← COURSEWARE HOME PURE MATHEMATICS CIE 9709 · PAPER 3

Cambridge International A & AS Level — Mathematics 9709

Pure Mathematics 3

A courseware index in nine chapters — sixty-five lessons, forty minutes each.

  • 9 CHAPTERS · §3.1–§3.9
  • 65 LESSONS
  • 40 MIN EACH
  • SELF-CONTAINED · WORKS OFFLINE
§3.1

Algebra

Six lessons on the algebra of Paper 3 — from the modulus function and polynomial division to partial fractions and the binomial expansion.

6 LESSONS
240 MIN
01 Modulus Function Defining |x|, sketching modulus graphs, and solving |f(x)| = g(x) by case analysis. GRAPHS · EQUATIONS · CASES 40′→ 02 Polynomial Division Dividing polynomials by linear and quadratic divisors — quotient and remainder by long division. LONG DIVISION · DEGREE · REMAINDER 40′→ 03 Factor & Remainder Theorem The Remainder Theorem gives r = f(a); the Factor Theorem turns roots into factors of cubics and beyond. FACTOR THEOREM · ROOTS · CUBICS 40′→ 04 Partial Fractions I Splitting proper fractions with distinct linear factors into partial fractions. LINEAR FACTORS · MULTIPLY THROUGH · IDENTITIES 40′→ 05 Partial Fractions II Partial fractions with repeated and quadratic factors in the denominator. REPEATED FACTORS · QUADRATIC FACTORS 40′→ 06 Binomial Expansion Expanding (1 + x)n for rational n — the binomial series and its range of validity. SERIES · VALIDITY · APPROXIMATION 40′→
§3.2

Logarithmic & Exponential Functions

Six lessons on logarithms and exponentials — the laws, ex and ln x, equations, linearised data and growth models.

6 LESSONS
240 MIN
01 Logarithms & Indices Index laws and logarithm laws — the same algebra written in two notations. LAWS OF LOGS · INDICES · CHANGE OF BASE 40′→ 02 ex & ln x The natural exponential and its inverse ln x — graphs, laws, and the inverse pair eln x = x. NATURAL LOG · INVERSE · GRAPHS 40′→ 03 Solving Exponential Equations Solving ax = b by matching bases or taking logarithms of both sides. EQUATIONS · SAME BASE · LOG BOTH SIDES 40′→ 04 Linearising Data with Logs Straightening power and exponential laws with logarithms — finding a and n from data. LOG-LINEAR · SEMI-LOG · GRADIENT & INTERCEPT 40′→ 05 Log & Exponential Applications Growth and decay models — formulating and solving N(t) = N₀ekt. GROWTH · DECAY · MODELLING 40′→ 06 Log & Exponential Review Consolidating §3.2 — mixed logarithmic and exponential problem solving across weeks one and two. REVISION · MIXED PRACTICE · EXAM SKILLS 40′→
§3.3

Trigonometry

Twelve lessons on the trigonometry of Paper 3 — reciprocal and Pythagorean identities, compound and double angle formulae, the R-form, identity proofs and equation solving.

12 LESSONS
480 MIN
01 Reciprocal Trig Functions Defining sec θ, cosec θ and cot θ as reciprocals — graphs, domains and ranges. SEC · COSEC · COT 40′→ 02 Pythagorean Identities Dividing sin²θ + cos²θ = 1 by cos²θ and sin²θ to obtain sec²θ = 1 + tan²θ and cosec²θ = 1 + cot²θ. IDENTITIES · SEC²θ · COSEC²θ 40′→ 03 Compound Angle Formulae I Expanding sin(A ± B) and cos(A ± B), and why sin(A + B) ≠ sinA + sinB. SIN(A±B) · COS(A±B) · EXPANSION 40′→ 04 Compound Angle Formulae II Deriving tan(A ± B) from sin and cos compound formulae, and simplifying expressions. TAN(A±B) · DERIVATION · SIMPLIFY 40′→ 05 Double Angle Formulae Setting B = A to obtain sin 2A, cos 2A in three forms and tan 2A. SIN 2A · COS 2A · TAN 2A 40′→ 06 Using Identities in Equations Rewriting equations such as tan θ + cot θ = 4 with identities, then solving within an interval. EQUATIONS · REWRITE · SOLVE 40′→ 07 The R-Form Writing a sin θ + b cos θ as R sin(θ + α) with R = √(a² + b²). R-FORM · AMPLITUDE · PHASE 40′→ 08 R-Form Applications I Solving equations of the form R sin(θ + α) = k and reading off maximum and minimum values. EQUATIONS · MAX & MIN 40′→ 09 R-Form Applications II The full R-form toolbox — combining expression, solving and extrema on comprehensive problems. TOOLBOX · PAST PAPER 40′→ 10 Identity Proofs Proving trigonometric identities rigorously by transforming one side into the other, avoiding invalid operations. PROOF · VALID OPERATIONS 40′→ 11 Equations Comprehensive Planning a route through multi-step trigonometric equations needing double angles, R-form and quadratics. STRATEGY · MULTI-STEP 40′→ 12 Trigonometry Review Consolidating §3.3 — a full recall of compound, double angle, R-form and identity techniques. REVISION · FORMULA RECALL 40′→
§3.4

