SS 1

Revision – First Term

Mathematics SS 1 First Term

📌 Welcome to Your First Term Revision!

This comprehensive revision covers all the topics from the First 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 First Term

Week	Topic	Key Concepts
1	Number Bases I	Definition of number bases, binary, octal, decimal, hexadecimal; place value; conversion to base 10
2	Number Bases II	Conversion from base 10 to other bases; addition, subtraction, multiplication in bases
3	Modular Arithmetic	Concept of modulo; congruence; addition, subtraction, multiplication in modulo
4	Indices I	Definition; laws of indices (multiplication, division, power of power, zero index)
5	Indices II	Negative indices; fractional indices; solving equations with indices
6	Logarithms I	Definition; conversion between exponential and logarithmic forms; laws of logarithms
7	Logarithms II	Characteristic and mantissa; calculations with logarithms; antilogarithms; solving equations
8	Surds I	Definition of surds; simplification; addition and subtraction of surds
9	Surds II	Multiplication and division of surds; rationalising the denominator; surd equations

First Term Topics – Summary

Topic 1: Number Bases I – Summary

🔑 Key Points:

Number Base: A system for representing numbers using a set of digits.
Common Bases: Binary (2), Octal (8), Decimal (10), Hexadecimal (16).
Expanded Form: N = dₙ×bⁿ + ... + d₂×b² + d₁×b¹ + d₀×b⁰
Conversion to Base 10: Use the expanded form method.

Topic 2: Number Bases II – Summary

🔑 Key Points:

Conversion from Base 10: Use the division method (read remainders bottom to top).
Addition: Carry when sum ≥ base.
Subtraction: Borrow when top digit < bottom digit.
Multiplication: Use the base's multiplication table; carry when product ≥ base.

Topic 3: Modular Arithmetic – Summary

🔑 Key Points:

Modulus (m): The number that defines the range of values.
Congruence: a ≡ b (mod m) if m divides (a − b).
Addition: (a + b) mod m
Subtraction: (a − b) mod m (if negative, add m).
Multiplication: (a × b) mod m

Topic 4: Indices I – Summary

🔑 Key Points:

Definition: aⁿ = a × a × ... (n times).
Law 1: aᵐ × aⁿ = aᵐ⁺ⁿ
Law 2: aᵐ ÷ aⁿ = aᵐ⁻ⁿ
Law 3: (aᵐ)ⁿ = aᵐⁿ
Law 4: a⁰ = 1 (a ≠ 0)

Topic 5: Indices II – Summary

🔑 Key Points:

Negative Index: a⁻ⁿ = 1/aⁿ
Fractional Index: a¹/ⁿ = ⁿ√a
General Fractional: aᵐ/ⁿ = ⁿ√(aᵐ) = (ⁿ√a)ᵐ
Solving Equations: Express both sides as powers of the same base, then equate indices.

Topic 6: Logarithms I – Summary

🔑 Key Points:

Definition: logₐ b = c ⇔ aᶜ = b
Product Law: logₐ (xy) = logₐ x + logₐ y
Quotient Law: logₐ (x/y) = logₐ x − logₐ y
Power Law: logₐ (xⁿ) = n logₐ x
Change of Base: logₐ x = log_b x / log_b a

Topic 7: Logarithms II – Summary

🔑 Key Points:

Characteristic: Integer part of log (digits − 1 for N ≥ 1; negative for N < 1).
Mantissa: Decimal part (found from log tables).
Calculations: Add logs for multiplication; subtract for division; multiply by power for powers; divide by root for roots.
Antilog: antilog y = 10ʸ

Topic 8: Surds I – Summary

🔑 Key Points:

Definition: Irrational numbers expressed as roots (e.g., √2, √3).
Simplification: √(a×b) = √a × √b; find perfect square factors.
Addition/Subtraction: Only similar surds can be added/subtracted.
Conjugate: a + √b and a − √b

Topic 9: Surds II – Summary

🔑 Key Points:

Multiplication: √a × √b = √(ab)
Division: √a ÷ √b = √(a/b)
Rationalising: Multiply numerator and denominator by the conjugate.
Solving Equations: Isolate the surd, square both sides, solve, check for extraneous solutions.

Key Formulas Summary

Topic	Formula	Variables
Number Bases	N = dₙ×bⁿ + ... + d₀×b⁰	b = base
Modular Arithmetic	a ≡ b (mod m) if m\|(a−b)	m = modulus
Indices	aᵐ × aⁿ = aᵐ⁺ⁿ	a = base
Negative Index	a⁻ⁿ = 1/aⁿ	n = positive integer
Fractional Index	a¹/ⁿ = ⁿ√a	n = root
Logarithms	logₐ b = c ⇔ aᶜ = b	a = base
Log Product	logₐ (xy) = logₐ x + logₐ y	a = base
Surds	√(a×b) = √a × √b	a, b ≥ 0

Revision Questions

Test your understanding of all First Term topics:

Section A: Multiple Choice (Choose the correct option)

Convert 1101₂ to base 10.
A) 11 B) 13 C) 15 D) 17
Evaluate: 7 + 8 (mod 6)
A) 1 B) 2 C) 3 D) 4
Simplify: 2³ × 2⁵
A) 2⁸ B) 2¹⁵ C) 2² D) 2⁷
What is log₂ 8?
A) 2 B) 3 C) 4 D) 5
Simplify: √12
A) 2√3 B) 3√2 C) 4√3 D) 2√6
Which of the following is a surd?
A) √4 B) √9 C) √16 D) √2
Evaluate: 5⁻²
A) 25 B) 1/25 C) -25 D) -1/25
What is the characteristic of log₁₀ 4567?
A) 1 B) 2 C) 3 D) 4
Convert 25₁₀ to binary.
A) 11001₂ B) 11010₂ C) 11100₂ D) 10101₂
Simplify: √8 + √18
A) 5√2 B) 4√2 C) 3√2 D) 2√2

