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• Statistics – Descriptive Statistics Chapter Map

  Descriptive statistics: population vs sample; measures of centre (mean, median, mode); dispersion (range, SD); relative position (Z-score, percentiles).

• Statistics – Point vs Interval Estimation

  Point estimation gives one value (e.g. x̄ = 75); interval estimation gives a range (e.g. [72,78] at 95% confidence); advantages of interval over point.

• Statistics – Confidence Level and Significance Level

  Confidence level (1−α) expresses % certainty; significance level α is its complement; table of Z-critical values: 1.645, 1.96, 2.576 for 90/95/99%.

• Statistics – Confidence Interval for Mean (Known Variance)

  Known-variance CI: X̄ ± Z_{α/2}·σ/√n; Z_{α/2}=1.96 for 95%; components: sample mean, Z-critical, population SD, sample size; worked example.

• Statistics – Confidence Interval for Mean (Unknown Variance)

  CI for mean (unknown variance): X̄ ± t_{α/2}·S/√n using t-distribution with df=n−1; t is wider than Z to account for extra uncertainty; t→Z for large n.

• Statistics – Confidence Interval for a Proportion

  Proportion CI: p̂ ± Z_{α/2}·√(p̂(1−p̂)/n); requires np̂≥5 and n(1−p̂)≥5; full worked example: 80 successes from n=200, p̂=0.4, 95% CI=[0.332,0.468].

• Statistics – CI for Difference Between Means

  CI for μ₁−μ₂: (X̄₁−X̄₂) ± t·SE with df=min(n₁−1,n₂−1); if 0 is inside CI there is no significant difference; if outside, significant difference.

• Statistics – CI for Difference Between Proportions

  Two-proportion CI: (p̂₁−p̂₂) ± Z·√(p̂₁(1−p̂₁)/n₁+p̂₂(1−p̂₂)/n₂); example: 60/100 vs 50/100; point difference = 0.1 (10%); 95% CI interpretation.

• Statistics – Confidence Interval for Variance

  CI for variance: uses χ² distribution (asymmetric, positive); formula ((n−1)S²/χ²_upper, (n−1)S²/χ²_lower); df=n−1; why the interval is asymmetric.

• Statistics – Setting Up Hypotheses

  Null H₀ (no change or difference); alternative H₁ (what researcher wants to prove); two-tailed (≠) and one-tailed (>, <); testing assumes H₀ true.

• Statistics – Type I and Type II Errors

  Type I (α): reject H₀ when true = false positive; Type II (β): fail to reject H₀ when false = false negative; decision table with four outcomes.

• Statistics – The P-Value

  P-value: probability of result this extreme or more given H₀ true; small P → reject H₀; rule P<α reject, P>α do not reject; common misconception.

• Statistics – HT for Mean with Known Variance

  Z-test for mean with known σ²: statistic Z=(X̄−μ₀)/(σ/√n); standard normal distribution; decision by critical value or P-value; worked example.

• Statistics – HT for Mean with Unknown Variance

  t-test for mean with unknown σ²: t=(X̄−μ₀)/(S/√n) with df=n−1; t has heavier tails than Z because S estimates σ; for large n the two converge.

• Statistics – HT for Difference of Two Independent Means

  HT for μ₁−μ₂ (independent samples): H₀ μ₁=μ₂; t-statistic with pooled SE; df=min(n₁−1,n₂−1); independent means separate groups, no subject link.

• Statistics – HT for Difference of Two Paired Means

  Paired t-test: compute differences dᵢ=X₁ᵢ−X₂ᵢ, test μ_d=0; df=n−1 (pairs); advantage: each subject is own control; use for before/after or matched pairs.

• Statistics – Hypothesis Test for a Proportion

  HT for proportion: Z=(p̂−p₀)/√(p₀(1−p₀)/n); uses p₀ from H₀ (unlike CI which uses p̂); conditions np₀≥5 and n(1−p₀)≥5; two-tailed and one-tailed.

• Statistics – HT for a Single Variance

  HT for single variance: χ² statistic with df=n−1; asymmetric distribution, always positive; rejection region depends on one/two-tailed test type.

• Statistics – HT for Ratio of Two Variances

  F-test for variance ratio: F=S₁²/S₂² (larger variance in numerator so F>1); F-distribution with df₁=n₁−1 and df₂=n₂−1; always place larger S² on top.

• Statistics – One-Tailed vs Two-Tailed Tests

  Two-tailed: H₁ uses ≠, rejection region α/2 each tail; one-tailed: H₁ gives direction (> or <), full α in one tail; choosing based on research question.

• Statistics – Statistical Power

  Statistical power = 1−β: probability of rejecting H₀ when false; affected by n, α, effect size; desired power ≥ 0.8; relation to Type II error β.

• Statistics – Random Variables: Definition

  Random variable maps outcomes to numbers; discrete (countable, e.g. die roll) vs continuous (any value, e.g. height); tip: can you list all values?

• Statistics – Expectation and Variance of a Random Variable

  E(X) = Σx·P(x) (weighted average); Var(X) = E(X²)−[E(X)]²; linear transformation rules: E(aX+b)=aE(X)+b, Var(aX+b)=a²Var(X); worked example.

• Statistics – The Binomial Distribution

  Binomial distribution: n fixed trials, binary outcome, constant p, independent; E(X)=np, Var(X)=np(1−p); examples; when NOT to use binomial.

• Statistics – Non-Parametric Tests: When and Why

  Non-parametric tests: for non-normal data, ordinal scales, or small samples; comparison table with parametric equivalents; advantages and disadvantages.

• Statistics – Chi-Square Goodness-of-Fit Test

  χ² goodness-of-fit: compares observed vs expected frequencies; df=k−1; worked example (fair die, 60 rolls); condition: all expected frequencies ≥ 5.

• Economics – The Keynesian Income Multiplier

  Income multiplier k=1/(1−b): each unit of G raises GDP by k; tax multiplier −b/(1−b); balanced-budget multiplier=1; traced 100-unit example; MPC=0.8.

• Economics – Output Gaps and the Paradox of Thrift

  Deflationary gap: unemployment, expansionary fix; inflationary gap: inflation, contractionary; gap=|Y_eq−Y_fe|; closing with multiplier; paradox of thrift.

• Statistics – Measures of Centre

  Mean (sum/n, sensitive to outliers), median (middle value, robust), mode (most frequent); salary example showing why median is better with outliers.

• Statistics – Measures of Dispersion

  Measures of dispersion: range (max−min), IQR (Q3−Q1), sample variance (n−1), SD (√variance); Bessel's correction; Z-score interpretation; visual example.

• Economics – GDP: The Basics

  GDP: total value of goods/services produced within borders; GDP vs GNP; value added; flow vs stock variables; expenditure formula GDP = C+I+G+(X−M).

• Statistics – Hypothesis Tests for Variance

  χ² test for one variance df=n−1; F-test for two variances, larger in numerator; worked example σ²=25 vs S²=40; notes on asymmetry and tail direction.

• Statistics – Z-Score: Interpretation, Calculation and Comparison

  Z-score Z=(X−μ)/σ: how far a value is from the mean in SDs; positive/negative/zero interpretation; three worked examples; comparing groups using Z-scores.

• Statistics – Normal Distribution: Four Problem Types

  Four normal distribution problem types: Type 1 value→probability, Type 2 interval, Type 3 inverse probability→value, Type 4 group comparison via Z-scores.

• Statistics – Z-Table Common Mistakes

  Z-table always gives left-tail area P(Z≤z); mistake 1: P(Z=z)=0 for continuous; mistake 2: right tail = 1−Φ(z); mistake 3: use symmetry Φ(−z)=1−Φ(z).

• Statistics – Normal Distribution Word Problems: Step by Step

  Normal word problems: identify distribution, convert to Z-score, interpret position relative to mean, link to probability via Z-table or empirical rule.

• Statistics – Empirical Rule: When to Use and When Not To

  Empirical 68-95-99.7 rule: for quick approximation only; do NOT use when exact probability, non-round percentages, or exact threshold values are needed.

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• Statistics – Linear Transformation

  Linear transformation Y=a+bX: mean and median shift by a and scale by b; SD multiplied by |b|; variance by b²; adding constant does not change dispersion.

• Analytic Geometry – The Coordinate System

  Coordinate system: perpendicular axes and origin O; four quadrants with sign rules; plotting (x,y); axes points (y=0 or x=0); (x,y)=(horizontal,vertical).

• Analytic Geometry – Lines Parallel to the Axes

  Horizontal y=c (slope=0, parallel x-axis); vertical x=c (slope undefined, parallel y-axis); why vertical slope undefined; three worked examples.

• Linear Equations — Solve ax + b = c

  Practice solving linear equations step by step.

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• Accumulation Function – GeoGebra Activity

  Explore the accumulation function with GeoGebra. Visualize area accumulation over an interval and understand key integral concepts in calculus.

• Integral – Area Accumulation Function and Its Properties

  Discover area accumulation function properties through GeoGebra. Drag the slider to see how accumulation changes across an interval and explore calculus.

• Adding Complex Numbers (GeoGebra Activity)

  Learn to add complex numbers algebraically and geometrically. GeoGebra activity demonstrates the parallelogram rule for z1 plus z2 on the complex plane.

• Multiplication by i – Complex Numbers

  Understand multiplying a complex number by i with GeoGebra. See a 90-degree counterclockwise rotation on the complex plane clearly visualized.

• Complex Numbers – Multiplication of Complex Numbers

  Explore complex number multiplication geometrically with GeoGebra. Visualize how moduli multiply and arguments add when computing z1 times z2.

• Complex Numbers – Roots of Unity

  Explore roots of unity interactively with GeoGebra. Use the slider to change n and see the nth roots arranged symmetrically on the unit circle.

• Complex Numbers – Roots of a Complex Number

  Visualize all roots of the equation z^n equals w on the complex plane. Toggle roots, connecting lines, and powers of z0 with this GeoGebra activity.

