\ldots\)
S-shaped curve. Up to 3 x-intercepts. Point of inflection possible.
Exam Tip

Draw both curves on the same axes. The x-coordinates of intersections are the solutions.

Chapter 6

Further Trigonometry

AdvancedTextbook p. 43

Extends trigonometry to non-right-angled triangles using the Sine Rule, Cosine Rule, and area formula.

Core Formulas
Sine Rule
\[\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}\]
Use when: two angles + one side, or two sides + non-included angle.
Cosine Rule
\[a^2=b^2+c^2-2bc\cos A\]
Use when: two sides + included angle, or all three sides.
Area of Triangle
\[\text{Area}=\frac{1}{2}ab\sin C\]
Use when you know two sides and the included angle.
Exam Tip

In the ambiguous case, if the diagram shows an obtuse angle, use \(180°-\theta\). Always state which case applies.

Chapter 7

Applications of Trigonometry

AdvancedTextbook p. 55

Real-world problems involving bearings, angles of elevation and depression, and 3D geometry.

Key Concepts
Bearings
Measured clockwise from North. Always 3 digits (e.g. 045°, 270°). Back-bearing = bearing ± 180°.
Elevation & Depression
Elevation: angle above horizontal. Depression: angle below horizontal. Both measured from the horizontal.
3D Problems
Identify right-angled triangles within the 3D figure. Work systematically — find intermediate lengths before the final answer.
Exam Tip

Draw a clear diagram for bearing and 3D problems. Label all known values first.

Chapter 8

Arc Length, Sector Area & Radian Measure

IntermediateTextbook p. 69

Radian measure, arc length, sector area, and circular segment area.

Formulas
Radian Conversion
\[\pi\text{ rad}=180°\]
degrees × \(\frac{\pi}{180}\) = radians
Arc Length
\[l=r\theta\]
\(\theta\) must be in radians.
Sector Area
\[A=\frac{1}{2}r^2\theta\]
\(\theta\) must be in radians.
Segment Area
\[A_{\text{seg}}=\frac{1}{2}r^2(\theta-\sin\theta)\]
Segment = Sector − Triangle
Exam Tip

Ensure your calculator is in RAD mode for radian problems.

Chapter 9

Congruence & Similarity

IntermediateTextbook p. 77

Congruent figures are identical in shape and size; similar figures share shape but differ in size. Scale factor extends to area and volume.

Key Relationships
Congruence Tests
SSS, SAS, AAS, RHS — any one proves congruence. State which test you are using.
Similarity Tests
AA (two equal angles), SSS (three sides proportional), SAS (two sides proportional + included angle equal).
Area Scale Factor
\[\frac{A_1}{A_2}=k^2\]
If linear scale factor is \(k\), area scale factor is \(k^2\).
Volume Scale Factor
\[\frac{V_1}{V_2}=k^3\]
If linear scale factor is \(k\), volume scale factor is \(k^3\).
Exam Tip

State the corresponding vertices in order: "Triangle ABC is similar to Triangle PQR" means \(A\leftrightarrow P\), \(B\leftrightarrow Q\), \(C\leftrightarrow R\).

Chapter 10

Geometrical Properties of Circles

AdvancedTextbook p. 87

Powerful geometric properties for finding unknown angles and proving geometric relationships. Know all theorems and when to apply each.

Circle Theorems
Angle at Centre
Angle at centre is twice the angle at the circumference subtended by the same arc.
Angle in Semicircle
Angle in a semicircle is always 90°.
Angles in Same Segment
Angles subtended by the same arc in the same segment are equal.
Cyclic Quadrilateral
Opposite angles of a cyclic quadrilateral are supplementary (sum to 180°).
Tangent-Radius
A tangent is perpendicular to the radius at the point of contact.
Tangents from External Point
Tangents from an external point to a circle are equal in length.
Exam Tip

State the circle theorem used as a reason. This earns the method mark even if the final answer is wrong.

Chapter 11

Sets

FoundationTextbook p. 99

Set notation and Venn diagrams for describing and visualising relationships between collections.

