Exam Prep Study Guides
Free, in-depth study guides with step-by-step explanations, worked examples, practice problems, and expert tips — for every major exam.
50 topics across 6 exams
A-Level
6 guidesStatistical Hypothesis Testing
Master A-Level hypothesis testing: null and alternative hypotheses, significance levels, binomial tests, critical regions, and one/two-tailed tests.
Differentiation from First Principles
Master differentiation from first principles: the limit definition, deriving the power rule, and finding gradients of tangent lines at A-Level.
Integration (Indefinite and Definite)
Learn indefinite and definite integration at A-Level: the power rule for integration, finding areas under curves, and the constant of integration.
Quadratic Functions and Equations
Master quadratic functions at A-Level: completing the square, the discriminant, sketching parabolas, and solving quadratic inequalities.
Vectors
A complete guide to A-Level vectors: position vectors, magnitude, unit vectors, the scalar (dot) product, and geometric proofs using vectors.
Trigonometric Identities and Equations
A comprehensive guide to A-Level trigonometric identities: Pythagorean identities, double angle formulae, solving trig equations in radians and degrees.
ACT
10 guidesSubject-Verb Agreement
Master subject-verb agreement for the ACT English section. Learn tricky subjects, intervening phrases, and inverted sentence patterns.
Rhetorical Strategy (Adding, Revising, Deleting)
Master ACT English rhetorical strategy questions: adding/deleting sentences, transitions, essay goals, and paragraph ordering.
Sentence Structure and Formation
Learn to fix fragments, run-ons, and comma splices on the ACT English section. Master parallel structure, modifiers, and coordination.
Punctuation Rules (Commas, Semicolons, Colons)
Master ACT English punctuation: comma rules, semicolons, colons, and dashes. Learn FANBOYS, appositives, and key strategies for test day.
Trigonometry Basics (SOH CAH TOA)
Master SOH CAH TOA, unit circle values, and right triangle trig for the ACT with worked examples and practice problems.
Functions and Function Notation
Learn f(x) notation, domain/range, composition, and transformations for ACT Math with worked examples and practice.
Coordinate Geometry
Learn slope, distance, midpoint, and line equations for ACT coordinate geometry with formulas, examples, and practice problems.
Plane Geometry (Angles, Triangles, Circles)
Master angle relationships, triangle properties, circle theorems, and area formulas for ACT Math with examples and practice.
Ratios, Proportions and Percentages
Master ratios, proportions, percent change, and successive percentages for the ACT with strategies and practice problems.
Linear Equations and Inequalities
Master solving linear equations, inequalities, and systems with ACT-specific strategies, worked examples, and practice problems.
IB
5 guidesVectors in 2D and 3D
Comprehensive guide to vectors for IB Maths AA HL. Covers vector operations, dot product, angle between vectors, vector lines, and applications.
Sequences and Series (Arithmetic & Geometric)
Master arithmetic and geometric sequences and series for IB Maths AA SL. Covers nth terms, sums, sigma notation, and real-world applications.
Derivatives and Their Applications
Master differentiation for IB Maths AA SL. Covers power rule, chain rule, product/quotient rule, tangent lines, optimisation, and curve sketching.
Probability Distributions (Binomial and Normal)
Complete guide to binomial and normal distributions for IB Maths AA SL. Covers conditions, formulas, z-scores, GDC techniques, and inverse normal.
Functions and Transformations
Comprehensive guide to functions and transformations for IB Maths AA SL. Covers domain, range, inverses, composites, and graph transformations.
GCSE
6 guidesArea and Perimeter of 2D Shapes
Calculate area and perimeter of rectangles, triangles, circles, trapeziums and composite shapes for GCSE Maths with worked examples.
Solving Linear Equations
Master solving linear equations for GCSE Maths. Covers one-step, two-step, brackets and unknowns on both sides with worked examples.
Probability Basics
Learn the basics of probability for GCSE Maths. Covers single events, sample spaces, mutually exclusive events and expected outcomes.
Expanding and Factorising
Master expanding brackets and factorising expressions for GCSE Maths. Covers single and double brackets, DOTS and quadratics.
Fractions, Decimals and Percentages
Convert between fractions, decimals and percentages for GCSE Maths. Covers FDP conversions, percentage change and fraction arithmetic.