• Geometric Loci in the Complex Plane

  Explore geometric loci in the complex plane with GeoGebra. Visualize distance conditions such as |z minus z1| equals R and Apollonius circle sets.

• Visual Solution of a Complex Number Problem

  Visual proof: three complex numbers on the unit circle summing to zero implies their squares also sum to zero. Explore this interactively with GeoGebra.

• Complex Numbers – Addition of Complex Numbers

  Add two complex numbers Z and W interactively on the complex plane. Drag the points and watch Q equals Z plus W update in real time with full geometry.

• Multiplication of Two Complex Numbers

  Multiply two complex numbers Z and W interactively on the complex plane. Observe Q equals WZ as you drag the points and see moduli and arguments change.

• Complex Numbers – Written Explanation

  Complete guide to complex numbers: definition, complex plane, addition, multiplication, polar form, De Moivre formula, and roots with worked examples.

• Vector Addition

  Learn vector addition step by step with GeoGebra. Copy vector MN, display the sum u plus v, and clearly visualize the parallelogram rule for vectors.

• Vectors – Dot Product

  Understand the dot product of two vectors with GeoGebra. Interactive activity shows projection and demonstrates v dot u equals |v||u| times cos alpha.

• Triangle Medians – Vectors

  Investigate triangle medians using vectors with GeoGebra. Find when GH is parallel to BC, prove your conjecture, and explain its geometric meaning.

• Points in Space – Vectors (GeoGebra Activity)

  Explore points and vectors in three-dimensional space with GeoGebra. Drag points A, B, C to observe how the algebraic components of vector P change.

• Parametric Equation of a Line in the Plane

  Learn the parametric equation of a line with an interactive GeoGebra activity. Choose parameter t and trace the line OP equals OA plus t times AB.

• 3D Trigonometry – Constructing a Right Pyramid with Triangular Base

  Build a right pyramid with a triangular base and apply 3D trigonometry with GeoGebra. Drag base vertices to reshape the pyramid and explore spatial angles.

• 3D Trigonometry – Right Pyramid with Right-Angle Triangular Base

  Explore a right pyramid with a right-angle triangular base using 3D trigonometry. Drag the triangle vertices to reshape the pyramid with GeoGebra.

• 3D Trigonometry – Pyramids and Prisms

  Compare pyramids and prisms with different base edges using an interactive slider. Watch both shapes transform and explore 3D geometry with GeoGebra.

• Angle Between a Line and a Plane

  Explore the angle between a line and a plane with GeoGebra. Drag the vertices to see how the angle between line BC and the gray plane changes.

• Angle Between Two Planes

  Investigate the angle between two planes interactively with GeoGebra. Drag point A to examine different plane orientations and measure the dihedral angle.

• 3D Trigonometry – Identifying Angles in a Box

  Explore angles in a rectangular box with GeoGebra. Drag points I and J, track angle IJB, find when it is a right angle, and calculate it for a cube.

• Perpendicular Planes – Are the Lines Also Perpendicular?

  Investigate whether two perpendicular planes imply perpendicular lines using a right prism with an isosceles triangular base. Explore interactively.

• Angles Between Parallel Lines and a Transversal

  Explore angles formed by parallel lines and a transversal through a dynamic GeoGebra worksheet. Discover the key angle theorems through guided inquiry.

• Optimization – Rectangle Perimeter

  Observe how the perimeter of a rectangle changes as a function of its dimensions. Drag slider Xa to move point A and explore optimization interactively.

• Optimization – Sum of Areas

  Find the length CE that maximizes the sum of areas of a square and a triangle. Interactive GeoGebra activity for optimization problems in calculus.

• Optimization – Maximum Area Enclosure

  Find the rectangular enclosure with the largest area using 6 meters of fence along three sides. A classic optimization problem with GeoGebra.

• Inscribed Angles – GeoGebra Activity

  Explore inscribed angles in a circle with GeoGebra. Visualize and measure inscribed angles and discover their key geometric properties interactively.

• Inscribed Angle and Central Angle on the Same Arc

  Compare an inscribed angle and a central angle subtended by the same arc. Interactive GeoGebra activity reveals the relationship between the two angles.

• Central Angles in a Circle

  Explore central angles in a circle interactively with GeoGebra. Visualize how central angles relate to arcs and discover their geometric properties.

• Trigonometric Functions on the Unit Circle

  Visualize sine, cosine, and tangent on the unit circle with GeoGebra. See how the trigonometric functions change as the angle varies around the circle.

• Tangent to a Parabola

  Learn to find the tangent to a parabola at a given point or slope. GeoGebra activity covers the tangent formula and worked examples for y squared = 2px.

• Parabola – Tangent and Normal

  Find the tangent and normal to a parabola step by step. Covers implicit differentiation, slope formula, and full worked example for the parabola y2 = 2px.

• Parabola – Tangent and Normal (Summary)

  Explore tangent and normal lines to a parabola with GeoGebra. Drag the point along y2 = 2px and watch the tangent and normal lines update in real time.

• GeoGebra Activity – Parabola, Tangent, Normal and Focus

  GeoGebra guided activity on parabola tangents, normals, and the focus. Drag point A along y2 = 2px to explore slopes, normals, and focal ray reflection.

• Inquiry Activity – Tangent, Normal and Focus in a Parabola

  Inquiry on tangent, normal, and focus of the parabola y squared = 2px. Drag point A, record slope values, and explore the focal reflection property.

• Inquiry Activity – Ellipse, Foci, Tangent and Reflection

  Inquiry on the ellipse, its foci, tangent, normal, and reflection property. Explore how a ray from one focus always reflects toward the other focus.

• Explanation – Ellipse as a Geometric Locus

  Clear explanation of the ellipse as a geometric locus. Covers the canonical equation, foci, semi-axes, and the constant-sum-of-distances definition.

• Inquiry – The Relationship Between a Function and Its Derivative

  Discover the relationship between a function and its derivative with GeoGebra. Explore how the slope sign relates to increasing, decreasing, and extrema.

• Investigation: Concavity and Inflection Points

  Learn how to investigate concavity and inflection points using the first and second derivative. Step-by-step visual explanation for high-school calculus.

• What Is a Function? Core Concepts

  Understand what a function is, how to define it algebraically and graphically, and why it is the foundation of calculus. Examples for every level.

• Slope and Tangent Line – Geometric Meaning of the Derivative

  Explore the geometric meaning of the derivative: the slope of the tangent line. Understand instantaneous rate of change with clear visual examples.

• Differentiation Rules – Polynomial Functions

  Master the differentiation rules for polynomial functions: power rule, constant rule, sum rule. Includes worked examples and practice for every term.

• Function Investigation – Increasing, Decreasing and Extreme Points

  Learn how to use the first derivative to find increasing and decreasing intervals and classify local maxima and minima. Full step-by-step guide.

• Symmetry – Even and Odd Functions

  Learn how to identify even and odd functions algebraically and graphically. Understand axis and origin symmetry with clear examples and practice problems.

• Concavity and Inflection Points

  Master concavity analysis using the second derivative. Learn to identify concave up and concave down intervals and locate inflection points with examples.

• Domain of a Rational Function

  Learn how to find the domain of a rational function by identifying where the denominator equals zero. Includes rules, examples, and interval notation.

• Vertical Asymptote and Removable Discontinuity

  Learn the difference between a vertical asymptote and a removable discontinuity. Identify each type by checking numerator and denominator with examples.

• Horizontal Asymptote

  Learn how to find horizontal asymptotes by comparing degrees of numerator and denominator. Includes worked examples, limit calculations, and exam tips.

• Derivative of a Quotient – Quotient Rule

  Master the quotient rule for rational functions. Learn the formula, work through three examples, and avoid common mistakes with this clear calculus guide.

• The Relationship Between the Graph of a Function and Its Derivative

  Understand the relationship between f, f-prime and f-double-prime. Learn to read increasing, decreasing, concavity and inflection from derivative graphs.

• Probability – Core Concepts

  Learn the basics of probability: sample space, events, complement, mutually exclusive events, and probability trees with multiplication and addition rules.

• Probability Tree

  Master probability trees: multiply along paths, add between paths. Learn with and without replacement, worked examples and self-check methods.

• Probability – Advanced Topics

  Learn conditional probability, independent vs mutually exclusive events, and the binomial distribution formula with worked examples and a summary table.

• Probability Table

  Learn to build and use a probability table. Covers conditional probability, independence testing, and worked examples with counts and proportions.

• Arithmetic Sequence

  Master arithmetic sequences: use the general term formula to find any term, locate a term by value, and test whether a number belongs to the sequence.

• Geometric Sequence

  Master geometric sequences: find any term using the general term formula, locate a term by value, find the common ratio, and test sequence membership.

• Logarithms – Laws and Formulas

  Master all logarithm laws: product, quotient and power rules, change of base, key identities, and growth and decay applications with worked examples.

• Exponential Functions and Equations

  Master exponential functions and equations: matching bases, logarithm method, substitution for quadratics, inequalities, and power laws with examples.

• Indefinite Integral

  Master indefinite integration: power rule, constant, ln, e^x and trig formulas. Includes linearity rules, worked examples and finding the constant C.

• Definite Integral

  Master the definite integral: Newton–Leibniz formula, area under a curve, signed areas, and area between two functions with worked examples.

• Integration by Substitution

  Master integration by substitution: 5-step method, quick linear formula, 5 worked examples including definite integrals, and a table of key results.

• Area Accumulation Function

  Master the area accumulation function: Fundamental Theorem, variable upper and lower limits, computing values, and finding extreme points with examples.

• Integrating a Ratio of Polynomials

  Master integrating ratios of polynomials: long division, Horner method, f-prime over f rule, and 5 worked examples including definite integrals.

• Integrals of Trigonometric Functions

  Master integrals of sin, cos and tan: basic formulas, linear expressions, double-angle identities, substitution patterns and 8 worked examples.

• Complex Numbers – Part 1

  Master complex numbers: imaginary unit i, powers of i, the form a+bi, equality, addition, multiplication, conjugate, and division with worked examples.