Differentiation

Six lessons on Paper 3 differentiation — the derivative table, the chain, product and quotient rules, and parametric and implicit differentiation.

6 LESSONS
240 MIN
01 Derivatives of ex, ln x & Trig Functions The originals: differentiating ex, ln x, sin x, cos x, tan x and tan⁻¹x. FIRST PRINCIPLES · DERIVATIVE TABLE 40′→ 02 The Chain Rule Differentiating composites — dy/dx = (dy/du)(du/dx) for functions like √(2x³ + 5). COMPOSITES · DY/DU × DU/DX 40′→ 03 The Product Rule d/dx(uv) = u dv/dx + v du/dx — for products such as x²ln x and xe1−x². PRODUCTS · UV′ + U′V 40′→ 04 The Quotient Rule d/dx(u/v) = (vdu/dx − udv/dx)/v² for rational functions. QUOTIENTS · (VU′ − UV′)/V² 40′→ 05 Parametric Differentiation dy/dx = (dy/dt)/(dx/dt) for parametric curves such as x = t − e2t. PARAMETRIC · DY/DT ÷ DX/DT 40′→ 06 Implicit Differentiation Differentiating x² + y² = xy + 7 implicitly — tangents and normals to implicit curves. IMPLICIT · TANGENTS & NORMALS 40′→
§3.5

Integration

Ten lessons on integration — reverse differentiation, special forms and trig identities, partial fractions and the kf′(x)/f(x) pattern, definite and improper integrals, integration by parts, substitution, and areas and volumes of revolution.

10 LESSONS
400 MIN
01 Reverse Differentiation Extended Integrating eax+b, 1/(ax+b), sin(ax+b) and cos(ax+b) — the reverse chain rule. REVERSE CHAIN RULE · STANDARD FORMS 40′→ 02 Special Integral Forms ∫sec²(ax+b) and ∫1/(x²+a²) — special forms such as ∫1/(2+3x²)dx. SEC²(AX+B) · 1/(X²+A²) 40′→ 03 Trig Identities in Integration Using double-angle formulae to integrate sin²x, cos²(2x) and tan²x. DOUBLE ANGLE · SIN²X · COS²X 40′→ 04 Partial Fractions in Integration Integrating rational functions — divide first, decompose into partial fractions, then integrate term by term. DECOMPOSE · RATIONAL FUNCTIONS 40′→ 05 The kf′(x)/f(x) Form Recognising ∫k f′(x)/f(x) dx = k ln|f(x)| — including ∫tan x dx. LOGARITHMIC PATTERN · ∫TAN X 40′→ 06 Definite & Improper Integrals A quick review of definite integrals, then improper integrals by limits — including ∫₁∞ 1/x² dx. DEFINITE · IMPROPER 40′→ 07 Integration by Parts I Reversing the product rule — ∫u dv = uv − ∫v du, choosing u and dv for ∫x sin 2x dx. ∫U DV · CHOOSING U 40′→ 08 Integration by Parts II Special cases — ∫ln x dx, ∫tan⁻¹x dx and cyclic integration by parts. ∫LN X · CYCLIC · DV = 1 40′→ 09 Substitution Method Reversing the chain rule with u = g(x) — changing limits and back-substitution. U = G(X) · CHANGE LIMITS 40′→ 10 Area & Volume of Revolution Areas between curves and volumes of revolution — ∫(f−g)dx and V = π∫y²dx. AREA · VOLUME · Π∫Y² 40′→
§3.6

Numerical Solution of Equations

One lesson on numerical methods — trapping a root with a sign change, then rearranging the equation and iterating to the required accuracy.

1 LESSON
40 MIN
01 Sign Change and Fixed-Point Iteration A sign change traps a root in an interval; rearranging f(x) = 0 into x = F(x) and iterating xn+1 = F(xn) then converges on the root. SIGN CHANGE · ITERATION · Xₙ₊₁ = F(Xₙ) 40′→
§3.7

Vectors

Six lessons on 3D vectors — components, magnitude and the scalar product, the vector equation of a line, intersecting, parallel and skew lines, and perpendicular distance.

