Study Notes • Grade 12 Pure Mathematics
Grade 12 Pure Maths Notes — Every Topic, Formula and Key Fact You Need
All ten CAPS topics covered in order of mark importance. Formulas explained, key facts listed, and common exam traps flagged — everything you need to revise every topic in Pure Maths before the NSC.
These notes cover the complete Grade 12 Pure Mathematics CAPS syllabus, organised by mark weight so you know exactly where to focus your time. Each topic includes the key formulas, what you must know from memory, the most common exam traps, and a link to the full in-depth guide for that topic.
Use this page as your master revision reference — work through it topic by topic, use the full guides for deeper understanding, and practise every formula against past paper questions before the exam.
How Marks Are Distributed — Know This Before You Revise
| Topic | Paper | Approx. Marks | Priority | Full Guide |
|---|---|---|---|---|
| Functions and Graphs | P1 | ~35 | Highest | Guide |
| Differential Calculus | P1 | ~35 | Highest | Guide |
| Euclidean Geometry | P2 | ~40 | Highest | Guide |
| Trigonometry | P2 | ~40 | Highest | Guide |
| Analytical Geometry | P2 | ~40 | Highest | Guide |
| Sequences and Series | P1 | ~25 | High | Guide |
| Algebra and Equations | P1 | ~25 | High | Guide |
| Finance, Growth and Decay | P1 | ~15 | High | Guide |
| Probability and Counting | P1 | ~15 | High | Guide |
| Statistics and Regression | P2 | ~20 | High | Guide |
Paper 1 — Functions and Graphs (~35 marks)
Functions and Graphs
~35 marks — Paper 1
Parabola (quadratic)
y = a(x − p)² + q or y = ax² + bx + c
Turning point at (p, q) in vertex form. Axis of symmetry: x = p. For standard form: x = −b/2a gives the axis of symmetry.
Hyperbola
y = a/(x − p) + q
Vertical asymptote: x = p. Horizontal asymptote: y = q. Axes of symmetry: y = x − p + q and y = −x + p + q.
Exponential
y = a · b^(x−p) + q , b > 0, b ≠ 1
Horizontal asymptote: y = q. If b > 1: increasing. If 0 < b < 1: decreasing. If a < 0: reflection over horizontal asymptote.
Logarithm (inverse of exponential)
y = log_b(x − p) + q is the inverse of y = b^(x−q) + p
Vertical asymptote: x = p. Domain: x > p. To find the inverse, swap x and y, then solve for y.
- The domain of a function is all valid x-values. The range is all resulting y-values.
- Increasing function: as x increases, y increases. Decreasing: as x increases, y decreases.
- To find x-intercepts: set y = 0 and solve. To find y-intercept: set x = 0.
- A function is one-to-one if each y-value corresponds to exactly one x-value — passes the horizontal line test.
- The inverse of f(x) is written f⁻¹(x). To find it: replace f(x) with y, swap x and y, solve for y.
Common exam trap
When sketching a hyperbola, students often forget to draw both branches. There are always two branches — one in each relevant quadrant. Also: the asymptotes are not part of the graph but must be shown as dotted lines.
Paper 1 — Differential Calculus (~35 marks)
Differential Calculus
~35 marks — Paper 1
First Principles
f'(x) = lim[h→0] [f(x+h) − f(x)] / h
Only use when the question specifically says "first principles." Otherwise use the power rule — it is much faster.
Power Rule
If f(x) = xⁿ, then f'(x) = nxⁿ⁻¹
Multiply by the exponent, reduce the exponent by 1. Constants differentiate to 0. Apply term by term.
Gradient at a Point
m = f'(a) — substitute x = a into the derivative
Equation of tangent at point (a, f(a)): y − f(a) = f'(a)(x − a)
Stationary Points
Set f'(x) = 0 and solve for x
Local maximum: f'(x) changes from + to −. Local minimum: f'(x) changes from − to +. Point of inflection: f''(x) = 0.
- Before differentiating: expand all brackets and convert roots/fractions to index notation (e.g. √x = x^½, 1/x² = x⁻²).
- Second derivative f''(x): differentiating f'(x) again. Used to confirm max/min and find points of inflection.
- Increasing: f'(x) > 0. Decreasing: f'(x) < 0. Stationary: f'(x) = 0.
- For cubic graph sketches: find x-intercepts (factorise), y-intercept (x=0), stationary points (f'(x) = 0), and end behaviour.
- Optimisation: set up an expression for the quantity to optimise, differentiate, set equal to zero, solve.
Common exam trap
Students differentiate 3x² as 3x instead of 6x — forgetting to multiply by the existing coefficient. The power rule applies to the x term only: d/dx[3x²] = 3 × 2x = 6x, not 3x. Write out each step clearly.
Paper 1 — Sequences and Series (~25 marks)
Sequences and Series
~25 marks — Paper 1
Arithmetic Sequence
T_n = a + (n−1)d | S_n = n/2(2a + (n−1)d) | S_n = n/2(a + l)
d = common difference = T₂ − T₁. Use last formula when last term l is known.