• Complex Numbers – Part 2

  Master complex numbers part 2: modulus formula, Pythagorean distance, circle equations, conjugate reflection and inequalities in the complex plane.

• Complex Numbers — Part C

  Quadratic equations with complex roots: square roots of negatives, the discriminant, polynomial factorisation, and the Fundamental Theorem of Algebra.

• Complex Numbers – Part 4

  Master polar form of complex numbers: modulus, argument, Cartesian-to-polar conversions, a special angles reference table and 3 fully worked examples.

• Complex Numbers – Part 5

  Master De Moivre's formula: multiplication, division, powers and roots in polar form with 5 worked examples including cube roots and roots of unity.

• Geometric Vectors – Part 1

  Master geometric vectors part 1: definition, graphical representation, notation, magnitude, equal vectors, zero vector, opposite and unit vectors.

• Geometric Vectors – Part 2

  Master geometric vector operations: triangle rule, parallelogram rule, polygon chain, subtraction and scalar multiplication with worked examples.

• Geometric Vectors – Part 3

  Master vectors in a coordinate system: components, position vector, magnitude, unit vectors, distance between points and parallel vector conditions.

• Geometric Vectors – Part 4

  Master the dot product: algebraic and geometric definitions, perpendicularity, angle between vectors, vector projection and worked examples.

• Geometric Vectors – Part 5

  Apply vectors to geometry: midpoint formula, section formula, centroid, proof techniques for parallel and perpendicular lines and parallelograms.

• Algebraic Vectors – Part 1

  Master algebraic vectors in 3D: ordered n-tuples, standard unit vectors, addition, scalar multiplication, magnitude norm and unit vector formula.

• Algebraic Vectors – Part 2

  Master dot product and cross product in 3D: definitions, perpendicularity, angle formula, parallelism and area of triangle and parallelogram.

• Algebraic Vectors – Part 3

  Master lines and planes in 3D space: parametric and symmetric line equations, plane equation, three-point plane and point-to-plane distance formula.

• Function Graph and Its Derivative

  Interactive GeoGebra tool exploring the relationship between a function graph and its derivative, illustrating increasing and decreasing intervals.

• Function Graph and Derivative – Interactive Tool

  Interactive GeoGebra tool exploring the relationship between a function graph and its derivative under shift operations and concavity analysis.

• The Relationship Between a Function and Its Derivative – Identify f, f', f''

  Interactive GeoGebra activity: identify which of three colored graphs represents the original function, its derivative and second derivative.

• Function Graph and Derivative – Dynamic Applet

  Explore how the derivative changes as the tangent line moves: increasing and decreasing intervals, maximum, minimum and inflection points interactively.

• Mathematical Induction – Principles and Structure

  Master mathematical induction: the three steps of base, inductive step and conclusion, the domino analogy and why examples alone are not a proof.

• Mathematical Induction – Pedagogical Insights

  Deep pedagogical guide to induction: base step vs examples, inductive hypothesis, independence of steps and a clear proof-writing template for students.

• Algebraic Technique A – Order of Operations

  Master order of operations with BODMAS: brackets, powers, multiplication, division, addition and subtraction, with six fully worked examples.

• Algebraic Technique B – What Is an Equation?

  Understand equations: definition, structure, solution, balance principle and types of linear and quadratic equations with complete worked examples.

• Algebraic Technique C – Dividing by the Coefficient

  Solve equations by dividing by the coefficient: positive, negative and fractional coefficients with ten fully worked examples and verification.

• Algebraic Technique D – Expanding Brackets

  Master the distributive law: expanding brackets, minus before brackets, multiplying two bracketed expressions and special algebraic identities.

• Algebraic Technique E – Like Terms and Transposing

  Collect like terms and transpose sides to solve equations: step-by-step method combining bracket expansion, collection and division by coefficient.

• Algebraic Technique F – Substitution

  Evaluate algebraic expressions by substitution: single variable, negative numbers in brackets, two-variable expressions and real-world formulas.

• Algebraic Technique G – Factorisation

  Master factorisation by extracting the greatest common factor: numeric GCF, variable GCF and combined examples with step-by-step verification.

• Algebraic Technique H – Systems of Equations

  Solve systems of two equations using substitution and elimination methods: four fully worked examples, comparison table and answer verification.

• Algebraic Technique I – Quadratic Equations

  Solve quadratic equations using the quadratic formula, factorisation, square roots and common factor; includes discriminant analysis and Vieta formulas.

• Algebraic Technique J – Percentages

  Master percentages: conversions, finding a percentage of a quantity, percentage increase and discount, and a useful reference table of common values.

• Algebraic Technique K – Ratio and Proportion

  Learn ratios and proportion: equivalent ratios, dividing a quantity, cross-multiplication, direct and inverse ratios with six fully worked examples.

• Mathematical Induction on Finite Sums – Proofs | Grade 11

  Learn to prove sum formulas with mathematical induction. Full proofs for natural numbers, squares, and geometric series with interactive worked examples.

• Infinite Geometric Series – Convergence Explained | Grade 11

  Understand infinite geometric series: convergence condition, sum formula, visual intuition, and common mistakes. Clear explanations for Grade 11 students.

• Common Mistakes in Mathematical Induction Proofs | Grade 11

  Identify and correct the five most common errors in induction proofs. Covers skipped base steps, faulty transitions, and the classic horse paradox.

• Visual Proofs in Mathematical Induction Explained | Grade 11

  Explore induction through visual proofs: domino chains, staircase diagrams, and triangle subdivisions that make the logic of induction transparent.

• Mathematical Induction Practice – Worked Solutions

  Practice induction with seven fully worked exercises covering inequalities, finite sums, geometric series, and identifying faulty proofs for Grade 11.

• Mathematical Induction – Inquiry and Deep-Thinking Questions

  Open-ended inquiry questions on induction: base step, inductive step, geometric cases, and creative process-based thinking for Grade 11 students.

• Mathematical Induction Mind Map – Deep Inquiry | Grade 11

  A visual mind map of mathematical induction covering base step, inductive step, conclusion, inductive hypothesis, and common failure cases for Grade 11.

• Mathematical Induction – Basic and Intermediate Exercises

  Twenty graded induction exercises for Grade 11: sum formulas, inequalities, divisibility, sequences, and Fibonacci at basic and intermediate levels.

• Mathematical Induction – Full Solutions: Basic Exercises

  Ten fully worked basic induction proofs: natural number sums, even sums, squares, inequalities, factorial bounds, sequences, and divisibility.

• Mathematical Induction – Full Solutions: Intermediate Exercises

  Ten fully worked intermediate induction proofs: products, cube sums, telescoping series, Fibonacci identity, and odd-number products for Grade 11.

• Factoring Expressions – All Methods | Grade 11 Algebra

  Master five factoring methods: common factor, special identities, trinomials, leading-coefficient trinomials, and grouping. Worked examples and exam tips.

• Algebraic Fractions – Simplify, Multiply, Add | Grade 11

  Work with algebraic fractions: find the domain, simplify, multiply, divide, add, and subtract. Covers complex fractions with 11 worked examples.

• Bi-quadratic Equations – Substitution Method | Grade 11

  Solve bi-quadratic equations using the substitution t=x². All solution cases (0–4) are covered with four fully worked examples and a summary table.

• Equations with Variables in the Denominator | Grade 11

  Solve equations with variables in the denominator: find the domain, clear denominators, solve, and check for extraneous solutions. Five examples.

• Irrational Equations – Radical Equations | Grade 11 Algebra

  Solve radical equations: find the domain, isolate the radical, square both sides, and verify solutions. Includes five fully worked examples.

• Nine Common Mistakes in Mathematical Induction Proofs

  Nine common student mistakes in induction proofs: skipping the base step, circular reasoning, unjustified jumps, and poor structure with corrections.

• Seven Common Mistakes in Mathematical Induction Proofs

  Seven common induction errors with pedagogical explanations: skipped base, circular reasoning, unjustified jumps, and faulty hypotheses — with corrections.

• Perfect Writing Template for a Mathematical Induction Proof

  A five-step writing template for induction proofs: state the claim, base step, inductive hypothesis, inductive step, and concluding sentence.

• Vectors – Introduction and Basic Properties | Grade 11

  Introduction to vectors: definition, geometric interpretation, addition, scalar multiplication, and vector space axioms. For Grade 11 students.

• Linear Dependence and Unique Representation | Grade 11

  Understand linear dependence and unique vector representation: linear combinations, spanning sets, independence conditions, and geometric applications.

• The Scalar Dot Product of Vectors | Grade 11 Math

  Master the dot product: definition, geometric interpretation, angles, projections, and proofs. Standard formulas with geometric and physical applications.

• Algebraic Representation of Vectors in Space | Grade 11

  Complete guide to vectors in 3D space: coordinates, parametric lines, plane equations, distances between points and planes, and angles between lines.

• Applications in Spatial Geometry | Grade 11 Math

  Apply vectors to spatial geometry: distances, angles, areas, and volumes in cylinders, cones, spheres, prisms, and pyramids. Pedagogical notes included.

• Parabola – Comprehensive Summary | Grade 11 Algebra

  Comprehensive parabola guide: geometric definition, canonical equation, symmetry, tangent formula, locus problems, and the optical reflection property.

• Ellipse – Comprehensive Summary | Grade 11 Algebra

  Comprehensive ellipse guide: geometric definition, canonical equation, symmetry, relative positions with lines and circles, and locus problems.

• Hyperbola – Full Visual Explanation | Grade 11 Algebra

  Visual guide to the hyperbola: definition, canonical equation, asymptotes, symmetry, relative positions, and locus problems for Grade 11 students.

• Hyperbola – Introduction and Basic Concepts | Grade 11

  Introduction to the hyperbola: definition, diagrams, parameters a/b/c, canonical equations for both axis types, with a worked example and exam tips.

• Hyperbola – Parameters and Formulas | Grade 11 Algebra

  Hyperbola parameters and formulas: eccentricity, asymptote equations, latus rectum, distances to foci, and a comprehensive formula table for Grade 11.

