H2 Maths Notes (JC 1-2): 00A) Section A -- Pure Mathematics Overview

> **How to use this guide**\\
> Section A contributes roughly 70% of the H2 Mathematics papers. Treat this overview as the scaffolding for the topic-by-topic notes that follow. Each subsection below lists the core skills MOE expects, links to the companion deep dives (coming next), and flags IP bridge topics you can revisit for quick wins.

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## 1 | Functions and Graphs (Topic 1)

- **Why it matters**: Functions frame almost every JC problem. You must be fluent with domain/range, inverse and composite functions, and graph transformations.
- **Workplan**
  1. Revisit IP checkpoints: surds, logarithms, graph sketching of quadratics, hyperbolas, and simple rational forms.
  2. Practise the "set-up ( \rightarrow ) algebra ( \rightarrow ) graph" workflow so you can confirm one-to-one behaviour before finding an inverse.
  3. Use the GC intentionally: plot, annotate roots or turning points, then justify algebraically.
- **Key reminders**
  - Verify inverses by checking ( f\bigl(f^{-1}(x)\bigr) = x ) and ( f^{-1}(f(x)) = x ).
  - Catalogue the four basic transformations (vertical stretch/shift, horizontal stretch/shift) and how they combine.
- **Linked articles**: `H2 Maths Notes (JC 1-2): 1) Functions and Graphs` plus sub-topic briefs (1.1-1.3).

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## 2 | Sequences and Series (Topic 2)

- **Focus areas**: Convergence criteria for geometric progressions, recurrence relations, and modelling contexts such as savings plans.
- **Habits to build**
  - Translate words into a recurrence ( u\_\{n+1} = f(u_n) ) quickly, then generate a few terms with the GC to spot behaviour.
  - Distinguish between the finite sum ( S_n ) and the sum to infinity ( S\_\infty ); justify convergence using ( \lvert r \rvert \< 1 ).
- **IP bridges**: Additional Maths arithmetic/geometric series and sigma notation basics.
- **Linked articles**: `H2 Maths Notes (JC 1-2): 2) Sequences and Series` and sub-topic `2.1`.

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## 3 | Vectors (Topic 3)

- **Syllabus emphasis**: Move from 2-D manipulation to 3-D geometry covering direction vectors, normals, shortest distance, and scalar/vector products.
- **Workflow**
  1. Anchor notation: columns for direction vectors, rows for normals.
  2. Solve line/plane intersections step-by-step (parameter substitution followed by consistency checks).
  3. Drill mixed-product questions such as areas of parallelograms or volumes of parallelepipeds.
- **Watch points**: state units clearly ((\text{units}), (\text{units}^2), (\text{units}^3)) and describe direction sense.
- **Linked articles**: `H2 Maths Notes (JC 1-2): 3) Vectors` plus sub-sections 3.1-3.3.

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## 4 | Complex Numbers (Topic 4)

- **Key skills**
  - Convert between Cartesian ( x + iy ) and polar ( r \bigl( \cos\theta + i\sin\theta \bigr) ) forms.
  - Interpret Argand diagram loci and transformations.
  - Apply De Moivre to handle powers/roots and derive trigonometric identities.
- **IP bridge**: complex arithmetic from enrichment or Olympiad exposure.
- **Linked articles**: `H2 Maths Notes (JC 1-2): 4) Complex Numbers` (sub-topic 4.1).

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## 5 | Calculus (Topic 5)

- **Breadth**: differentiation, Maclaurin series, integration techniques, definite integrals, differential equations.
- **Study cadence**
  - **JC1 Term 1-2**: consolidate differentiation rules, Maclaurin series up to third order, and baseline integrals.
  - **JC1 Term 3 - JC2 Term 1**: tackle integration techniques (partial fractions, by parts) and definite integral applications.
  - **JC2 Term 1-2**: solve separable differential equations and interpret steady-state solutions.
- **Essentials**
  - Memorise the baseline Maclaurin expansions: ( e^x ), ( \sin x ), ( \cos x ), and ( (1 + x)^n ) for small ( x ).
  - Differential equations typically reduce to separable form; practise rewriting into ( \frac\{dy}\{dx} = f(x) g(y) ).
- **Linked articles**: deep dives for sub-topics 5.1-5.5.

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## 6 | Integrating Pure Math into Exam Skills

- **Paper structure**: Both Paper 1 and Paper 2 include Pure Math questions mixing short items with long structured tasks.
- **Revision ladder**
  1. Drill fundamentals weekly (derivative rules, vector identities, binomial manipulations).
  2. Rotate through mixed-topic timed sets (functions + calculus + vectors) in 45-minute blocks.
  3. Post-mortem with mark schemes; highlight presentation issues such as missing units or incomplete reasoning.
- **Tools**: maintain a Formula & Methods notebook, practise GC workflows in Exam Mode, annotate common mistakes.

Next steps: dive into Topic 1 (Functions and Graphs) where we unpack the official sub-topics with worked examples and exam-style problems.