Grade 12 Mathematics — Full Year Study Notes CAPS Aligned | Full Year | South Africa
GRADE 12
MATHEMATICS
Complete Full-Year Study Notes
Notes · Worked Examples · Practice Exercises · Answers
Term Topics Covered
Term 1 Algebra & Equations, Number Patterns, Functions & Graphs
Term 2 Trigonometry, Analytical Geometry, Euclidean Geometry
Term 3 Statistics, Counting & Probability, Finance
Term 4 Calculus (Differentiation & Applications), Exam Revision
CAPS Aligned · Full Year · Worked Examples Included · Exercises with Answers
© Grade 12 Mathematics Study Notes — All rights reserved
,Grade 12 Mathematics — Full Year Study Notes CAPS Aligned | Full Year | South Africa
1. Algebra & Equations (Term 1)
Equations form the foundation of all Grade 12 Mathematics. Master these before moving on.
1.1 Quadratic Equations
Method 1 — Factorisation
• Move all terms to one side: ax² + bx + c = 0
• Find two numbers that multiply to ac and add to b.
• Write in factored form and solve each bracket = 0.
Example: Solve x² + 5x + 6 = 0
→ Find two numbers: multiply to 6, add to 5 → 2 and 3
→ (x + 2)(x + 3) = 0
→ x = −2 or x = −3
Method 2 — Quadratic Formula
x = [−b ± √(b²−4ac)] / 2a
Example: Solve 2x² − 3x − 2 = 0 (a=2, b=−3, c=−2)
→ x = [3 ± √(9 + 16)] / 4 = [3 ± √25] / 4 = [3 ± 5] / 4
→ x = 8/4 = 2 or x = −2/4 = −½
The Discriminant (Δ = b²−4ac)
Condition Roots Graph Meaning
Δ>0 Two different real roots (unequal) Graph cuts x-axis at two points
Δ=0 Two equal real roots (one solution) Graph touches x-axis at one point
Δ<0 No real roots Graph does not touch x-axis
EXAM TIP: Always calculate the discriminant FIRST before deciding which method to use. If Δ
is a perfect square → use factorisation. Otherwise → use the formula.
1.2 Simultaneous Equations
Substitution Method (one linear, one quadratic)
1. Rearrange the linear equation to express one variable (e.g. y = mx + c).
2. Substitute into the other equation.
3. Solve the resulting equation.
4. Substitute back to find the other variable.
© Grade 12 Mathematics Study Notes — All rights reserved
, Grade 12 Mathematics — Full Year Study Notes CAPS Aligned | Full Year | South Africa
Example: Solve y = x + 2 and x² + y² = 10
→ Substitute y = x + 2: x² + (x+2)² = 10
→ x² + x² + 4x + 4 = 10
→ 2x² + 4x − 6 = 0 → x² + 2x − 3 = 0
→ (x+3)(x−1) = 0 → x = −3 or x = 1
→ When x = −3: y = −1. When x = 1: y = 3
→ Solutions: (−3, −1) and (1, 3)
1.3 Inequalities
• Solve like a normal equation BUT: when you multiply/divide by a negative number → FLIP the
inequality sign.
• For quadratic inequalities: find roots first, then test regions.
Example: Solve x² − x − 6 > 0
→ Factorise: (x−3)(x+2) > 0
→ Roots: x = 3 and x = −2
→ Test regions: x < −2 (positive ✓), −2 < x < 3 (negative ✗), x > 3 (positive ✓)
→ Answer: x < −2 or x > 3
REMEMBER: For ax² + bx + c > 0 (a>0): answer is the OUTSIDE regions (less than smaller root
or greater than larger root). For < 0: answer is the BETWEEN region.