• Hyperbola – Building the Equation from Data | Grade 11

  Six worked examples for building the hyperbola equation from given foci, vertices, eccentricity, asymptotes, and points. Includes parameter identification.

• Hyperbola and Line – Intersections and Tangents | Grade 11

  Hyperbola and line: find intersections, tangency conditions, tangent equations at a point, and the special case of a line parallel to an asymptote.

• Hyperbola – Advanced Topics | Grade 11 Algebra

  Advanced hyperbola topics: conjugate hyperbola, point position test, optical property, chord length, translated hyperbola, and a complete formula table.

• Hyperbola – Practice Exercises | Grade 11 Algebra

  Hyperbola practice: identify parameters from equations, sketch graphs, build equations from given data, and find asymptotes. Self-practice included.

• Hyperbola – Advanced Practice Exercises | Grade 11

  Advanced hyperbola practice: intersections, tangents, distances to foci, conjugate hyperbola, point position, and matriculation-style summary questions.

• Statistics – Basic Concepts | Grade 11 Mathematics

  Introduction to statistics: population vs sample, statistical variables, and the three variable types — qualitative, discrete, and continuous. Grade 11.

• Statistics – Grouping Data in a Frequency Table | Grade 11

  Frequency and grouped frequency tables: class width, midpoints, true boundaries, rules for building grouped tables, and when to use each type.

• Statistics – All Frequency Types | Grade 11 Mathematics

  All four frequency types: absolute, relative, cumulative, and relative cumulative frequency. Includes a full worked example and application questions.

• Statistics – Types of Diagrams | Grade 11 Mathematics

  Types of statistical diagrams: bar chart, histogram, pie chart, frequency polygon, and cumulative ogive. Includes construction rules and exam tips.

• Statistics – Arithmetic Mean | Grade 11 Mathematics

  Arithmetic mean for raw data, frequency tables, and grouped data. Covers weighted mean, mean properties, and five worked examples. Grade 11.

• Statistics – Median and Mode | Grade 11 Mathematics

  Median and mode: formulas for raw data, frequency tables, and grouped data. Comparison of all three measures of centre with worked examples.

• Statistics – Measures of Spread | Grade 11 Mathematics

  Measures of spread: range, variance, and standard deviation. Includes deviations, shortcut formulas, a full frequency-table example, and properties.

• Statistics – Percentiles and Quartiles | Grade 11 Math

  Percentiles, quartiles, and IQR: definitions, computation for discrete and grouped data, outlier detection with the 1.5×IQR rule, and deciles.

• Statistics – Comprehensive Practice | Grade 11 Math

  Comprehensive statistics practice: variable classification, frequency tables, measures of centre, grouped data, spread, and quartiles with full solutions.

• Combinatorics – Basic Counting Principles | Grade 11

  Basic counting principles: multiplication rule, addition rule, complement principle, and tree diagrams. Nine worked examples with exam tips.

• Combinatorics – Factorial and Permutations | Grade 11

  Factorial and permutations: definition, permutations with conditions (adjacent or not), repeated elements, circular arrangements, and exam tips.

• Combinatorics – Partial Permutations P(n,k) | Grade 11

  Partial permutations P(n,k): definition, formula, six worked examples including conditions on digits and letters, and comparison with combinations.

• Combinatorics – Combinations C(n,k) | Grade 11 Math

  Combinations C(n,k): formula, properties, Pascal's triangle, and six examples covering committees, polygon diagonals, and the shortest grid paths.

• Combinatorics – Combinations with Repetitions | Grade 11

  Combinations with repetitions, stars and bars method, minimum conditions, group division, and counting equation solutions. Includes a full summary table.

• Newton's Binomial Theorem | Grade 11 Mathematics

  Newton's binomial theorem: expansion formula, finding specific coefficients, important identities, alternating sums, and the difference (a-b)^n.

• Statistics – Chi-Square, Phi, Cramér's V, Lambda | G11

  Correlation measures for nominal variables: chi-square, Phi, Cramér's V, and Lambda. Includes cross-tabulation, expected frequencies, and worked examples.

• Statistics – Spearman Correlation | Grade 11 Math

  Spearman's rank correlation: formula, full worked example, handling tied ranks, interpretation table, and comparison with Pearson's correlation.

• Statistics – Eta and Pearson Correlation | Grade 11

  Eta and Pearson correlation for interval variables: formulas, worked examples, coefficient of determination r², and comparison between the two measures.

• Arithmetic Sequence – Basics and Common Difference

  Introduction to arithmetic sequences: definition, common difference d, real-life examples, exercises for completing sequences, and general term formula.

• Arithmetic Sequence – Sum of Terms | Grade 11 Math

  Sum of arithmetic sequence terms: two formulas, a basic example, six exercises, and two word problems (savings and ladder). Grade 11 algebra.

• Arithmetic Sequence – General Term from Sum Formula

  Finding the general term of an arithmetic sequence from its sum formula. Includes three worked examples, an identification rule, and exercises.

• Arithmetic Sequence – Sum of Last Terms | Grade 11

  Computing the sum of last terms in an arithmetic sequence: two methods (subtracting sums and new sequence), a full example, and comparison table.

• Arithmetic Sequence – Even and Odd Positions | G11

  Sum of even- and odd-positioned terms in arithmetic sequences: key observation, two cases (even/odd n), a summary table, and three exercises.

• Powers – Definition, Operations, and Laws | Grade 10

  Powers: definition, base and exponent, special powers, order of operations, and six laws of exponents. Includes summary table and exam tips.

• Roots – Definition, Operations, and Laws | Grade 10

  Roots: definition, even/odd order roots, order of operations, five laws of roots, inserting and extracting factors, and rationalising the denominator.

• Inequalities — First-Degree, AND/OR Systems, and Quadratic

  Practice inequalities: first-degree, AND/OR systems, and second-degree (parabola method). Step-by-step explanations with worked examples.

• Pre-Calculus – Introduction to Functions | Grade 11

  Introduction to functions: domain and range, axis intercepts, positivity and negativity, intervals of increase and decrease, and local extrema.

• Pre-Calculus – Even and Odd Functions | Grade 11

  Even and odd functions: definitions, geometric symmetry, step-by-step testing, detailed examples, comparison table, and product/sum properties.

• Pre-Calculus – Transformations of Functions | G11

  Transformations of functions: vertical and horizontal shifts, reflections about the x- and y-axis, a summary table, and a combined step-by-step example.

• Pre-Calculus – Power Functions and Polynomials | G11

  Power functions and polynomial analysis: even vs odd degree, root multiplicity, end behaviour, and a step-by-step qualitative investigation method.

• Pre-Calculus – Absolute Value Function | Grade 11

  Absolute value function: definition, V-shaped graph, transformations, removing absolute value, equations and inequalities, and the graph of y=|f(x)|.

• Exponential Functions – Definition and Properties

  Exponential functions: definition, two types (growth and decay), shared properties, graphical comparison, the reflection relationship, and key values.

• Exponential Equations – Four Types | Grade 11 Math

  Exponential equations: four types — same base, common base, hidden quadratic (substitution), and logarithms. Includes examples and exam tips.

• Exponential Inequalities | Grade 11 Math

  Exponential inequalities: the key sign-flip rule, three types (a>1, a<1, hidden quadratic substitution), a summary table, and exam tips for grade 11.

• Percentages – Increases and Decreases | Grade 9 Math

  Percentages: calculating a percentage of a quantity, price increases and decreases, finding the original price, percentage change, and successive changes.

• Exponential Growth and Decay – Introduction | G11

  Exponential growth and decay: formula f(t)=f(0)·qᵗ, growth/decay factor q, graph properties, and three worked examples (population, car, interest).

• Exponential Growth – Finding q, t, f(0) | Grade 11

  Finding unknowns in exponential growth/decay: f(t), initial value f(0), factor q, and time t using logarithms. Six worked examples included.

• Half-Life and Doubling Time | Grade 11 Mathematics

  Half-life and doubling time: definitions, formulas, worked examples, using the rule of 70, and a comparison summary table for exponential processes.

• Geometric Sequence – Introduction | Grade 11 Math

  Geometric sequences: definition, common ratio q, recurrence relation, general term formula, finding a term position, and the geometric condition b²=ac.

• Geometric Sequence – Sum of Terms | Grade 11 Math

  Sum of a geometric sequence: formula, derivation, equivalent form, formula using the last term, and inverse problems. Includes three worked examples.

• Geometric Sequence – General Term from Sum | G11

  Finding the general term aₙ from the sum formula Sₙ: four-step method, two worked examples, and the special case when the formula is invalid for n=1.

• Geometric Sequence – Sum of Last Terms | Grade 11

  Sum of the last k terms of a geometric sequence: three methods — subtracting sums, new sequence, and the qⁿ⁻ᵏ·Sₖ relationship. Includes a worked example.

• Geometric Sequence – Even and Odd Positions | G11

  Sum of even- and odd-positioned terms in a geometric sequence: key observation (ratio q²), formulas, two worked examples, and a useful sum relationship.

• Geometric Sequence – Infinite Series | Grade 11

  Infinite geometric series: convergence condition, sum formula, worked examples, recurring decimal application, inverse problems, and full formula summary.

• Normal Distribution – Z-Score Introduction | G11

  Z-score: definition, formula, sign meaning, key properties (unit-free, mean=0, SD=1), inverse formula, and effect of linear transformations.

• Normal Distribution – Properties | Grade 11 Math

  Normal distribution: properties, discrete vs continuous variables, bell-curve parameters, symmetry rule, area as probability, and the 68-95-99.7 rule.

• Normal Distribution – Z-Table | Grade 11 Math

  Z-table: how to read it, convert decimal to percentage, find area to the right, area between two Z-scores, and find Z from a given percentage.

• Normal Distribution – Raw Score to Probability | G11

  Converting raw scores to probability: three-step process, four worked examples (below, above, between, symmetric), and a question-type summary table.