Answers: 1-B, 2-C, 3-A, 4-B, 5-A, 6-D, 7-B, 8-C, 9-A, 10-A

Section B: Short Answer Questions

Convert 101011₂ to base 10.
Evaluate: 4 − 9 (mod 5).
Simplify: (2³ × 2⁵) ÷ 2⁴.
Evaluate: 27²/³.
Simplify: log₂ 16 + log₂ 4.
Simplify: √27 + √12.
Rationalise: 1/√3.
Solve: 2ˣ = 64.
Find the characteristic of log₁₀ 0.0456.
Expand: (√2 + √3)².

Section C: Calculation Questions

Add: 101₂ + 111₂.
Subtract: 1101₂ − 101₂.
Solve: log₂ x = 5.
Calculate: 3⁻² × 3⁴.
Simplify: √12 × √3.
Rationalise: 1/(2 − √3).
Evaluate: log₂ 32 − log₂ 4.
Solve: √(2x + 1) = 5.
Simplify: 8√15 ÷ 2√3.
Solve: log₃ (2x + 1) = 2.

Answers to Section B

1. 101011₂ = 1×32 + 0×16 + 1×8 + 0×4 + 1×2 + 1×1 = 43₁₀

2. 4 − 9 = −5; −5 + 5 = 0

3. (2³ × 2⁵) ÷ 2⁴ = 2⁸ ÷ 2⁴ = 2⁴

4. 27²/³ = (³√27)² = 3² = 9

5. log₂ 16 + log₂ 4 = 4 + 2 = 6

6. √27 = 3√3, √12 = 2√3, sum = 5√3

7. 1/√3 × √3/√3 = √3/3

8. 64 = 2⁶, so x = 6

9. 0.0456 has 2 zeros → characteristic = −2

10. (√2)² + 2(√2)(√3) + (√3)² = 2 + 2√6 + 3 = 5 + 2√6

Answers to Section C

1. 101₂ + 111₂ = 1100₂

2. 1101₂ − 101₂ = 1000₂

3. 2⁵ = x → x = 32

4. 3⁻² × 3⁴ = 3² = 9

5. √12 × √3 = √36 = 6

6. (2 + √3)/(4 − 3) = 2 + √3

7. log₂ 32 − log₂ 4 = 5 − 2 = 3

8. 2x + 1 = 25 → 2x = 24 → x = 12

9. 8√15 ÷ 2√3 = 4√5

10. 3² = 2x + 1 → 9 = 2x + 1 → x = 4

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 number bases – binary, octal, and hexadecimal conversions.
Practice surd operations – simplification, rationalisation, and equations.
Time management – allocate time for each section during the exam.
Read questions carefully – identify what is being asked before answering.

Board Summary

╔════════════════════════════════════════════════════════════════════════════╗ ║ SS1 MATHEMATICS – FIRST TERM REVISION ║ ╠════════════════════════════════════════════════════════════════════════════╣ ║ ║ ║ WEEK 1-2: NUMBER BASES ║ ║ ───────────────────────────────────────────────────────────────────────── ║ ║ • Definition: System using a set of digits ║ ║ • Common bases: Binary (2), Octal (8), Decimal (10), Hex (16) ║ ║ • Conversions: Expanded form (to base 10), Division method (from base 10)║ ║ • Operations: Addition, subtraction, multiplication in bases ║ ║ ║ ║ WEEK 3: MODULAR ARITHMETIC ║ ║ ───────────────────────────────────────────────────────────────────────── ║ ║ • Modulus m defines the range ║ ║ • Congruence: a ≡ b (mod m) if m divides (a−b) ║ ║ • Operations: addition, subtraction, multiplication ║ ║ ║ ║ WEEK 4-5: INDICES ║ ║ ───────────────────────────────────────────────────────────────────────── ║ ║ • aᵐ × aⁿ = aᵐ⁺ⁿ, aᵐ ÷ aⁿ = aᵐ⁻ⁿ, (aᵐ)ⁿ = aᵐⁿ ║ ║ • a⁰ = 1, a⁻ⁿ = 1/aⁿ, a¹/ⁿ = ⁿ√a, aᵐ/ⁿ = ⁿ√(aᵐ) ║ ║ ║ ║ WEEK 6-7: LOGARITHMS ║ ║ ───────────────────────────────────────────────────────────────────────── ║ ║ • logₐ b = c ⇔ aᶜ = b ║ ║ • Product: logₐ (xy) = logₐ x + logₐ y ║ ║ • Quotient: logₐ (x/y) = logₐ x − logₐ y ║ ║ • Power: logₐ (xⁿ) = n logₐ x ║ ║ • logₐ a = 1, logₐ 1 = 0 ║ ║ • Characteristic + Mantissa ║ ║ ║ ║ WEEK 8-9: SURDS ║ ║ ───────────────────────────────────────────────────────────────────────── ║ ║ • √(a×b) = √a × √b ║ ║ • Only similar surds can be added/subtracted ║ ║ • Rationalise by multiplying by conjugate ║ ║ • Conjugate: a + b√x and a − b√x ║ ║ ║ ╚════════════════════════════════════════════════════════════════════════════╝

✅ Good luck with your examination!

Remember to stay calm, read all questions carefully, and show your working clearly. You've got this! 💪