Grade 12 Pure Maths Notes - Every Topic, Formula and Key Fact You Need

Grade 12 Pure Maths Notes - Every Topic, Formula and Key Fact You Need

This is your one-page reference for the entire Grade 12 Maths CAPS syllabus. Every topic. Every formula. Every key fact. Bookmark this page and use it every time you sit down to study. Whether you are preparing for June exams, prelims, or the NSC finals, everything you need is right here.

This is Question 1 in every Paper 1. It is your warm-up and your mark bank. It covers factorisation, quadratic equations (three methods), simultaneous equations (one linear, one quadratic), surd equations, quadratic inequalities, and the nature of roots using the discriminant.

The three methods for quadratics: factorise when it is clean, use the quadratic formula when it is ugly, and complete the square when the question asks for it or you need exact answers. The discriminant (b² - 4ac) tells you how many real roots exist before you solve. If the question asks "for which values of k does the equation have real roots," set the discriminant ≥ 0 and solve the inequality.

For simultaneous equations, always substitute into the linear equation, not the quadratic. For surds, square both sides but always check your answers because squaring introduces extraneous solutions. For inequalities, find the critical values, test intervals on a number line, and write the solution as an interval.

Key formulas:

x = [-b ± √(b² - 4ac)] / 2a

Δ = b² - 4ac (discriminant)

This is one of the most predictable topics in the exam. You need to identify whether a sequence is arithmetic (constant difference) or geometric (constant ratio), then apply the right formula. The exam always asks for a specific term, a sum, or a proof that a sequence is arithmetic or geometric.

Sigma notation is shorthand for "add up these terms." If you see Σ, substitute each value of k and add. For sum to infinity, check that |r| < 1 before applying the formula. If |r| ≥ 1, the series diverges and has no sum to infinity. The exam loves asking "show that the series converges" which means prove |r| < 1.

Key formulas:

Arithmetic: Tn = a + (n-1)d | Sn = n/2[2a + (n-1)d]

Geometric: Tn = ar^(n-1) | Sn = a(r^n - 1)/(r - 1)

Sum to infinity: S∞ = a/(1-r), |r| < 1

One of the two biggest topics in Paper 1. You must know four graph types: parabola y = a(x-p)² + q, hyperbola y = a/(x-p) + q, exponential y = a.b^(x-p) + q, and the logarithmic function y = log_b(x) which is the inverse of the exponential.

For every function, you need to find intercepts, turning points or asymptotes, domain, range, and be able to sketch it accurately. The inverse function (logarithmic) appears every year since it was added to CAPS. You must be able to sketch a function and its inverse on the same axes, reflected in the line y = x.

The "read from graph" questions are often the easiest marks. "For which values of x is f(x) > 0?" means where is the graph above the x-axis. "For which values of x is f(x) > g(x)?" means where is f higher than g. These are 2 to 3 marks each and require reading, not calculating.

Key facts: Turning point form: (p, q). For y = a(x-p)²+q, if you see (x-3), then p = 3 (not -3). Domain of hyperbola excludes the vertical asymptote. Range of parabola starts at the turning point. The log function has domain x > 0 and vertical asymptote at x = 0.

The formulas are on your formula sheet. Your job is knowing which one to use and getting the calculator input right. Simple growth/decay uses A = P(1 ± in). Compound growth/decay uses A = P(1 ± i)^n. The annuity formulas (future value Fv and present value Pv) are for regular payments over time like sinking funds, loans, and retirement savings.

The nominal-effective interest rate conversion is tested regularly. To convert nominal to effective: i_eff = (1 + i_nom/m)^m - 1, where m is the number of compounding periods per year. Always make sure your n (number of periods) matches your i (interest rate per period). If interest is monthly, convert i to monthly and n to months.

Calculator trap: Always bracket (1 + i) or (1 - i) before raising to the power. Typing 1 + 0.08^12 gives a completely different answer to (1.08)^12.

Joint biggest topic in Paper 1 with functions. Calculus covers differentiation from first principles (the limit definition), the power rule, finding equations of tangent lines, sketching cubic graphs, and optimisation or rate of change application problems.