• Normal Distribution – Inverse Problems | Grade 11

  Inverse normal problems: finding a raw score from a percentage (below or above), percentiles, and finding a missing mean or standard deviation.

• Normal Distribution – Transformations | Grade 11

  Transformations on a normal distribution: adding/subtracting a constant, multiplying/dividing by a constant, and properties of asymmetric distributions.

• Sampling Distribution – Basic Concepts | Grade 11

  Sampling basics: population vs sample, parameter vs statistic, why statistics are random variables, and the definition of sampling distribution.

• Sampling Distribution – Sample Mean and CLT | G11

  Sampling distribution of the sample mean: properties, standard error, CLT, when to apply it, Z-score formula, and a detailed worked example.

• Normal Approximation to Binomial | Grade 11 Math

  Normal approximation to the binomial: conditions (np≥5, nq≥5), continuity correction, two worked examples, and approximation for sample proportions.

• Analytic Geometry – Slope and y-Intercept | Grade 10

  Identifying slope m and y-intercept b from an equation and from a graph: reading coefficients, converting to slope-intercept form, and a summary table.

• Analytic Geometry – Slope Formula | Grade 10 Math

  Slope of a line: definition, rise-over-run formula, graphical illustration, five worked examples including undefined slope, and comparing steepness.

• Analytic Geometry – Line from Two Points | Grade 10

  Finding a line equation from two points: two-step method, four worked examples, special cases (horizontal and vertical), and the two-point formula.

• Analytic Geometry – Line from Point and Slope | G10

  Finding a line equation from a point and slope: direct substitution method, point-slope formula, four worked examples, and lines through the origin.

• Analytic Geometry – Distance Formula | Grade 10

  Distance between two points: distance formula derivation from Pythagoras, four worked examples, common Pythagorean triples, and distance from the origin.

• Analytic Geometry – Distance Along Axes | Grade 10

  Distance between points on horizontal or vertical lines: simplified formulas, points on axes, case identification table, and quick practice exercises.

• Analytic Geometry – Midpoint Formula | Grade 10

  Midpoint formula and finding the other endpoint: formula, graphical illustration, three examples for each direction, and the equation-writing method.

• Analytic Geometry – Parallel and Perpendicular | G10

  Parallel and perpendicular lines: conditions on slopes, three worked examples, special cases (horizontal and vertical), and a summary table.

• Analytic Geometry – Quadrilaterals | Grade 10 Math

  Analytic geometry proofs of quadrilateral properties: tools (slope, distance, midpoint), four types of quadrilaterals, and a property comparison table.

• Analytic Geometry – Trapezoid and Kite | Grade 10

  Trapezoid and kite in analytic geometry: definitions, proof methods, properties of the isosceles trapezoid and kite, and a full worked example.

• Analytic Geometry – Circle Equation | Grade 10

  Circle equation in standard form: identifying centre and radius from the equation or a graph, converting from expanded form, and writing circle equations.

• Analytic Geometry – Point and Circle | Grade 10

  Determining a point's position relative to a circle: substitution method, worked examples, the distance method, and a summary table of the three cases.

• Analytic Geometry – Circle and Axes | Grade 10

  Finding intersection points of a circle with the axes: substituting y=0 or x=0, using the discriminant, and a full example with four intersection points.

• Analytic Geometry – Line and Circle | Grade 10

  Intersection of a line with a circle: substitution method, three worked examples (secant, tangent, no intersection), and computing the chord length.

• Analytic Geometry – Tangent Line | Grade 10 Math

  Tangent line to a circle: tangent formula at a point, two worked examples, the perpendicularity method, and horizontal and vertical tangents.

• Analytic Geometry – Two Circles | Grade 10 Math

  Two circles: five possible configurations, the distance-of-centres condition, finding intersection points by subtraction, and tangent circle examples.

• Analytic Geometry – Circle Through 3 Points | G10

  Circle through 3 points using a system of equations; circle with a given diameter (midpoint and half-diameter); inscribed angle on diameter = 90°.

• Directed Numbers – Introduction | Grade 7 Math

  Introduction to directed numbers: positive and negative numbers, zero, absolute value, opposite numbers, and how to write negatives in expressions.

• Directed Numbers – Comparing and Ordering | Grade 7

  Comparing and ordering directed numbers: three comparison rules, negative vs positive, ordering on the number line, and a summary table of all cases.

• Directed Numbers – Adding Directed Numbers | Grade 7

  Adding directed numbers: the number-line movement model, same-sign rule, different-sign rule, a rules table with examples, and real-life analogies.

• Directed Numbers – Subtracting Directed Numbers | G7

  Subtracting directed numbers: the golden rule (subtract = add the opposite), subtracting a negative, a quick conversion table, and practice problems.

• Directed Numbers – Multiplying Directed Numbers | G7

  Multiplying directed numbers: sign rules, sign table, multiply by zero, powers of negatives, the (−3)² vs −3² distinction, and products of many numbers.

• Directed Numbers – Dividing Directed Numbers | G7

  Dividing directed numbers: sign rules (same as multiplication), fractions with signs, division by zero, and a summary table of all four operations.

• Pre-Calculus Graph Reading – Basics | Grade 11

  Reading a function graph: domain, range, x-intercepts (roots), y-intercept, and the sign of the function. Includes a summary table of all five concepts.

• Pre-Calculus – Monotonicity | Grade 11 Math

  Monotonicity of functions: increasing, decreasing, and constant intervals; how to identify them from a graph; and a full worked example with extrema.

• Pre-Calculus – Extrema | Grade 11 Math

  Extrema of functions: maximum and minimum points, local vs global extrema, how to identify them from a graph, notation, and a full worked example.

• Pre-Calculus – Vertical Asymptote | Grade 11 Math

  Vertical asymptote: definition, graph appearance, behaviour near the asymptote, four cases, how to identify one, and the classic y=1/x example.

• Pre-Calculus – Horizontal Asymptote | Grade 11

  Horizontal asymptote: definition, graph appearance, three behaviour types at the ends, different asymptotes left and right, and the y=1/x example.

• Pre-Calculus – Full Graph Analysis | Grade 11

  Full graph analysis: nine-point checklist, complete worked example covering domain, range, intercepts, sign, monotonicity, extrema, and asymptotes.

• Special Functions – The Parabola y = x² | Grade 10

  The parabola y = x²: graph, values table, properties (domain, range, vertex, symmetry), monotonicity, direction, and the meaning of the vertex.

• Special Functions – Parabola Transformations | G10

  Parabola family y=a(x−h)²+k: vertical and horizontal shifts, stretch and compress, flip, vertex form, and the meaning of the three parameters.

• Special Functions – Square Root Function | Grade 10

  Square root function y=√x: graph, values table, properties (domain, range, monotonicity), relationship to the parabola, and domain restrictions.

• Special Functions – Square Root Transformations | G10

  Square root family y=√(x−h)+k: vertical and horizontal shifts, domain changes, reflections over x and y axes, starting point, and summary table.

• Special Functions – Reciprocal Function | Grade 10

  Reciprocal function y=1/x: hyperbola graph, properties, sign, special behaviour near zero and at infinity, and point symmetry about the origin.

• Special Functions – Graphing 1/f(x) | Grade 10

  How to graph 1/f(x) from f(x): six rules covering fixed points, vertical asymptotes, sign, monotonicity, plus a summary table and two worked examples.

• Special Functions – Absolute Value | Grade 10 Math

  Absolute value function y=|x|: V-shape, domain, range, vertex, monotonicity, shifts, flip, and applications including distance and equations.

• Domain of Definition – Introduction | Grade 10 Math

  Introduction to domain of definition: real-life examples, why it matters, three major restrictions, function type table, and interval notation.

• Domain of Square Root Functions | Grade 10 Math

  Finding the domain of square root functions: the central rule, linear and quadratic expressions under the root, higher degrees, and ten worked examples.

• Domain of Rational Functions | Grade 10 Math

  Domain of rational functions: the central rule, linear and quadratic denominators, factored denominators, and the cancellation special case.

• Domain – Root and Rational Combined | Grade 10

  Domain when root and rational are combined: all conditions must hold, root in denominator, root in numerator, two roots, and complex combinations.

• Domain of Trigonometric Functions | Grade 11

  Domain of trigonometric functions: sin and cos defined everywhere, tan restrictions, roots with trig, rational with trig, and complex combinations.

• Domain of Exponential Functions | Grade 11 Math

  Domain of exponential functions: always defined on ℝ, combined with roots or rational functions, and complex combinations with fifteen examples.

• Domain of Logarithmic Functions | Grade 11 Math

  Domain of logarithmic functions: expression inside log must be > 0, quadratic cases, combined with root or rational, seventeen worked examples.

• Geometry Theorems – Angles and Triangles | G9

  Grade 9 geometry theorems: supplementary angles (180°), vertical angles, triangle angle sum, exterior angle, sides and angles, and triangle inequality.

• Geometry Theorems – Isosceles Triangle | Grade 9

  Isosceles triangle theorems: equal base angles, coincident altitude/median/bisector, five theorems for identification, and a worked example.

• Geometry Theorems – Midsegment and Parallels | G9

  Midsegment and parallel lines: midsegment properties, bisecting theorem, corresponding, alternate, and co-interior angles, and tests for parallel lines.

• Geometry Theorems – Parallelogram | Grade 9 Math

  Parallelogram properties: opposite sides equal, opposite angles equal, diagonals bisect each other; includes rhombus and rectangle special cases.

• Geometry Theorems – Trapezoid and Kite | Grade 9

  Trapezoid and kite theorems: isosceles trapezoid base angles and equal diagonals, midsegment formula (a+b)/2, and kite main diagonal properties.

• Geometry – Medians, Bisectors, Altitudes | Grade 9

  Triangle special lines: medians to centroid (ratio 2:1), angle bisectors to incentre, perpendicular bisectors to circumcentre, altitudes to orthocentre.