Pythagoras' Theorem
Learn Pythagoras' Theorem for GCSE Maths. Find the hypotenuse, shorter sides, and tackle 3D problems with step-by-step worked examples.
AP Calculus AB
5 guidesChain Rule
Master the chain rule for composite functions, including nested derivatives and combining with product and quotient rules for AP Calc AB.
Related Rates
Solve related rates problems with step-by-step strategies for setting up equations, implicit differentiation, and classic AP exam scenarios.
Limits and Continuity
Master limit laws, algebraic evaluation techniques, one-sided limits, and the formal definition of continuity for AP Calculus AB.
Fundamental Theorem of Calculus
Understand FTC Part 1 and Part 2, evaluate definite integrals, and work with accumulation functions for the AP Calculus AB exam.
Derivative Rules (Power, Product, Quotient)
Learn the power rule, product rule, quotient rule, and trig derivatives essential for AP Calculus AB differentiation problems.
SAT
18 guidesTransition Words and Phrases
Transitions are the "glue" of the Digital SAT Reading and Writing section. While they may seem like simple vocabulary questions, they are actually testing
Analyzing Author's Purpose
In the Digital SAT Reading and Writing section, the "Craft and Structure" domain accounts for approximately 28% of your total score. Within this domain, "T
Vocabulary in Context
The "Words in Context" question type is a cornerstone of the Digital SAT Reading and Writing section, appearing within the **Craft and Structure** domain.
Comma Rules (Essential vs. Nonessential Clauses)
On the Digital SAT, the "Standard English Conventions" domain accounts for approximately 26% of your total score. Within this domain, punctuation—specifica
Pronoun Agreement and Clarity
Pronoun Agreement and Clarity is a cornerstone of the **Standard English Conventions** domain on the Digital SAT. While it may seem like a basic grammar to
Subject-Verb Agreement
Subject-verb agreement (SVA) is a cornerstone of the **Standard English Conventions** domain on the Digital SAT. While it may seem like a basic grammar rul
Identifying the Main Idea
The "Central Ideas and Details" question type is a cornerstone of the Information and Ideas domain on the Digital SAT. While it may seem straightforward, i
Sentence Boundaries: Fragments and Run-Ons
Welcome to one of the most critical sections of the Digital SAT Reading and Writing (R&W) module. If you are aiming for a 700+ score, mastering **Sentence
Mean, Median, and Mode
Welcome to one of the most high-yield topics in the Digital SAT Math section: **Measures of Center**. While you likely learned about mean, median, and mode
Percent Increase and Decrease
Welcome to one of the most high-yield topics on the Digital SAT. If you are aiming for a 700+ score, mastering **Percent Increase and Decrease** is non-neg
Trigonometric Ratios (SOH-CAH-TOA)
Trigonometry often strikes fear into the hearts of SAT test-takers, but on the Digital SAT, it is one of the most predictable and "hackable" topics in the
Special Right Triangles (30-60-90 and 45-45-90)
In the world of the Digital SAT, efficiency is just as important as accuracy. While the Pythagorean Theorem ($a^2 + b^2 = c^2$) is a reliable tool for solv
The Pythagorean Theorem
Welcome to one of the most essential pillars of the SAT Math section: The Pythagorean Theorem. If you are aiming for a 700+ score, you cannot simply "know"
Solving Systems by Elimination
Systems of linear equations are a cornerstone of the Digital SAT Math section, appearing frequently within the **Algebra** domain. On any given test, you c
Slope-Intercept Form (y = mx + b)
Welcome to one of the most critical pillars of the Digital SAT Math section. If you want to score a 700+ on the SAT, you must master the slope-intercept fo
Linear Equations from Word Problems
In the landscape of the Digital SAT, the ability to translate real-world scenarios into mathematical language is perhaps the most critical skill you can de
The Quadratic Formula
In the landscape of the Digital SAT, the Quadratic Formula is your ultimate "safety net." While many quadratic equations you encountered in Algebra 1 could
Factoring Quadratic Expressions
Factoring quadratic expressions is one of the most fundamental "power skills" you need for the Digital SAT. While it might seem like a basic algebra topic
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