Geometric Sequence
T_n = ar^(n−1) | S_n = a(rⁿ − 1)/(r − 1) | S_∞ = a/(1−r), |r| < 1
r = common ratio = T₂ ÷ T₁. S_∞ only exists when |r| < 1. Always state this condition.
Sigma Notation
Σ from k=1 to n of T_k = T₁ + T₂ + T₃ + ... + T_n
The sigma sign means sum. Identify whether the sequence is arithmetic or geometric, then use the appropriate sum formula.
- T_n = the nth TERM. S_n = the SUM of n terms. These are different — do not confuse them.
- To find how many terms: set T_n equal to the last term given and solve for n.
- For sum of infinite geometric series: the ball bouncing problem. Total distance = first drop + 2 × S_∞ (for all bounces).
- Quadratic sequences: second differences are constant. T_n = an² + bn + c. Find a first (a = second difference ÷ 2).
Paper 1 — Algebra and Equations (~25 marks)
Algebra, Equations and Inequalities
~25 marks — Paper 1
Quadratic Formula
x = (−b ± √(b² − 4ac)) / 2a
Use when factorisation is not obvious. The discriminant Δ = b² − 4ac: Δ > 0 gives two real roots, Δ = 0 gives equal roots, Δ < 0 gives no real roots.
Nature of Roots (Discriminant)
Δ = b² − 4ac
Δ > 0 and a perfect square: two rational unequal roots. Δ > 0 but not a perfect square: two irrational roots. Δ = 0: equal rational roots. Δ < 0: no real roots.
- Factor theorem: if f(a) = 0 then (x − a) is a factor. Use to factorise cubic expressions.
- Remainder theorem: if f(x) is divided by (x − a), the remainder = f(a).
- Simultaneous equations with a linear and quadratic: substitute the linear equation into the quadratic.
- Exponential equations: get same base on both sides, then equate exponents. If impossible, use logarithms.
- Surds: rationalise denominators by multiplying by the conjugate. Never leave a surd in a denominator.
- For inequalities with quadratics: factorise, draw a number line, test each region.
Paper 1 — Finance, Growth and Decay (~15 marks)
Finance, Growth and Decay
~15 marks — Paper 1
Simple and Compound Growth/Decay
A = P(1 + in) | A = P(1 + i)ⁿ | A = P(1 − i)ⁿ
Simple: interest on original amount only. Compound: interest on growing total. Compound decay: replace + with −.
Annuities
F = x[(1+i)ⁿ − 1]/i | P = x[1 − (1+i)^(−n)]/i
F = future value (savings/sinking fund). P = present value (loan/bond repayment). x = regular payment. Always convert annual rate to period rate.
- For monthly compounding: divide annual rate by 12 and multiply n by 12.
- Effective annual rate: (1 + i_nominal/m)^m − 1 where m = number of compounding periods per year.
- Home loan/bond = present value formula. Retirement annuity = future value formula.
- Sinking fund: saving to replace an asset. Use future value formula.
- Outstanding balance = present value of remaining payments.
Paper 1 — Probability and Counting (~15 marks)
Probability and Counting Principle
~15 marks — Paper 1
Addition Rule
P(A or B) = P(A) + P(B) − P(A and B)
For mutually exclusive events: P(A and B) = 0, so P(A or B) = P(A) + P(B).
Complementary Events
P(not A) = 1 − P(A)
Multiplication Rule (independent events)
P(A and B) = P(A) × P(B)
Only valid when A and B are independent — the occurrence of A does not affect P(B).
Counting — Arrangements and Selections
n! = n × (n−1) × (n−2) × ... × 2 × 1
Arrangements of n different objects: n! | With restriction: fix the restricted items first, then arrange the rest. | Identical objects: divide by factorial of identical count.
- Mutually exclusive events cannot happen at the same time: P(A and B) = 0.
- Independent events: P(A) is not affected by B occurring. Test: P(A and B) = P(A) × P(B).
- Fundamental counting principle: if event A can happen in m ways and event B in n ways, total = m × n.
- Tree diagrams: useful for two or three-stage experiments. Each branch represents an outcome.
- Venn diagrams: use for two or three overlapping events. Label all regions including outside both circles.
Paper 2 — Euclidean Geometry (~40 marks)
Euclidean Geometry and Proofs
~40 marks — Paper 2
- Theorem 1 — Line from centre perpendicular to chord bisects the chord. Converse: line from centre to midpoint of chord is perpendicular to the chord.
- Theorem 2 — Angle at centre = 2 × angle at circumference (subtended by the same arc).
- Theorem 3 — Angles in the same segment are equal. Angles subtended by the same chord on the same side.
- Theorem 4 — Opposite angles of a cyclic quadrilateral are supplementary (add up to 180°).
- Theorem 5 — Tangent-radius is perpendicular. A tangent to a circle is perpendicular to the radius at the point of contact.
- Theorem 6 — Tangent-chord angle = inscribed angle in alternate segment (tan-chord theorem).
- Theorem 7 — Two tangents from same external point are equal in length.
- Proportionality theorem: A line parallel to one side of a triangle divides the other two sides proportionally.