• Geometry – Circle: Central and Inscribed Angles | G9

  Circle theorems: central angle ↔ arc ↔ chord equivalence, inscribed angle = half the central angle, and inscribed angle on a diameter = 90°.

• Geometry – Tangents and Chords in a Circle | G9

  Tangent and chord theorems: tangent ⊥ radius, tangent-chord angle, two equal tangents from one point, equal chords equidistant from centre, two circles.

• Geometry – Pythagorean Theorem | Grade 9 Math

  Pythagorean theorem (a²+b²=c²), converse for identifying right triangles, median to hypotenuse theorem, and 30°-60°-90° triangle side ratios.

• Geometry – Thales' Theorem and Similarity | Grade 9

  Thales' theorem, extended and converse forms, angle bisector theorem, AA/SAS/SSS similarity, and similar triangle properties including area ratio k².

• Geometry – Chords, Secants, Polygons | Grade 9 Math

  Circle theorems: intersecting chords, two secants, secant-tangent products; geometric mean in right triangles; polygon angle sums; cyclic quadrilaterals.

• Plane Geometry – Geometric Proofs Introduction | G9

  Introduction to geometric proofs: structure of a proof (statement + justification), useful claims (common side, vertical angles), and exam tips.

• Plane Geometry – Angles Summary | Grade 9 Math

  Angles summary: types by size (acute, right, obtuse, straight), supplementary and vertical angles, complementary angles, and angles around a point.

• Plane Geometry – Triangles Summary | Grade 9 Math

  Triangle summary: types by sides and angles, isosceles properties, angle sum 180°, exterior angle, side-angle relationship, and triangle inequality.

• Plane Geometry – Triangle Congruence Summary | G9

  Triangle congruence summary: the four theorems (SAS, ASA, SSS, SSA), the SSA warning, and tips for finding common sides, angles, and vertical angles.

• Plane Geometry – Sides and Angles in a Triangle | G9

  Sides and angles in a triangle: longer side opposite larger angle, isosceles implication, angle sum 180°, exterior angle, and triangle inequality.

• Plane Geometry – Right Triangle and Pythagoras | G9

  Right triangle summary: Pythagorean theorem a²+b²=c², its converse, median to hypotenuse = ½ hypotenuse, and 30°-60°-90° triangle side ratios.

• Plane Geometry – Area and Perimeter Summary | G9

  Area and perimeter formulas: triangle, parallelogram, rhombus, trapezoid, and circle. All formulas usable without proof in the matriculation exam.

• Plane Geometry – Quadrilateral Family Summary | G9

  Quadrilateral family summary: hierarchy of shapes, kite (main diagonal properties), isosceles trapezoid, and trapezoid midsegment formula (a+b)/2.

• Plane Geometry – Parallel Lines Summary | Grade 9

  Parallel lines summary: corresponding angles equal, alternate angles equal, co-interior angles sum to 180°, and three tests for proving lines parallel.

• Plane Geometry – Parallelogram Summary | Grade 9

  Parallelogram properties and identification: opposite sides, angles, diagonals (bisect each other), adjacent angles 180°, and five identification tests.

• Plane Geometry – Rectangle, Rhombus, Square | G9

  Rectangle, rhombus, and square: special properties, identification tests, and the square as the combination of rectangle and rhombus properties.

• Plane Geometry – Triangle Midsegment Summary | G9

  Triangle midsegment summary: the midsegment is parallel to the third side and equals half its length, plus related parallel-bisecting theorems.

• Plane Geometry – Thales and Similarity Summary | G9

  Thales' theorem, extended form, angle bisector theorem (BD/DC = AB/AC), AA/SAS/SSS similarity, and similar triangle side, perimeter, area ratios.

• Plane Geometry – Circle: Angles and Arcs | Grade 9

  Circle summary: basic concepts, inscribed angle = ½ central angle, inscribed angle on diameter = 90°, tangent properties, and cyclic quadrilateral.

• Circle – Perpendicular Bisector and Circumscribed | G9

  Perpendicular bisector theorem and its converse: a point is equidistant from two endpoints exactly when it lies on their perpendicular bisector.

• Circle – Circumscribed Circle of a Triangle | G9

  Circumscribed circle of a triangle: the three perpendicular bisectors meet at the circumcentre, with full proof and location by triangle type.

• Circle – Altitudes and Inscribed Circle | Grade 9

  Altitudes meet at the orthocentre; the four special triangle points (centroid, incentre, circumcentre, orthocentre); inscribed vs circumscribed circle.

• Circle – Cyclic and Tangential Quadrilaterals | G9

  Cyclic quadrilateral (opposite angles = 180°) and tangential quadrilateral (opposite sides equal sums), with proofs, comparison, and worked examples.

• Circle – Regular Polygons and Their Circles | Grade 9

  Regular polygons: interior angle formula (n−2)×180°/n; circumscribed and inscribed circles share the same centre; apothem; real-life examples.

• Circle – Central Angles, Arcs, and Chords | G9

  Central angles, arcs, and chords: triple equivalence (equal angles ↔ arcs ↔ chords), proofs using SAS, summary table, and real-life example.

• Circle – Chord Distances from the Centre | Grade 9

  Chord distance theorems: equal chords are equidistant; longer chord is closer to centre; the diameter has distance zero. Proof using Pythagoras.

• Circle – Perpendicular from Centre to Chord | G9

  Perpendicular from centre to chord bisects the chord, arc, and central angle. Converse theorem, finding the centre, and a numerical worked example.

• Circle – Inscribed Angle and Central Angle | G9

  Inscribed angle = ½ central angle on the same arc. Proof for the simple case, key conclusions, equal inscribed angles on the same arc, and examples.

• Circle – Inscribed Angle on a Diameter | Grade 9

  Inscribed angle on a diameter = 90°, its converse, connection to the right triangle, applications, real-life examples, and a worked numerical exercise.

• Circle – Interior and Exterior Angles | Grade 9

  Circle angle types: interior angle with vertex inside = ½(sum of arcs); exterior angle with vertex outside = ½(difference of arcs). Proofs and examples.

• Circle – Tangent to a Circle | Grade 9 Math

  Tangent to a circle: tangent ⊥ radius, two equal tangents from an external point, bisector of the angle between tangents, and tangent-chord angle.

• Circle – Two Circles: Common Chord and Tangency | G9

  Two intersecting circles: line of centres bisects and is perpendicular to the common chord. Tangent circles: external and internal tangency distances.

• Circle – Power of a Point Theorems | Grade 9 Math

  Three power-of-a-point theorems: intersecting chords (AP·PB=CP·PD), two secants, secant-tangent (PA·PB=PT²), with proofs, examples, and applications.

• Trigonometry Fundamentals – Angles | Grade 10

  Angle types: acute, right, obtuse, straight, reflex. Complementary, supplementary, and vertical angles. Degrees-to-radians conversion with examples.

• Trigonometry Fundamentals – Triangle Family | G10

  Triangle family: angle sum 180°, classification by sides (equilateral, isosceles, scalene) and angles (acute, right, obtuse), isosceles properties.

• Trigonometry Fundamentals – Right Triangle | G10

  Right triangle: hypotenuse opposite 90°, complementary acute angles, median to hypotenuse, and side ratios for 45-45-90 and 30-60-90 triangles.

• Trig Fundamentals – Pythagorean Theorem | Grade 10

  Pythagorean theorem a²+b²=c²: finding the hypotenuse and legs, common Pythagorean triples (3-4-5, 5-12-13), and converse test with examples.

• Trig Fundamentals – Quadrilateral Family | Grade 10

  Quadrilateral family: parallelogram, rectangle (equal diagonals), rhombus (perpendicular diagonals), square, and trapezoid with area formulas.

• Trig in Right Triangle – sin, cos, tan | Grade 10

  sin, cos, and tan as side ratios: SOH-CAH-TOA memory trick, special values for 30°, 45°, 60°, and the complementary angle identity sin(α) = cos(90°−α).

• Trig Applications – Isosceles and Equilateral | G10

  Trigonometry applied to isosceles and equilateral triangles: altitude and base formulas using sin and cos, equilateral triangle height and area formulas.

• Trig Applications – Quadrilaterals | Grade 10 Math

  Trigonometry in quadrilaterals: rectangle diagonal, rhombus diagonals and area, square diagonal, trapezoid altitude and area using sin and Pythagoras.

• Unit Circle – Introduction and Motivation | Grade 10

  Why the unit circle extends sin and cos beyond 90° to any angle, negative angles, and beyond 360°. Coordinates on the unit circle are sin and cos.

• Unit Circle – Defining sin and cos | Grade 10 Math

  Unit circle definition of sin and cos as coordinates of a point; special-angle values; quadrant sign rules — positive above x-axis for sin, right for cos.

• Unit Circle – Periodicity and Symmetry | Grade 10

  Period 2π for sin and cos; sin is odd sin(−α)=−sin(α); cos is even cos(−α)=cos(α); symmetry about x-axis, y-axis, and origin on the unit circle.

• Unit Circle – First Trig Identity sin²+cos²=1 | G10

  The Pythagorean identity sin²α + cos²α = 1: proof from the unit circle, derived formulas, and a worked example finding cos from sin in a given quadrant.

• Unit Circle – Graphs of sin and cos | Grade 10

  Graphs of y=sin(x) and y=cos(x): domain, range [−1,1], period 2π, zeros, maxima and minima, and the π/2 phase shift relationship between the two graphs.

• Unit Circle – Trig Graph Transformations | Grade 10

  General form A·sin(Bx+C)+D: amplitude |A|, period 2π/|B|, horizontal shift −C/B, vertical shift D. Rules, examples, and a complete worked exercise.

• Unit Circle – The tan(x) Function | Grade 10 Math

  The tan(x) function: definition as sin/cos, domain excluding π/2+πn (vertical asymptotes), period π, range ℝ, odd function, and special values.

• Trig Equations – Solving sin(ax+b) = m | Grade 10

  Solving sin(ax+b)=m: necessary condition |m|≤1, two-family general solution α=α₀+2πn and α=(π−α₀)+2πn, step-by-step algorithm, and a worked example.