Practice — Algebra & Equations
5. Solve: x² − 7x + 12 = 0
6. Solve using the formula: 3x² + 5x − 2 = 0
7. For what values of k does x² + kx + 9 = 0 have equal roots?
8. Solve simultaneously: y = 2x − 1 and x² + y = 5
9. Solve: x² − x − 12 < 0
Answers:
10. x = 3 or x = 4
11. x = 1/3 or x = −2
12. k = ±6 (Δ = 0 → k² − 36 = 0)
13. x = 2, y = 3 and x = −3, y = −7
14. −3 < x < 4
2. Number Patterns — Sequences & Series (Term 1)
2.1 Arithmetic Sequences
General term: Tₙ = a + (n−1)d where a = first term, d = common difference
© Grade 12 Mathematics Study Notes — All rights reserved
GRADE 12
MATHEMATICS
Complete Full-Year Study Notes
Notes · Worked Examples · Practice Exercises · Answers
Term Topics Covered
Term 1 Algebra & Equations, Number Patterns, Functions & Graphs
Term 2 Trigonometry, Analytical Geometry, Euclidean Geometry
Term 3 Statistics, Counting & Probability, Finance
Term 4 Calculus (Differentiation & Applications), Exam Revision
CAPS Aligned · Full Year · Worked Examples Included · Exercises with Answers
© Grade 12 Mathematics Study Notes — All rights reserved
,Grade 12 Mathematics — Full Year Study Notes CAPS Aligned | Full Year | South Africa
1. Algebra & Equations (Term 1)
Equations form the foundation of all Grade 12 Mathematics. Master these before moving on.
1.1 Quadratic Equations
Method 1 — Factorisation
• Move all terms to one side: ax² + bx + c = 0
• Find two numbers that multiply to ac and add to b.
• Write in factored form and solve each bracket = 0.
Example: Solve x² + 5x + 6 = 0
→ Find two numbers: multiply to 6, add to 5 → 2 and 3
→ (x + 2)(x + 3) = 0
→ x = −2 or x = −3
Method 2 — Quadratic Formula
x = [−b ± √(b²−4ac)] / 2a
Example: Solve 2x² − 3x − 2 = 0 (a=2, b=−3, c=−2)
→ x = [3 ± √(9 + 16)] / 4 = [3 ± √25] / 4 = [3 ± 5] / 4
→ x = 8/4 = 2 or x = −2/4 = −½
The Discriminant (Δ = b²−4ac)
Condition Roots Graph Meaning
Δ>0 Two different real roots (unequal) Graph cuts x-axis at two points
Δ=0 Two equal real roots (one solution) Graph touches x-axis at one point
Δ<0 No real roots Graph does not touch x-axis
EXAM TIP: Always calculate the discriminant FIRST before deciding which method to use. If Δ
is a perfect square → use factorisation. Otherwise → use the formula.
1.2 Simultaneous Equations
Substitution Method (one linear, one quadratic)
1. Rearrange the linear equation to express one variable (e.g. y = mx + c).
2. Substitute into the other equation.
3. Solve the resulting equation.
4. Substitute back to find the other variable.
© Grade 12 Mathematics Study Notes — All rights reserved
, Grade 12 Mathematics — Full Year Study Notes CAPS Aligned | Full Year | South Africa
Example: Solve y = x + 2 and x² + y² = 10
→ Substitute y = x + 2: x² + (x+2)² = 10
→ x² + x² + 4x + 4 = 10
→ 2x² + 4x − 6 = 0 → x² + 2x − 3 = 0
→ (x+3)(x−1) = 0 → x = −3 or x = 1
→ When x = −3: y = −1. When x = 1: y = 3
→ Solutions: (−3, −1) and (1, 3)
1.3 Inequalities
• Solve like a normal equation BUT: when you multiply/divide by a negative number → FLIP the
inequality sign.
• For quadratic inequalities: find roots first, then test regions.
Example: Solve x² − x − 6 > 0
→ Factorise: (x−3)(x+2) > 0
→ Roots: x = 3 and x = −2
→ Test regions: x < −2 (positive ✓), −2 < x < 3 (negative ✗), x > 3 (positive ✓)
→ Answer: x < −2 or x > 3
REMEMBER: For ax² + bx + c > 0 (a>0): answer is the OUTSIDE regions (less than smaller root
or greater than larger root). For < 0: answer is the BETWEEN region.
Practice — Algebra & Equations
5. Solve: x² − 7x + 12 = 0
6. Solve using the formula: 3x² + 5x − 2 = 0
7. For what values of k does x² + kx + 9 = 0 have equal roots?
8. Solve simultaneously: y = 2x − 1 and x² + y = 5
9. Solve: x² − x − 12 < 0
Answers:
10. x = 3 or x = 4
11. x = 1/3 or x = −2
12. k = ±6 (Δ = 0 → k² − 36 = 0)
13. x = 2, y = 3 and x = −3, y = −7
14. −3 < x < 4
2. Number Patterns — Sequences & Series (Term 1)
2.1 Arithmetic Sequences
General term: Tₙ = a + (n−1)d where a = first term, d = common difference
© Grade 12 Mathematics Study Notes — All rights reserved
