A/L Science Stream

A/L Combined Mathematics (NIE syllabus)

G.C.E. A/L Combined Mathematics in Sri Lanka follows the National Institute of Education (NIE) syllabus for Grades 12 and 13. The NIE publishes it in two parts: Combined Mathematics I, the pure mathematics (algebra, trigonometry, coordinate geometry, calculus, complex numbers and matrices), and Combined Mathematics II, the applied mathematics: vectors, forces, motion, probability and statistics. The NIE allocates 600 teaching periods, 309 in Grade 12 and 291 in Grade 13. The largest topics are derivatives (30 periods), integration (28) and quadratic functions and equations (25). This is the Sri Lankan G.C.E. (A/L) syllabus, not Cambridge or Edexcel Further Mathematics.

Find your topic, not your term

The NIE calls this a proposed term-wise breakdown, so your class may take the topics in a different order — find the topic you are on.

Grade 12: the NIE's own topics, in the term it proposes for each, with the teaching periods it gives them. Part I is Combined Mathematics I (pure), Part II is Combined Mathematics II (applied).

Grade · term (NIE)	Competency levels	NIE subject topic	Periods	Part	On this site	Source
Grade 12 · Term 1	1.1, 1.2	Real numbers	2	I	no page yet	[S1] p. 3
Grade 12 · Term 1	2.1, 2.2	Functions	4	I	no page yet	[S1] p. 3
Grade 12 · Term 1	8.1, 8.2	Angular measurements	2	I	no page yet	[S1] p. 3
Grade 12 · Term 1	17.1, 17.2	Rectangular cartesian system, straight line	3	I	Notes · Practice	[S1] p. 3; [S3]
Grade 12 · Term 1	9.1–9.4	Circular functions	12	I	Notes · Practice	[S1] p. 3; [S3]
Grade 12 · Term 1	11.1	Sine rule, cosine rule	1	I	Notes · Practice	[S1] p. 3; [S3]
Grade 12 · Term 1	4.1–4.3	Polynomials	7	I	Notes · Practice	[S1] p. 3; [S3]
Grade 12 · Term 1	10.1–10.4	Trigonometric identities	14	I	no page yet	[S1] p. 3
Grade 12 · Term 1	5.1	Rational functions (partial fractions)	6	I	Notes · Practice	[S1] p. 3; [S3]
Grade 12 · Term 1	6.1	Index laws and logarithmic laws	1	I	Notes · Practice	[S1] p. 3; [S3]
Grade 12 · Term 1	7.1–7.3	Inequalities	14	I	Notes · Practice	[S1] p. 3; [S3]
Grade 12 · Term 1	9.5	Solving trigonometric equations	4	I	no page yet	[S1] p. 3
Grade 12 · Term 1	1.1–1.4	Vectors	14	II	no page yet	[S1] p. 3
Grade 12 · Term 1	2.1–2.3	Systems of coplanar forces acting at a point	11	II	Notes · Practice	[S1] p. 3; [S3]
Grade 12 · Term 2	3.1, 3.2	Quadratic functions and quadratic equations	25	I	Notes · Practice	[S1] p. 3; [S3]
Grade 12 · Term 2	12.1–12.3	Inverse trigonometric functions	8	I	no page yet	[S1] pp. 3–4
Grade 12 · Term 2	11.2	Sine rule, cosine rule (continued)	6	I	Notes · Practice	[S1] p. 3; [S3]
Grade 12 · Term 2	13.1–13.8	Limits	18	I	Notes · Practice	[S1] p. 4; [S3]
Grade 12 · Term 2	2.4–2.7	Systems of coplanar forces on a rigid body	23	II	Notes · Practice	[S1] p. 4; [S3]
Grade 12 · Term 2	3.1–3.3	Motion in a straight line	23	II	Notes · Practice	[S1] p. 4; [S3]
Grade 12 · Term 3	14.1–14.8	Derivatives	30	I	Notes · Practice	[S1] p. 4; [S3]
Grade 12 · Term 3	15.1–15.4	Applications of derivatives	15	I	no page yet	[S1] p. 4
Grade 12 · Term 3	3.7	Projectiles	10	II	no page yet	[S1] p. 4
Grade 12 · Term 3	2.8	Equilibrium of three coplanar forces	8	II	no page yet	[S1] p. 4
Grade 12 · Term 3	2.9	Friction	10	II	no page yet	[S1] p. 4
Grade 12 · Term 3	3.4–3.6	Relative motion	22	II	no page yet	[S1] p. 4
Grade 12 · Term 3	3.8	Newton's laws of motion	17	II	Notes · Practice	[S1] p. 4; [S3]