• Trig Equations – Solving cos(ax+b) = m | Grade 10

  Solving cos(ax+b)=m: condition |m|≤1 is required, general solution uses ± because cosine is even (α=±α₀+2πn), and a worked example is included.

• Trig Equations – Solving tan(ax+b) = m | Grade 10

  Solving tan(ax+b)=m: always has a solution, one-family general solution α=α₀+πn, period π, comparison table with sin and cos, worked example.

• Trig Equations – Complex Equations | Grade 10 Math

  Complex trig equations: quadratic substitution (t=sin/cos), factoring out (never divide by a trig function), and using sin²+cos²=1 for conversion.

• Trig – Triangle Area S = ½ab·sin(γ) | Grade 10

  Triangle area formula S = ½ab·sin(γ): proof, three versions, special cases (90°, equilateral), and a worked example. Works for any triangle.

• Trigonometry – The Law of Sines | Grade 10 Math

  Law of sines: a/sin A = b/sin B = c/sin C = 2R. Proof using area formula, when to use (AAS, SSA), ambiguous case warning, and worked example.

• Trig – Finding Side or Angle in Right Triangle | G10

  Finding a side or angle in a right triangle: identify opposite/adjacent/hypotenuse, choose sin/cos/tan, solve. Use inverse functions for angles.

• Trigonometry – The Law of Cosines | Grade 10 Math

  Law of cosines: c²=a²+b²−2ab·cos(C). Three forms, generalises Pythagoras; use for SAS or SSS cases; angle formula and comparison with the sine rule.

• Trig Summary – Triangle Area S = ½ab·sinC | Grade 10

  Trigonometric area formula S=½ab·sinC: three forms, proof, when to use (SAS/ASA/SSS), worked examples, quadrilateral area, and additional area formulas.

• Trig Summary – The Unit Circle | Grade 10 Math

  Unit circle reference: definition, degrees-to-radians table, quadrant sign rules (ASTC), supplementary angle identities, parity, and periodicity.

• Trig Summary – Graphs of Trig Functions | Grade 10

  Graphs of sin, cos, and tan: properties table, transformations A·sin(Bx+C)+D (amplitude, period, shift), worked example, tips for sketching.

• Trig Summary – Trigonometric Identities | Grade 10

  Trigonometric identities: Pythagorean, angle sum/difference, double angle, power-reduction, and tangent identities with a worked proof example.

• Trig Summary – Trigonometric Equations | Grade 10

  Trig equations summary: general solutions for sin x=m, cos x=m, and tan x=m; ax+b form algorithm; equation types table; domain restrictions.

• Trig Summary – Law of Sines | Grade 10 Math

  Law of sines reference: theorem a/sinA=2R, when to use, ambiguous case (SSA) with decision criteria, three worked examples, and circumradius.

• Trig Summary – Law of Cosines | Grade 10 Math

  Law of cosines summary: c²=a²+b²−2ab·cosC; three forms; SAS and SSS cases; Pythagorean link; angle type from cosine sign; examples and comparison table.

• Trig Summary – Area S = ½ab·sinC | 5 Math Units

  Trig area S=½ab·sinC: three forms, proof, usage guide (SAS/ASA/SSS), three worked examples, and quadrilateral and parallelogram area formulas.

• Plane Geometry – Special Lines in a Triangle

  The four special lines of a triangle: median (centroid), altitude (orthocentre), angle bisector (incentre), and perpendicular bisector (circumcentre).

• Plane Geometry – Medians of a Triangle

  Medians: three meet at the centroid (2:1 ratio), each median divides the triangle into two equal areas; median to hypotenuse equals half the hypotenuse.

• Plane Geometry – Angle Bisectors of a Triangle

  Angle bisectors: locus equidistant from sides, three bisectors meet at incentre (inscribed circle), internal bisector theorem BD/DC = AB/AC, and converse.

• Plane Geometry – Perpendicular Bisectors

  Perpendicular bisectors: locus equidistant from endpoints, three bisectors meet at the circumcentre, OA=OB=OC, and comparison with the inscribed circle.

• Plane Geometry – Altitudes of a Triangle

  Triangle altitudes: definition, foot of altitude, orthocentre position changes with triangle type — inside (acute), at vertex (right), or outside (obtuse).

• Plane Geometry – What Is Triangle Similarity?

  Triangle similarity: definition, similarity ratio k, corresponding sides and angles, similarity vs congruence (k=1), and a numerical example.

• Plane Geometry – AA Similarity Theorem

  AA similarity: two equal angles imply similar triangles. When to use: shared angle, parallel lines, right angles. Real-life tree height example included.

• Plane Geometry – SSS and SAS Similarity Theorems

  SSS similarity theorem (all three sides in equal ratio) and SAS similarity theorem (two sides in ratio plus equal included angle), with worked examples.

• Plane Geometry – Ratios in Similar Triangles

  Ratios in similar triangles: perimeters scale by k, special lines by k, and areas by k² — with worked examples and a photo enlargement exercise.

• Plane Geometry – Summary and Applications

  Plane geometry chapter summary: four special lines, concurrency points, three similarity theorems (AA/SSS/SAS), ratios, and three practice exercises.

• Algebra – Laws of Exponents

  Five exponent laws: product (add), quotient (subtract), power of power (multiply), power of product/quotient (distribute) — examples and mistake table.

• Economics – Basic Concepts in Microeconomics

  Intro to microeconomics: scarcity, three economic questions (what/how/for whom), economic goods, factors of production, and the concept of choice.

• Economics – Production Possibility Frontier

  The PPF model: four assumptions, diagram, production combinations (efficient, inefficient, infeasible), four components, and a numerical example.

• Economics – Comparative and Absolute Advantage

  Comparative vs absolute advantage: definitions, opportunity cost calculations, key rules, linear vs concave PPF, and steps for building a PPF.

• Economics – International Trade

  International trade: why trade benefits both sides, two types (bilateral and world market), terms-of-trade range, and specialisation gains with examples.

• Economics – Opportunity Costs

  Opportunity costs on the PPF: total (Ymax−Y), average ((Ymax−Y)/X), and marginal (ΔY/ΔX) — with a full numerical example and the inverse cost relationship.

• Economics – Production Function and Resource Allocation

  Production function: TP, AP and MP defined, MP–TP and MP–AP rules, law of diminishing marginal product, and worker allocation by highest MP.

• Economics – Production Costs

  Production costs: fixed costs FC (sunk and non-sunk), variable costs VC, total costs TC = FC + VC, marginal cost MC, total revenue TR and profit π.

• Economics – Average Costs and Their Relationships

  Average costs: AVC (= VC/Q), AFC (= FC/Q, always falling), ATC (= AVC + AFC), and the MC golden rule — MC intersects AVC and ATC at their minimum points.

• Economics – Producer Supply and Producer Surplus

  Producer surplus (PS = TR − VC), its link to profit, MR = P in competition, optimal quantity rule MR = MC, and marginal profit interpretation.

• Economics – Supply Curve: Short Run and Long Run

  Supply curve in short and long run: sunk vs non-sunk FC, three producer decision cases (always / P≥AVC / P≥ATC), and supply curve shifts explained.

• Economics – Area Calculations for Microeconomics

  Area calculations for microeconomics: triangle formula (½·base·height) for surplus, rectangle for TR, and trapezoid for surplus changes and taxes.

• Economics – Isolating Variables and Inverting Formulas

  How to isolate variables in economics: converting inverse demand P=f(Q) to Q=f(P), supply function inversion, order of operations, and worked examples.

• Economics – Operations with Fractions

  Fractions for economics: multiply, divide (flip and multiply), add/subtract (common denominator), fraction to decimal conversion, and elasticity.

• Economics – The Demand Curve

  The demand curve: downward slope (law of demand), movement along vs shift of the curve, factors that shift demand including income and other prices.

• Economics – Types of Goods by Income

  Types of goods by income: normal (demand rises), inferior (demand falls), neutral (unchanged) — definitions, examples, identification method.

• Economics – Relationships Between Goods

  Product relationships: substitutes (price of Y up → demand for X up), complements (opposite direction), and independents — with examples and summary table.

• Economics – Consumer Surplus

  Consumer surplus (CS): graphical triangle below demand curve, formula for linear demand CS=½·Q₀·(Pmax−P₀), numerical example, and effect of price changes.

• Economics – Aggregate Curves and Market Equilibrium

  Aggregate demand and supply (horizontal summation), equilibrium price where QD=QS, excess demand/supply dynamics, and algebraic equilibrium calculation.

• Economics – Changes in Market Equilibrium

  Equilibrium shifts: supply changes (input price, technology), demand changes (income, preferences), and simultaneous shifts with ambiguous outcomes.

• Economics – Demand and Supply Elasticity

  Price elasticity: elastic (|E|>1), inelastic (|E|<1), unit elastic (|E|=1); effect on total revenue; special cases; arc elasticity with example.

• Economics – Social Welfare

  Social welfare SW = CS + PS: consumer/producer surplus, graphical triangles, numerical worked example, and why competitive equilibrium maximises SW.

• Economics – Labour Market and Derived Demand

  The labour market: VMPL = P × MPL as the labour demand curve, optimal employment rule VMPL = W, and the law of diminishing marginal product.

• Economics – Product Market and Labour Market Interactions

  Product–labour market interactions: four scenarios — product demand rise/fall, technology, workers leaving — combined changes and exam points.

• Economics – Key Areas in the Labour Market

  Labour market areas: employer surplus (below VMPL, above wage), worker surplus (above supply, below wage), total payments, and a numerical example.

• Economics – Taxation in a Closed Economy

  Taxation: supply shifts left by tax amount, consumer price rises, producer price falls, burden distributed by elasticity, deadweight loss formula.

• Economics – Subsidies in a Closed Economy

  Subsidies: supply shifts right, consumer price falls, producer price rises, subsidy distribution by elasticity, DWL formula, and comparison with taxes.