Set Notation
Union
\[A\cup B\]
All elements in A or B (or both).
Intersection
\[A\cap B\]
Elements in both A and B.
Complement
\[A'\]
All elements in \(\xi\) that are not in A.
Subset
\[A\subseteq B\]
Every element of A is also in B.
Exam Tip

Fill a Venn diagram from the innermost region outward: start with \(A\cap B\), then each set, then outside.

Chapter 12

Probability of Combined Events

IntermediateTextbook p. 109

Tree diagrams and sample space diagrams for organising and calculating probabilities of combined events.

Key Rules
Addition (Mutually Exclusive)
\[P(A\cup B)=P(A)+P(B)\]
General Addition Rule
\[P(A\cup B)=P(A)+P(B)-P(A\cap B)\]
Multiplication (Independent)
\[P(A\cap B)=P(A)\times P(B)\]
Complementary Events
\[P(A')=1-P(A)\]
Exam Tip

Without replacement: the denominator changes for the second draw. Update probabilities at each branch.

Chapter 13

Statistical Data Analysis

IntermediateTextbook p. 117

Measures of central tendency and spread, cumulative frequency curves, and box-and-whisker plots.

Measures of Central Tendency & Spread
Mean (Grouped Data)
\[\bar{x}=\frac{\sum fx}{\sum f}\]
\(f\) = frequency, \(x\) = midpoint of class interval.
Standard Deviation
\[\sigma=\sqrt{\frac{\sum f(x-\bar{x})^2}{\sum f}}\]
Larger \(\sigma\) = more spread.
Interquartile Range
\[\text{IQR}=Q_3-Q_1\]
Spread of the middle 50% of data. Less affected by outliers than range.
Exam Tip

Compare both central tendency (mean/median) and spread (SD/IQR). A complete answer addresses both.

Chapter 14

Matrices

AdvancedTextbook p. 131

Rectangular arrays used to represent and solve systems of equations. Covers operations, determinants, and inverse matrices.

Key Operations & Formulas
Determinant of 2×2
\[\det\begin{pmatrix}a&b\\c&d\end{pmatrix}=ad-bc\]
Inverse of 2×2
\[\begin{pmatrix}a&b\\c&d\end{pmatrix}^{-1}=\frac{1}{ad-bc}\begin{pmatrix}d&-b\\-c&a\end{pmatrix}\]
Only exists when \(\det\neq0\).
Solving Simultaneous Equations
\[AX=B\Rightarrow X=A^{-1}B\]
Matrix Multiplication
Only defined when columns of first = rows of second. Note: \(AB\neq BA\) in general.
Exam Tip

Work row by column systematically. A sign error in the determinant cascades through the entire inverse.

Chapter 15

Vectors

AdvancedTextbook p. 143

Quantities with magnitude and direction. Covers notation, operations, position vectors, and geometric proofs.

Key Concepts & Formulas
Vector Addition
\[\overrightarrow{AB}+\overrightarrow{BC}=\overrightarrow{AC}\]
Magnitude
\[|\mathbf{a}|=\sqrt{x^2+y^2}\]
Position Vector
\[\overrightarrow{AB}=\mathbf{b}-\mathbf{a}\]
Collinearity
Points A, B, C are collinear if \(\overrightarrow{AB}=k\overrightarrow{AC}\) for some scalar \(k\). State the scalar relationship and the shared point.
Exam Tip

To prove collinearity, show that \(\overrightarrow{AB}\) is a scalar multiple of \(\overrightarrow{AC}\), and state that they share a common point A. Both conditions must be stated for full marks.

Exam Prep

Papers & Real-World Contexts

Exam Practicep. 155–300

Full specimen papers and worked solutions. Use under timed conditions.

What's Included
Real-World Contexts (p. 155)
Multi-topic problems in real-world scenarios. Identify which mathematical tools apply.
Specimen Paper A (p. 161)
Full exam-format paper. Attempt under timed conditions.
Specimen Paper B (p. 199)
Second full practice paper for additional simulation.
Worked Solutions (p. 227)
Complete step-by-step solutions. Review after attempting.
Exam Strategy

Timed practice is the most effective final preparation. Attempt each paper without notes. After marking, categorise errors: careless mistakes, concept gaps, or time management. Address each differently.

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