Topics, competency levels, periods and terms from [S1] §7.0, pp. 3–4; the pages are this site's own [S3]. The sine rule and cosine rule appear twice because the NIE splits them across two terms (11.1 and 11.2). Newton's laws of motion (3.8) and motion in a straight line (3.1–3.3) share one page. The NIE's summary gives Grade 12 309 periods, one fewer than these rows add up to: it prints 24 for the two first-term Combined Mathematics II topics, which add up to 25 ([S1] pp. 3, 7).

Grade 13

Grade 13: the NIE's own topics, in the term it proposes for each, with the teaching periods it gives them.

Grade · term (NIE)	Competency levels	NIE subject topic	Periods	Part	On this site	Source
Grade 13 · Term 1	18.1–18.5	Straight line	16	I	Notes · Practice	[S1] p. 5; [S3]
Grade 13 · Term 1	16.1–16.9	Integration	28	I	Notes · Practice	[S1] p. 5; [S3]
Grade 13 · Term 1	2.10	Jointed rods	10	II	no page yet	[S1] p. 5
Grade 13 · Term 1	2.11	Frame works	10	II	Notes · Practice	[S1] p. 5; [S3]
Grade 13 · Term 1	3.11–3.13	Impulse and collision	16	II	no page yet	[S1] p. 5
Grade 13 · Term 1	3.9, 3.10	Work, power, energy	10	II	no page yet	[S1] p. 5
Grade 13 · Term 1	3.14–3.16	Circular motion	20	II	Notes · Practice	[S1] p. 5; [S3]
Grade 13 · Term 2	26.1, 27.1–27.5	Circle	15	I	Notes · Practice	[S1] p. 5; [S3]
Grade 13 · Term 2	24.1–24.3, 24.5	Permutations and combinations	15	I	Notes · Practice	[S1] p. 5; [S3]
Grade 13 · Term 2	19.1	Principle of mathematical induction	5	I	Notes · Practice	[S1] p. 5; [S3]
Grade 13 · Term 2	20.1, 20.2, 21.1, 21.2	Series	18	I	Notes · Practice	[S1] p. 5; [S3]
Grade 13 · Term 2	4.1, 4.2	Probability	10	II	Notes · Practice	[S1] p. 6; [S3]
Grade 13 · Term 2	3.17–3.19	Simple harmonic motion	18	II	Notes · Practice	[S1] p. 6; [S3]
Grade 13 · Term 2	2.12–2.17	Centre of mass	20	II	Notes · Practice	[S1] p. 6; [S3]
Grade 13 · Term 3	22.1–22.3	Binomial expansion	12	I	Notes · Practice	[S1] p. 6; [S3]
Grade 13 · Term 3	23.1–23.6	Complex numbers	18	I	Notes · Practice	[S1] p. 6; [S3]
Grade 13 · Term 3	25.1–25.4	Matrices	14	I	Notes · Practice	[S1] p. 6; [S3]
Grade 13 · Term 3	4.3–4.5	Probability (continued)	18	II	Notes · Practice	[S1] p. 6; [S3]
Grade 13 · Term 3	5.1–5.9	Statistics	18	II	Notes · Practice	[S1] p. 6; [S3]

Topics, competency levels, periods and terms from [S1] §7.0, pp. 5–6; the pages are this site's own [S3]. Where a topic spans several pages (circle, series, probability, statistics, circular motion, simple harmonic motion, centre of mass, complex numbers), the row links the first and its term page lists the rest.

