📌 Welcome to Your Second Term Revision!
This comprehensive revision covers all the topics from the Second Term. Use this to:
- Review key concepts from each topic
- Practice with revision questions
- Prepare for your end-of-term examination
- Identify areas that need more attention
Topics Covered in Second Term
| Week |
Topic |
Key Concepts |
| 1 |
Revision of First Term |
Review of Number Bases, Modular Arithmetic, Indices, Logarithms, Surds |
| 2 |
Sets I |
Definition, elements, roster and set-builder methods, types of sets, cardinality, subsets |
| 3 |
Sets II |
Union (∪), Intersection (∩), Difference (−), Complement (′) |
| 4 |
Sets III – Venn Diagrams |
Two-set and three-set Venn diagrams, word problems, inclusion-exclusion principle |
| 5 |
Algebraic Expressions |
Variables, coefficients, constants, like terms, expanding brackets, substitution |
| 6 |
Linear Equations |
Solving equations with variables on both sides, brackets, fractions, word problems |
| 7 |
Simultaneous Equations |
Elimination, substitution, graphical methods; word problems |
| 8 |
Quadratic Equations I |
Factorization method, difference of two squares |
| 9 |
Quadratic Equations II |
Completing the square, quadratic formula, discriminant, nature of roots |
Second Term Topics – Summary
Topic 1: Sets I – Summary
🔑 Key Points:
- Set: A well-defined collection of distinct objects.
- Methods: Roster (list) and Set-builder (rule).
- Types: Finite, infinite, empty (∅), universal (U).
- Cardinality: n(A) = number of elements in A.
- Subsets: A ⊆ B if all elements of A are in B. Number of subsets = 2ⁿ.
Topic 2: Sets II – Summary
🔑 Key Points:
- Union (∪): A ∪ B = {x | x ∈ A or x ∈ B}
- Intersection (∩): A ∩ B = {x | x ∈ A and x ∈ B}
- Difference (−): A − B = {x | x ∈ A and x ∉ B}
- Complement (′): A′ = {x ∈ U | x ∉ A}
Topic 3: Sets III – Venn Diagrams – Summary
🔑 Key Points:
- Two-set regions: A only, A∩B, B only, neither.
- Three-set regions: 8 regions (A only, B only, C only, intersections, triple intersection).
- Inclusion-Exclusion (2 sets): n(A∪B) = n(A) + n(B) − n(A∩B)
- Inclusion-Exclusion (3 sets): n(A∪B∪C) = n(A) + n(B) + n(C) − n(A∩B) − n(A∩C) − n(B∩C) + n(A∩B∩C)
Topic 4: Algebraic Expressions – Summary
🔑 Key Points:
- Terms: Parts separated by + or −.
- Like Terms: Same variable and exponent.
- Expanding: a(b + c) = ab + ac
- FOIL: (a + b)(c + d) = ac + ad + bc + bd
- Special Products: (a+b)(a−b) = a²−b², (a+b)² = a²+2ab+b²
Topic 5: Linear Equations – Summary
🔑 Key Points:
- Definition: ax + b = 0 (highest power 1)
- Steps: Expand brackets → collect like terms → move variables to one side → divide by coefficient.
- Types: Simple, variables on both sides, with brackets, with fractions.
Topic 6: Simultaneous Equations – Summary
🔑 Key Points:
- Methods: Elimination (add/subtract), Substitution, Graphical.
- Types of Solutions: Unique (intersecting), No solution (parallel), Infinite (coincident).
- Word Problems: Form two equations from the information given.
