AdvanceReview
Path to 300

Mathematics / Topic coverage

Calculus

Build a pipeline: limits/continuity → derivatives/AOD → DE → definite integrals/area.

Weight highIntensity Core · dailyTrend Falling share300-target 28.9 /100high
Path to 300

Mark budget for this portion

Full 300 plan →
At ~28.5% of math weight, this portion should reliably deliver ~30 of your 100 math marks — treat misses here as rank-droppers.
21.7Floor marks
28.9Target marks
33.3Stretch marks
28.5%Of subject weight
100Subject target
At a glance

Coverage metrics

69Tagged questions
28.5%Of Mathematics
9.86Average / year
2020Peak year (16 Q)
7/7Years present
Official · 2026

Syllabus map for this portion

Full Mathematics syllabus →
Maps to Differential Calculus + Integral Calculus (including first-order DEs and area).

Must cover from syllabus

  • L’Hôpital and continuity of composites
  • Rolle / Lagrange MVT geometric meaning
  • Definite integral properties + FTC
  • Area under simple curves
  • Homogeneous, separable, and linear first-order DEs

Gaps / exclusions to watch

  • Implicit differentiation up to order two
  • Formation of ODEs from word/geometry stems
Unit 01

Differential Calculus

high

Exam focusLimits with L’Hôpital, continuity, maxima/minima, MVT/Rolle, tangents.

Syllabus points

  • Limits and continuity (sum/difference/product/quotient); L’Hôpital
  • Continuity of composites; intermediate value property
  • Derivatives: sum/product/quotient/chain; standard function families
  • Tangents and normals; increasing/decreasing; second derivatives; max/min
  • Rolle’s and Lagrange’s MVT; geometric interpretation; implicit derivatives up to order two

Do / check

  • State domain/continuity before differentiating piecewise
  • Check endpoints separately for extrema on closed intervals
Unit 02

Integral Calculus

high

Exam focusDefinite integral properties, area, first-order DEs (homogeneous, separable, linear).

Syllabus points

  • Indefinite integrals of standard functions
  • Definite integrals as limit of sums; properties; FTC
  • Integration by parts, substitution, partial fractions
  • Area bounded by simple curves
  • Formation of ODEs; homogeneous first-order; separation of variables; linear first-order DE

Do / check

  • Use definite-integral properties before expanding
  • Identify DE type before applying a method
2020–2026

Year-wise appearance

Counts of tagged questions for this topic each year.

2016
219
229
239
248
258
2610

Reading the pattern

  • Calculus accounts for ~28.5% of tagged math questions (2020–2026).
  • Trend signal: falling (early-window avg 11.3/yr vs recent avg 8.7/yr).
  • Appears in 7/7 tagged years; peak 2020 (16 Q), low 2024 (8 Q).
  • Dominant subtopics: AOD (18), Definite Integrals (11), Continuity/Differentiability (9), Functions (7).
  • Question-type mix: Numerical 39.1%, Single correct 29.0%, Multi correct 26.1%, Matching 5.8%.
Internal breakdown

Subtopic coverage

Start with the top subtopics — they usually carry most of this portion’s weight.

  • AOD18 · 26.1%
  • Definite Integrals11 · 15.9%
  • Continuity/Differentiability9 · 13%
  • Functions7 · 10.1%
  • Limits7 · 10.1%
  • Differential Equations7 · 10.1%
  • Area6 · 8.7%
  • Differentiation4 · 5.8%
SubtopicCountShareYears (count)
AOD1826.1%2020·3 2021·2 2022·3 2023·2 2024·2 2025·5 2026·1
Definite Integrals1115.9%2020·2 2021·5 2022·1 2024·1 2025·1 2026·1
Continuity/Differentiability913%2020·2 2023·3 2024·1 2026·3
Functions710.1%2020·3 2023·1 2024·2 2026·1
Limits710.1%2020·2 2022·2 2023·1 2024·1 2025·1
Differential Equations710.1%2021·1 2022·2 2023·1 2025·1 2026·2
Area68.7%2020·1 2021·1 2022·1 2023·1 2024·1 2026·1
Differentiation45.8%2020·3 2026·1
Exam form

Type & difficulty mix

Question types

  • Numerical — 27 questions (39.1%)
  • Single correct — 20 questions (29%)
  • Multi correct — 18 questions (26.1%)
  • Matching — 4 questions (5.8%)

Difficulty (heuristic)

  • M — 39 questions
  • H — 30 questions

Use for planning load only — not an official difficulty label.