6 LESSONS
240 MIN
01 Introduction to 3D Vectors Component form, magnitude, direction, unit vectors, and addition and subtraction in three dimensions. COMPONENTS · MAGNITUDE · UNIT VECTORS 40′→ 02 Scalar Product & Angles a·b = a₁b₁ + a₂b₂ + a₃b₃ — angles between vectors and testing perpendicularity. DOT PRODUCT · PERPENDICULAR 40′→ 03 3D Geometry Applications Vectors in cubes, cuboids, prisms and roofs — coordinates, edges and diagonals, and angles in real 3D solids. CUBOIDS · DIAGONALS · 3D ANGLES 40′→ 04 Vector Equation of a Line r = a + λd — points on a line, midpoints and division in a given ratio. R = A + ΛD · RATIO DIVISION 40′→ 05 Intersecting, Parallel & Skew Lines Classifying pairs of lines as parallel, intersecting or skew, and finding intersection points. PARALLEL · INTERSECT · SKEW 40′→ 06 Perpendicular Distance & Applications The foot of the perpendicular and point-to-line distance, with a comprehensive vectors review. PERP DISTANCE · REVIEW 40′→
§3.8

Differential Equations

Six lessons on differential equations — formulating them from rates of change, separating variables, finding particular solutions and interpreting solutions in context.

6 LESSONS
240 MIN
01 Formulating Differential Equations Translating rate-of-change descriptions into equations — “rate proportional to” becomes dy/dt = k·f(y). RATE OF CHANGE · MODELLING 40′→ 02 Separable Variables Separation of variables — solving dy/dx = f(x)g(y) with the integration techniques of §3.5. SEPARATE · GENERAL SOLUTION 40′→ 03 Particular Solution Finding the constant C by substituting an initial condition into the general solution. INITIAL CONDITIONS · PARTICULAR 40′→ 04 Interpreting Solutions in Context Growth and decay — the real-world meaning of the constant C and what the solution actually tells us. GROWTH · DECAY · CONTEXT 40′→ 05 Comprehensive Application The complete modelling workflow — formulate, solve, apply initial conditions, and interpret the result. MODEL · SOLVE · INTERPRET 40′→ 06 Integration & DE Review Consolidating integration techniques, numerical methods and differential equations into one connected toolkit. W6–W9 · CONSOLIDATION 40′→
§3.9

Complex Numbers

Twelve lessons on complex numbers — the Cartesian form, the Argand diagram, polynomial equations, polar form, square roots and loci.

12 LESSONS
480 MIN
01 Complex Numbers in Cartesian Form z = x + iy — Re(z), Im(z), |z|, arg z, z* and equality. X + IY · |Z| · ARG Z 40′→ 02 Complex Operations in Cartesian Form Addition, subtraction, multiplication and division of x + iy, with the properties of conjugates. ± × ÷ · CONJUGATES 40′→ 03 The Argand Diagram Plotting z = x + iy and the geometric effects of conjugation, addition and multiplication. ARGAND · GEOMETRY 40′→ 04 Polynomial Equations with Complex Roots The conjugate root theorem — complex roots of real polynomials come in pairs — and factoring. CONJUGATE ROOTS · FACTORING 40′→ 05 Complex Operations Comprehensive Mixed operations, simplification strategies and the modulus–argument form. MIXED PRACTICE · MODULUS–ARGUMENT 40′→ 06 Cartesian Form Review Consolidating all of the Week 10 complex number skills in mixed problems. WEEK 10 · REVIEW 40′→ 07 Polar Form r(cos θ + i sin θ) ≡ reiθ — modulus and argument properties. POLAR · EULER FORM 40′→ 08 Polar Operations Multiplication and division in polar form — moduli multiply and divide, arguments add and subtract. MULTIPLY · DIVIDE · ARGS ADD 40′→ 09 Square Roots of Complex Numbers Finding hidden roots — solving (a + ib)² = z for numbers like 5 + 12i. (A+IB)² = Z · HIDDEN ROOTS 40′→ 10 Loci in the Argand Diagram I Circles and perpendicular bisectors — |z − a| = k and |z − a| = |z − b|. CIRCLES · PERP BISECTORS 40′→ 11 Loci in the Argand Diagram II Half-lines arg(z − a) = α and shaded regions described by inequalities. HALF-LINES · REGIONS 40′→ 12 Complex Numbers Comprehensive Review A full P3 review — algebra, trigonometry, calculus and complex numbers together. P3 · FULL REVIEW 40′→