Topic 7: Quadratic Equations I – Summary
🔑 Key Points:
- Standard Form: ax² + bx + c = 0
- Factorization: x² + bx + c = (x+p)(x+q) where p+q=b and p×q=c
- Difference of Squares: a² − b² = (a−b)(a+b)
Topic 8: Quadratic Equations II – Summary
🔑 Key Points:
- Quadratic Formula: x = [−b ± √(b² − 4ac)] / 2a
- Completing the Square: (x + b/2)² = (b/2)² − c
- Discriminant: Δ = b² − 4ac
- Nature of Roots: Δ > 0 (real & distinct), Δ = 0 (real & equal), Δ < 0 (no real roots)
Key Formulas Summary
| Topic |
Formula |
Variables |
| Sets – Subsets |
Number of subsets = 2ⁿ |
n = number of elements |
| Sets – Union |
n(A∪B) = n(A) + n(B) − n(A∩B) |
A, B = sets |
| Sets – Inclusion-Exclusion (3) |
n(A∪B∪C) = n(A) + n(B) + n(C) − n(A∩B) − n(A∩C) − n(B∩C) + n(A∩B∩C) |
A, B, C = sets |
| Algebra – Expansion |
a(b + c) = ab + ac |
a, b, c = constants/variables |
| Algebra – Double Brackets |
(a + b)(c + d) = ac + ad + bc + bd |
a, b, c, d = constants/variables |
| Algebra – Difference of Squares |
(a + b)(a − b) = a² − b² |
a, b = constants/variables |
| Algebra – Perfect Square |
(a + b)² = a² + 2ab + b² |
a, b = constants/variables |
| Quadratic – Formula |
x = [−b ± √(b² − 4ac)] / 2a |
a, b, c from ax² + bx + c = 0 |
| Quadratic – Discriminant |
Δ = b² − 4ac |
a, b, c from ax² + bx + c = 0 |
Revision Questions
Test your understanding of all Second Term topics:
Section A: Multiple Choice (Choose the correct option)
- If A = {1, 2, 3, 4} and B = {3, 4, 5, 6}, find A ∪ B.
A) {1,2,3,4,5,6} B) {3,4} C) {1,2,5,6} D) {1,2,3,4}
- Simplify: 3x + 5x − 2x
A) 6x B) 8x C) 10x D) 4x
- Solve: 2x + 3 = 11
A) 3 B) 4 C) 5 D) 6
- Solve: 3x + 2 = 2x + 7
A) 3 B) 4 C) 5 D) 6
- Solve: x² − 5x + 6 = 0
A) 2, 3 B) 1, 6 C) 2, 4 D) 3, 4
- The discriminant of x² + x + 1 = 0 is:
A) 3 B) 1 C) −3 D) 0
- If A = {1, 2, 3} and B = {3, 4, 5}, find A ∩ B.
A) {1,2,3,4,5} B) {3} C) {1,2} D) {4,5}
- Expand: (x + 3)(x + 2)
A) x² + 5x + 6 B) x² + 6x + 5 C) x² + 5x + 5 D) x² + 6x + 6
- How many subsets does a set of 4 elements have?
A) 8 B) 12 C) 16 D) 20
- Solve by completing the square: x² + 6x + 5 = 0
A) −1, −5 B) 1, 5 C) −1, 5 D) 1, −5
Answers: 1-A, 2-A, 3-B, 4-C, 5-A, 6-C, 7-B, 8-A, 9-C, 10-A
Section B: Short Answer Questions
- State the union and intersection laws for sets.
- Expand: 3(2x + 5)
- Solve: 4x − 3 = 2x + 9
- Factorize: x² + 7x + 12
- Solve using the quadratic formula: x² − 4x + 3 = 0
- If U = {1,2,3,4,5,6,7,8} and A = {1,3,5,7}, find A′.
- Simplify: (x + 4)(x − 4)
- Solve the simultaneous equations: x + y = 7, x − y = 3
- State the quadratic formula.
- Find the discriminant of 2x² − 3x + 1 = 0 and state the nature of roots.
Section C: Calculation Questions
- If A = {1, 2, 3, 4, 5} and B = {4, 5, 6, 7, 8}, find (a) A ∪ B (b) A ∩ B (c) A − B
- Simplify: 4a + 3b − 2a + 5b
- Solve: 3(x − 2) = 15
- Solve: 2x + 5 = 3x − 1
- Solve using elimination: 3x + 2y = 12, x − 2y = 4
- Solve: x² − 9 = 0
- Solve using the quadratic formula: 2x² − 5x + 2 = 0
- In a class of 40 students, 25 play football, 18 play basketball, and 8 play both. How many play neither sport?
- Find the discriminant of x² − 6x + 9 = 0 and state the nature of roots.