Focused study

Plan for this portion

Must know

  • Floor/piecewise continuity checklists
  • Monotonicity & extremum without graphing panic
  • Standard DE forms (linear, separable, homogeneous)
  • Property ∫₀ᵃ f(x)/(f(x)+f(a−x))
  • Area between curves from intersection search

Common traps

  • Missing non-differentiability at corners
  • Wrong sign in linear DE integrating factor
  • Area without absolute value / wrong interval

Weekly drill checklist

  • 3 continuity/AOD problems
  • 2 DE
  • 2 definite integral property drills
  • 1 linked area stem

Prerequisite chain

Functions & graphs→limits→continuity→differentiability→AOD→integrals→DE

Often joins with

Coordinate Geometry (area/conics)Trigonometry (integrals / equations)Algebra (functions counting)
Paper pointers

Sample tagged questions

Use these as deliberate practice targets from past papers.

YearPaperQSubtopicTypeDiff
2020P1Q2FunctionsSCM
2020P1Q3AreaSCM
2020P1Q6AODSCM
2020P1Q7Continuity/DifferentiabilityMCH
2020P1Q12Definite IntegralsMCH
2020P1Q13AODNUMM
Connections

Related mixed joins

Where this topic must connect with another concept in one stem.

Mathematics2026 · Paper 2 · Q15–Q16linked-numerical

Exponential curves + area

Calculus – Curves+Definite Integration – Area+Trigonometric equations

Knowledge bridge

Step 1: set y = e^{-x} equal to y = e^{-x}(sin x + cos x) and cancel the common exponential to get a trig equation for intersection abscissae α_i. Step 2: order those roots and set up the definite integral of the difference of the two curves between α_1 and α_4 for the enclosed area β. Step 3: simplify the given log expression in β to a clean numerical value.

Why it feels hard

The stem hides trig solving inside an exponential envelope; students often integrate before finding exact limits.

What to practise

Drill ‘cancel common positive factor → trig roots → area integral’ as one reflex chain.

Open 2026 year review →
Mathematics2025 · Paper 2 · Q1–Q2multi-concept

Limit/derivative structure + area inequalities

Calculus – Limits / AOD+Definite Integration – Area+Inequalities of regions

Knowledge bridge

Step 1: analyse the given limit expression to identify the correct f(x) / derivative behaviour. Step 2: on a related (or following) stem, sketch the region defined by inequalities. Step 3: integrate carefully (logs appear from 1/x-type boundaries).

Why it feels hard

Back-to-back calculus depth: analytic limit then geometric area.

What to practise

Pair every limit/AOD drill with one area-of-region numerical the same day.

Open 2025 year review →
Mathematics2023 · Paper 1 · Q7–Q9multi-concept

Parabola/ellipse tangents + area numerical

Conics – common tangents+Calculus – Area+Trigonometric equations

Knowledge bridge

Step 1: for the ellipse–parabola common-tangent MC, write both tangent conditions and compare. Step 2: later area NUM — find intersection limits first. Step 3: integrate difference of functions; keep trig-equation NUM technique nearby for neighbouring items.

Why it feels hard

Classic Advanced join: geometry conditions feeding calculus numerics in the same paper.

What to practise

Any week you revise conics, end with one area-between-curves NUM.

Open 2023 year review →
Mathematics2020 · Paper 1 · Q3–Q4single-correct

Area calculus beside parabola geometry

Calculus – Area+Coordinate Geometry – Parabola+Single-correct pair

Knowledge bridge

Step 1: for the area item, find intersection limits and integrate. Step 2: for the parabola item, apply focus–directrix or tangent properties. Step 3: notice how often Advanced seats Calculus and Conics as neighbours — train that adjacency.

Why it feels hard

2020’s Calculus weight makes area items feel easy until limits are wrong.

What to practise

Always find limits from geometry before writing ∫.

Open 2020 year review →
Mathematics

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