• Economics – Open Economy: Imports and Exports

  Open economy: Pw below P* → imports = Qd−Qs (CS rises, PS falls); Pw above P* → exports = Qs−Qd (PS rises, CS falls); SW higher in both cases.

• Economics – Import Tariff and Export Subsidy

  Import tariff (raises price, two DWL triangles, government revenue) and export subsidy (lowers domestic consumption, government expenditure) compared.

• Economics – Monopoly

  Monopoly: single seller, MR < P, profit maximisation at MR = MC, higher price and lower quantity than competition, DWL, government intervention.

• Economics – Public Goods

  Public goods (non-rival, non-excludable): four-quadrant classification, free-rider problem, government provision, vertical demand aggregation.

• Economics – Externalities

  Externalities: negative (MSC > MPC → overproduction, DWL), positive (MSB > MPB → underproduction), Pigouvian tax, subsidy solution, and Coase theorem.

• Economics – Game Theory: Basic Concepts

  Game theory: payoff matrix, dominant strategy (best regardless of rival), dominated strategy (never chosen), and dominant-strategy equilibrium explained.

• Economics – Prisoner's Dilemma and Nash Equilibrium

  Prisoner's dilemma (both betray despite cooperation being better) and Nash equilibrium: best-response method, multiple equilibria, not always efficient.

• Economics – Behavioural Economics

  Behavioural economics: cognitive biases (loss aversion, framing, anchoring, availability, endowment), mental accounting, ultimatum game, and nudge theory.

• Economics – Time Value of Money: Core Concepts

  Time value of money: why a shekel today exceeds a future shekel; compounding FV = PV×(1+r)ⁿ; discounting PV = FV/(1+r)ⁿ; investment decision rule.

• Economics – Discounting

  Discounting: formula PV = FV/(1+r)ⁿ, discount factor table, worked two-year example at 10% and 5%, and how interest rate affects investment viability.

• Economics – Net Present Value (NPV)

  NPV: formula, decision rule (NPV>0 → invest), full numerical example at 10%, comparison across interest rates, and comparing two projects by highest NPV.

• Economics – Introduction to Macroeconomics

  Macroeconomics basics: micro vs macro comparison, flow variables (GDP, income) vs stock variables (national debt, capital stock), endogenous vs exogenous.

• Economics – Output and Value Added

  Output measures: GDP (domestic) vs GNP (national), gross vs net, value added = sales minus intermediate inputs; no double-counting; factor income.

• Economics – Profit and Loss Statement

  P&L statement for macroeconomics: GVA = sales + inventory − inputs; NVA = GVA − depreciation; four production factors; factor income identity.

• Economics – Government Budget and Sources-and-Uses

  Government budget (G+Ig vs T+BD), deficit, government value added, and the GDP identity: Y = C+G+I (closed) and Y = C+G+I+(Ex−Im) (open economy).

• Economics – Capital Formation and Nominal/Real Output

  Capital formation (S = I closed economy); three saving types; nominal vs real output; price index; real output per capita for cross-year comparisons.

• Economics – The Investment Function

  Investment function: gross/net investment, business/public investment, determinants (interest rate, expectations, output), autonomous vs induced.

• Economics – Consumption and Saving Function

  Consumption function C = C₀ + MPC·Yd; four propensities MPC APC MPS APS; saving function; negative saving; factors affecting autonomous consumption.

• Economics – Consumption Curve Movements and Aggregate Consumption

  Consumption curve: movement along (Yd changes) vs shift (interest/expectations); effect on saving; aggregate MPC; income distribution impact.

• Economics – Investment Viability: Net Present Value

  NPV for investment decisions: PV formula, decision rule (NPV>0 → invest), worked example at 10%, inverse rate–NPV relationship, and key tips.

• Economics – The Simple Keynesian Model

  Simple Keynesian model: assumptions, AD = C+I+G, equilibrium Y = k·A₀, income multiplier k = 1/(1−MPC), and multiplier mechanism with numerical example.

• Economics – Equilibrium, Output Gaps and Paradox of Thrift

  Full-employment output YF, deflationary gap (gap/k), inflationary gap, and paradox of thrift: saving intent rises but total saving stays at I.

• Economics – Fiscal Policy

  Fiscal policy: G multiplier k_G, tax multiplier k_T = −MPC/(1−MPC), balanced budget multiplier = 1, and proportional tax as automatic stabiliser.

• Economics – Money and the Banking System

  Banking system: monetary base B = CA+RZ; money supply M = CA+D; reserve ratio R; deposit multiplier 1/R; bank balance sheet; money creation rounds.

• Economics – Internal and External Injections

  Internal injections (shift CA↔D, B fixed); external injections (ΔB, multiplier effect); ΔM = injection × 1/R through reserves; bank run mechanics.

• Economics – Central Bank Instruments

  Central bank instruments: bond/FX operations (ΔM = injection×1/R), reserve ratio change, monetary rate; expansionary vs contractionary policy; formulas.

• Economics – Central Bank Instruments (Full)

  Central bank instruments: bond/FX ops (ΔM = injection×1/R), monetary rate, reserve ratio change (B fixed); expansionary vs contractionary policy.

• Economics – The Money Market

  Money market: nominal balances M = CA+D, real balances M/P, interest rate as price of money, vertical supply curve, downward-sloping demand curve L.

• Economics – The Money Demand Curve

  Money demand: rate ↑→demand ↓ (movement along); output ↑→L shifts right; preference shifts; movement along vs shift; exercise: output fall ⟹ r falls.

• Economics – Money Supply and Market Equilibrium

  Money supply curve (vertical); shifts from ΔM or ΔP; equilibrium r* where L(Y,r) = Mˢ/P; central bank controls r; money market shift summary.

• Economics – The IS-LM Model

  IS-LM model: Y and r shared by both markets; G ↑ → Y ↑ → r ↑ → partial crowding out; 4-step analysis; fiscal vs monetary policy instruments.

• Economics – IS-LM: Closed Economy with Unemployment

  IS-LM with unemployment (Y<YF): fiscal (G↑ → r↑, partial crowding out), monetary (M↑ → r↓ → I↑), bond vs tax financing, special no-crowding-out cases.

• Economics – IS-LM: Full-Employment Economy

  IS-LM at full employment: fiscal policy (G ↑) raises P only (full crowding out); money neutrality: M ↑ raises P proportionally, real variables fixed.

• Economics – Balance of Payments

  Balance of payments: double-entry principle (debit = FX outflow, credit = inflow), current account, capital account, FX reserves; recording examples.

• Economics – Foreign Exchange Market Fundamentals

  FX market: nominal rate e, real rate e/P, depreciation vs appreciation, demand (imports), supply S = Sₓ + CF, capital inflows CF depend on r − r*.

• Economics – FX Market Equilibrium and Events

  FX equilibrium (D = S); event analysis: Y ↑ → depreciation, r ↑ → appreciation, r* ↑ → depreciation; opposing events → indeterminate; rules of thumb.

• Economics – FX Market in Real Terms

  FX market in real terms: real rate e/P, demand D = imports, supply S = Sₓ + CF; movement along vs shift; case analysis table; link to combined model.

• Economics – Combined Model: Open Economy with Unemployment

  Open-economy combined model (unemployment): G ↑ → r ↑, appreciation; M ↑ → r ↓, depreciation; bond vs tax financing; special cases no-crowding-out.

• Economics – Combined Model: Full-Employment Open Economy

  Open-economy IS-LM at full employment: G ↑ → full crowding out → P ↑; M ↑ → money neutral, P rises proportionally; r* ↑ → depreciation → P ↑.

• Statistics – Core Concepts

  Core statistics: descriptive vs inferential; population vs sample; nominal, ordinal, discrete, continuous variables; four scales of measurement.

• Statistics – Measures of Central Tendency

  Mean, median, mode: formulas, weighted mean, median for odd/even n, comparison table, combined mean, correcting mean after data entry error.

• Statistics – Measures of Dispersion

  Measures of dispersion: range, sample variance (n−1), standard deviation, shortcut formula, coefficient of variation CV, and linear transformation rules.

• Statistics – Introduction to the Normal Distribution

  Normal distribution: bell-curve shape, properties (symmetric, mean=median=mode), parameters μ and σ, 68-95-99.7 empirical rule, real-life examples.

• Statistics – Z-Score and the Standard Normal Table

  Z-score = (X−μ)/σ; Z-table gives cumulative left-tail P(Z≤z); use 1−Φ(z) for right tail; symmetry for negatives; complete numerical example.

• Statistics – Inverse Normal Problems

  Inverse normal problems: X = μ + Zσ; three problem types (left-tail, right-tail, below 0.5); common Z-values table; common mistakes; process summary.

• Statistics – Association Measures for Nominal Variables

  Lambda (λ): prediction-error-based; Cramér's V: χ²-based for any table; Phi: 2×2 special case; value interpretation table; Lambda is asymmetric.

• Statistics – Pearson Correlation and Regression

  Pearson r: covariance/sₓsᵧ; regression line ŷ=a+bx; slope b=r·sᵧ/sₓ; R² = % variance explained; correlation vs causation; association measures comparison.

• Statistics – Spearman's Rank Correlation

  Spearman's rₛ measures monotonic rank correlation; formula 1−6Σd²/n(n²−1); tied ranks; interpretation table; use when ordinal or non-normal.

• Maths – Parabola Sign Chart Examples

  Parabola sign chart: smiling (a>0) expression positive outside roots; frowning (a<0) positive between roots; worked examples; summary table.

• Maths – Probability Tree With Replacement

  Probability tree (with replacement): P unchanged across draws; one, two, three draws; exactly 2 red = C(3,2)×(3/5)²×(2/5); difficulty summary.

• Maths – Probability Tree Without Replacement

  Probability tree without replacement: P changes after each draw; two-draw example; comparison table (with vs without); golden rule formulas.

• Maths – Domain Sign Chart Review

  Domain sign chart: number line examples (x≥3, x≤4); smiling parabola (a>0) domain outside roots; frowning (a<0) domain between roots; worked examples.