For cubic graph sketching, follow this method every time: (1) find f'(x) and set it to 0 for turning points, (2) substitute back to find y-values of turning points, (3) find y-intercept by setting x = 0, (4) find x-intercepts by setting f(x) = 0, (5) determine concavity using f''(x), (6) plot all points and sketch. Show every step. The marks are in the working, not the final sketch.

The optimisation question is usually the last sub-part and carries 5 to 6 marks. You are given a word problem, need to set up the equation, differentiate, and find the maximum or minimum. Read the question twice. Draw a diagram. Label everything. Then differentiate.

Key rules: Power rule: if f(x) = x^n then f'(x) = nx^(n-1). Gradient of tangent at a point = f'(a). At a turning point f'(x) = 0. Point of inflection: f''(x) = 0.

Most students skip this. That is 15 free marks they are giving away. The counting principle says: multiply the number of choices at each step. For arrangements with restrictions, deal with the restricted position first. If letters are repeated, divide by the factorial of each repeated group.

For probability: P(A or B) = P(A) + P(B) - P(A and B). For "at least one" questions, use the complement: P(at least one) = 1 - P(none). Mutually exclusive means P(A and B) = 0. Independent means P(A and B) = P(A) x P(B). The exam always asks you to test for independence using this formula. Venn diagram questions: fill in the overlap first, then work outward.

Key fact: Mutually exclusive events are NEVER independent (unless one has probability 0). Students confuse these every year.

The easiest topic in Paper 2. Your calculator does most of the work. You need to calculate mean and standard deviation (use σ, not s), draw or interpret scatter plots, calculate the regression line equation (ŷ = a + bx), and interpret the correlation coefficient (r). The mean point (x̄, ȳ) always lies on the regression line. Use this to check your line.

Interpolation (predicting within the data range) is reliable. Extrapolation (predicting outside) is unreliable. The exam asks "is this prediction reliable?" almost every year. Know the difference and explain why.

Circle equation: (x-a)² + (y-b)² = r². If in general form, complete the square to find the centre and radius. To test if a point is inside/on/outside a circle, substitute into the left side and compare to r². Tangent to a circle: find gradient of radius (centre to point), then tangent gradient = negative reciprocal (m₁ x m₂ = -1). Use point-gradient form for the equation.

The grade 10/11 formulas (distance, midpoint, gradient) are still essential. The golden rule: tangent is perpendicular to radius at the point of tangency. This single fact unlocks every tangent question.

Three sub-sections: compound/double angle identities, general solutions, and 2D/3D trig problems. For identities, pick the more complicated side and simplify. Use cos2A = cos²A - sin²A (or the other two forms depending on context). "Sin keeps the sign, cos changes" helps remember compound angle signs.

General solutions: sinθ = sinα gives θ = α + k.360° or θ = (180° - α) + k.360°. cosθ = cosα gives θ = α + k.360° or θ = -α + k.360°. tanθ = tanα gives θ = α + k.180°. Always test k = -1, 0, and 1 when finding specific solutions in a restricted range.

Key formulas:

sin(A±B) = sinAcosB ± cosAsinB

cos(A±B) = cosAcosB ∓ sinAsinB

sin2A = 2sinAcosA | cos2A = cos²A - sin²A

The biggest single topic. It covers circle theorems, cyclic quadrilateral properties, tangent theorems (including tan-chord), the proportionality theorem, and similarity of triangles. Every statement in a proof needs a reason in brackets. No reason = no mark.

The proportionality theorem proof has appeared in almost every NSC exam since it was added to CAPS. It uses areas of triangles and is worth 6 to 8 marks. Memorise it. Write it from memory. That is free bookwork marks. The tan-chord theorem is the most tested theorem overall. If you see a tangent and a chord from the same point, immediately apply it.

For riders: mark all equal angles on the diagram first, identify cyclic quads and tangents, then write statements with reasons. Even partial attempts earn marks. Never leave a rider blank.

Key reasons: ∠ at centre = 2 × ∠ at circumference | ∠s in same segment | opp ∠s cyclic quad supp | tan-chord angle | line ∥ one side of △ | equiangular △s