- Similar triangles: Corresponding angles equal AND corresponding sides in proportion.
How proofs are marked
Every step in a geometry proof needs two parts: the statement (what is true) and the reason (why it is true). Writing a statement without a reason or a reason without a statement loses half the marks for that step. Common accepted reasons: "∠s in same seg", "opp ∠s cyc quad", "tan ⊥ rad", "tan-chord theorem", "prop theorem". Know the short forms.
Paper 2 — Trigonometry (~40 marks)
Trigonometry — Compound Angles, General Solutions
~40 marks — Paper 2
Compound Angle Formulas (on formula sheet)
sin(A±B) = sinA cosB ± cosA sinB
cos(A±B) = cosA cosB ∓ sinA sinB
Double Angle Formulas (on formula sheet)
sin2A = 2sinA cosA
cos2A = cos²A − sin²A = 2cos²A − 1 = 1 − 2sin²A
General Solution
sin θ = k → θ = arcsin(k) + 360°n OR θ = 180° − arcsin(k) + 360°n
cos θ = k → θ = ±arccos(k) + 360°n
tan θ = k → θ = arctan(k) + 180°n
n ∈ Z (n is any integer). For a specific interval, substitute n = 0, ±1, ±2 etc. and check which solutions fall in the interval.
- Identities you must know: sin²θ + cos²θ = 1, tanθ = sinθ/cosθ, 1 + tan²θ = sec²θ.
- Reduction formulas: sin(180° − θ) = sinθ, cos(180° − θ) = −cosθ, sin(90° − θ) = cosθ, cos(90° − θ) = sinθ.
- CAST diagram: All positive in Q1, Sin positive in Q2, Tan positive in Q3, Cos positive in Q4.
- For proofs: start on the more complex side. Work toward the simpler side. Never move things across the equals sign.
- Exact values: sin30°=½, cos30°=√3/2, tan30°=1/√3, sin45°=cos45°=1/√2, sin60°=√3/2, cos60°=½.
Paper 2 — Analytical Geometry (~40 marks)
Analytical Geometry — Lines, Circles, Tangents
~40 marks — Paper 2
Core Formulas (on formula sheet)
d = √[(x₂−x₁)² + (y₂−y₁)²] | M = ((x₁+x₂)/2, (y₁+y₂)/2)
m = (y₂−y₁)/(x₂−x₁) | (x−a)² + (y−b)² = r²
Line Equations (NOT on formula sheet — memorise)
y = mx + c | y − y₁ = m(x − x₁)
Parallel lines: equal gradients (m₁ = m₂). Perpendicular lines: m₁ × m₂ = −1.
- To find the equation of a circle: need centre (a, b) and radius r. Substitute one point to find r if not given.
- To find centre and radius from expanded form: complete the square for x and y separately.
- Tangent to a circle: perpendicular to the radius at the point of contact. Use m_radius × m_tangent = −1.
- Chord of a circle: connect two points on the circle. Perpendicular bisector of a chord passes through the centre.
- To prove a point lies on a circle: substitute coordinates into the circle equation — if LHS = RHS, it lies on the circle.
Paper 2 — Statistics and Regression (~20 marks)
Statistics, Regression and the Line of Best Fit
~20 marks — Paper 2
- Mean: x̄ = Σx/n. Use calculator STAT mode — enter data, read x̄ directly.
- Standard deviation: Use calculator STAT mode — read σ_x. Do not calculate by hand.
- Variance: σ² = (standard deviation)². Not usually required directly but understand the concept.
- Regression line (line of best fit): ŷ = a + bx. Calculated using calculator STAT regression function. Always check your calculator mode is LINEAR.
- Correlation coefficient (r): −1 ≤ r ≤ 1. r close to 1 = strong positive. r close to −1 = strong negative. r close to 0 = weak/no linear relationship.
- Interpolation: predicting within the data range. Extrapolation: predicting outside the data range — less reliable.
- Histograms, frequency polygons, ogives (cumulative frequency curves): know how to draw and interpret each.
- Quartiles: Q1 = lower quartile (25th percentile), Q2 = median (50th), Q3 = upper quartile (75th). IQR = Q3 − Q1.
- Outliers: values more than 1.5 × IQR below Q1 or above Q3.
Calculator method — memorise this
To find mean, standard deviation, and regression line: Mode → STAT → enter data in L1 and L2 → CALC → choose linear regression (a + bx). Practice this on your calculator before the exam — do it until it is automatic.
How to use these notes effectively: Read through the topic, identify the formulas you do not yet know by heart, then immediately do past paper questions on that topic. Reading notes without doing questions builds very little exam skill. The notes show you what to know — the past papers show you whether you actually know it.
For past papers to practise against, read our guide on where to download Grade 12 Maths past papers free. For the formula sheet used in the exam, read our full Grade 12 Maths formula sheet guide.
Notes are the foundation. Live lessons build the skill.
A-Game Academy teaches Grade 12 Pure Maths live via Zoom — every topic on this page, every question type the NSC uses. Mr Sawaya — 30 years experience, SACE registered, CAPS specialist.Past paper practice built into every term.
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