Term-by-Term Breakdown

Try one question — no account needed

Model question — written by Idasara, modelled on the G.C.E. (A/L) Combined Mathematics examination pattern. Combined Mathematics I, competency 3: quadratic functions and quadratic equations, 25 periods.

If α and β are the distinct real roots of the quadratic equation x² - 5x + 3 = 0, what is the exact value of α² + β²?

A) 19
B) 22
C) 25
D) 31

Check my answer

Correct answer: A) 19

From Vieta's formulas for ax² + bx + c = 0: Sum of roots α + β = -b/a = 5, Product of roots αβ = c/a = 3. Using the algebraic identity α² + β² = (α + β)² - 2αβ = (5)² - 2(3) = 25 - 6 = 19.

Show worked solution

Step 1: Identify root relationships: α + β = 5, αβ = 3. Step 2: Use identity: α² + β² = (α + β)² - 2αβ. Step 3: Calculate: 5² - 2(3) = 25 - 6 = 19. Answer: 19.

Get your own answer marked

How the exam works

G.C.E. A/L Combined Mathematics is set by the Department of Examinations in two papers (the structure for the examination from 2019 onwards). Each paper takes 3 hours, plus 10 minutes' reading time, and has two parts: Part A, 10 compulsory questions at 25 marks each (250); Part B, answer 5 of 7 questions at 150 marks each (750) — 1,000 marks a paper. The final mark is the two papers' total divided by 20, out of 100. (Department of Examinations structure booklet, p. 115.)

Department of Examinations — Structure of Question Papers, G.C.E. (A/L) 2019 and afterwards, Volume 1 (English, PDF)

Where should I start?

One short block per reader, one link each. These are suggestions, not rules; the tables above are the map.

Starting Grade 12 — The NIE's first term is twelve short pure-maths topics plus vectors and coplanar forces; trigonometric identities, inequalities and vectors carry the most time in it, 14 periods each, then circular functions at 12. → Circular functions notes
In Grade 13 — The year opens with the straight line and integration, 44 of its first term's 110 periods. → Integration notes
Behind on the applied part — The NIE's summary gives Combined Mathematics II 287 of the 600 periods, nearly half the course. Start where its time is: coplanar forces on a rigid body and motion in a straight line, 23 periods each. → Coplanar forces on a rigid body notes
Sitting the examination again — Practise topic by topic, then get your answers marked. → All A/L Combined Maths practice
Studying in Sinhala → සිංහල
Studying in Tamil → தமிழ்
A parent — What your child gets, and what it costs. → Pricing · For parents

Free study resources by topic

Guides

Questions students ask

Is Combined Mathematics one subject or two?

One subject in two parts: Combined Mathematics I, the pure mathematics, and Combined Mathematics II, the applied mathematics. The NIE's syllabus is published in those two parts. ([S1] §7.0)

How much teaching time does it get?

The NIE's summary allocates 600 periods over the two years: 309 in Grade 12 (94, 103 and 112 by term) and 291 in Grade 13 (110, 101 and 80). ([S1] p. 7)

Which topics carry the most time?

Derivatives at 30 periods, integration at 28, quadratic functions and quadratic equations at 25, then two applied topics at 23 each — systems of coplanar forces on a rigid body, and motion in a straight line — and relative motion at 22. ([S1] §7.0, pp. 3–6)

When do projectiles and circular motion come?

Projectiles in Grade 12, third term, for 10 periods; circular motion in Grade 13, first term, for 20. ([S1] pp. 4–5)

Do I have to follow the NIE's order?

No. The NIE calls it a proposed term-wise breakdown, so your class may take the topics in a different order — which is why this page lists the topic, not just the term. ([S1] §7.0 heading)

Is this the same as Cambridge Further Mathematics?

No. This is the Sri Lankan G.C.E. (A/L) syllabus published by the NIE, with its own two parts, its own period weights and its own examination. ([S1] title page)