- Factorize: 2x² + 7x + 3
Answers to Section B
1. Union: A∪B = {x | x∈A or x∈B}. Intersection: A∩B = {x | x∈A and x∈B}.
2. 3(2x + 5) = 6x + 15
3. 4x − 2x = 9 + 3 → 2x = 12 → x = 6
4. x² + 7x + 12 = (x + 3)(x + 4)
5. x = [4 ± √(16−12)]/2 = [4 ± 2]/2 → x = 3 or 1
6. A′ = {2, 4, 6, 8}
7. (x + 4)(x − 4) = x² − 16
8. Add: 2x = 10 → x = 5, y = 2 → x = 5, y = 2
9. x = [−b ± √(b² − 4ac)] / 2a
10. Δ = 9 − 8 = 1 > 0 → Two real and distinct roots
Answers to Section C
1. (a) {1,2,3,4,5,6,7,8} (b) {4,5} (c) {1,2,3}
2. 4a − 2a + 3b + 5b = 2a + 8b
3. 3x − 6 = 15 → 3x = 21 → x = 7
4. 2x + 5 = 3x − 1 → 5 + 1 = 3x − 2x → 6 = x → x = 6
5. Add: 4x = 16 → x = 4, 3(4) + 2y = 12 → 12 + 2y = 12 → y = 0 → x = 4, y = 0
6. x² − 9 = 0 → (x − 3)(x + 3) = 0 → x = 3 or −3
7. x = [5 ± √(25−16)]/4 = [5 ± 3]/4 → x = 2 or 1/2
8. Football only = 25−8=17, Basketball only = 18−8=10. Neither = 40 − (17+8+10) = 5
9. Δ = 36 − 36 = 0 → Two real and equal roots
10. 2x² + 7x + 3 = (2x + 1)(x + 3)
Examination Preparation Tips
📌 Tips for Success:
- Review all formulas – make a formula sheet and memorise them.
- Understand key definitions – be able to explain concepts in your own words.
- Practice calculations – work through all examples and assignment questions.
- Know your set operations – union, intersection, complement, Venn diagrams.
- Practice algebraic manipulation – expansion, factorization, solving equations.
- Quadratic equations – be comfortable with all three methods (factorization, completing the square, formula).
- Time management – allocate time for each section during the exam.
- Read questions carefully – identify what is being asked before answering.
Board Summary
╔════════════════════════════════════════════════════════════════════════════╗
║ SS1 MATHEMATICS – SECOND TERM REVISION ║
╠════════════════════════════════════════════════════════════════════════════╣
║ ║
║ WEEK 2-4: SETS & VENN DIAGRAMS ║
║ ───────────────────────────────────────────────────────────────────────── ║
║ • Union: A∪B, Intersection: A∩B, Difference: A−B, Complement: A′ ║
║ • Venn diagrams: 2-set (4 regions), 3-set (8 regions) ║
║ • n(A∪B) = n(A) + n(B) − n(A∩B) ║
║ ║
║ WEEK 5: ALGEBRAIC EXPRESSIONS ║
║ ───────────────────────────────────────────────────────────────────────── ║
║ • Like terms: ax + bx = (a+b)x ║
║ • Expand: a(b+c) = ab+ac ║
║ • FOIL: (a+b)(c+d) = ac+ad+bc+bd ║
║ ║
║ WEEK 6: LINEAR EQUATIONS ║
║ ───────────────────────────────────────────────────────────────────────── ║
║ • Form: ax + b = 0 ║
║ • Isolate variable using inverse operations. ║
║ ║
║ WEEK 7: SIMULTANEOUS EQUATIONS ║
║ ───────────────────────────────────────────────────────────────────────── ║
║ • Methods: Elimination, Substitution, Graphical ║
║ • Types: Unique (intersecting), No solution (parallel), Infinite ║
║ ║
║ WEEK 8-9: QUADRATIC EQUATIONS ║
║ ───────────────────────────────────────────────────────────────────────── ║
║ • Standard: ax² + bx + c = 0 ║
║ • Methods: Factorization, Completing the square, Quadratic formula ║
║ • Quadratic formula: x = [−b ± √(b²−4ac)] / 2a ║
║ • Discriminant: Δ = b² − 4ac ║
║ ║
║ KEY FORMULAS: ║
║ ───────────────────────────────────────────────────────────────────────── ║
║ • (a+b)² = a² + 2ab + b² ║
║ • (a−b)² = a² − 2ab + b² ║
║ • (a+b)(a−b) = a² − b² ║
║ ║
╚════════════════════════════════════════════════════════════════════════════╝
✅ Good luck with your examination!
Remember to stay calm, read all questions carefully, and show your working clearly. You've got this! 💪