IGCSE Mathematics (0580) — Complete Revision Notes (Core & Extended)

<h1 class="notes-h1">IGCSE Mathematics (0580) — Complete Revision Notes</h1>
<h2 class="notes-h2">Cambridge Assessment International Education</h2>
<h3 class="notes-h3">Syllabus Code: 0580 | Core & Extended</h3>
<hr class="section-divider">
<p><strong>Prepared by:</strong> CBC Edu Kenya | cbcedukenya.com</p>
<p><strong>Syllabus Version:</strong> 0580 (2023–2025 and 2026 onwards)</p>
<p><strong>Coverage:</strong> All topics — Core and Extended</p>
<p><strong>Note:</strong> These are original revision notes aligned to the Cambridge IGCSE Mathematics (0580) syllabus. They are not official Cambridge materials. For official past papers and mark schemes, visit cambridgeinternational.org.</p>
<hr class="section-divider">
<h2 class="notes-h2">HOW TO USE THESE NOTES</h2>
<ul class="notes-list">
<li><strong>Core</strong> content is required for all candidates (grades C–G).</li>
<li><strong>Extended</strong> content is required for Extended candidates only (grades A*–E).</li>
<li>Extended-only sections are marked with <strong>[E]</strong>.</li>
<li>Past paper references are noted as e.g. <strong>(MJ23 P2 Q5)</strong> = May/June 2023, Paper 2, Question 5.</li>
<li>Work through each section, then attempt the practice questions before checking answers.</li>
</ul>
<hr class="section-divider">
<h2 class="notes-h2">PAPER OVERVIEW</h2>
<div class="table-wrap"><table class="notes-table">
<thead>
<tr><th>Paper</th><th>Type</th><th>Duration</th><th>Marks</th><th>Notes</th></tr>
</thead><tbody>
<tr><td>Paper 1</td><td>Core — Short answer</td><td>1 hr</td><td>56</td><td>Non-calculator</td></tr>
<tr><td>Paper 2</td><td>Extended — Short answer</td><td>1 hr 30 min</td><td>70</td><td>Non-calculator</td></tr>
<tr><td>Paper 3</td><td>Core — Structured</td><td>2 hrs</td><td>104</td><td>Calculator allowed</td></tr>
<tr><td>Paper 4</td><td>Extended — Structured</td><td>2 hrs 30 min</td><td>130</td><td>Calculator allowed</td></tr>
</tbody></table></div>
<p><strong>Key tip:</strong> Papers 1 and 3 are for Core candidates. Papers 2 and 4 are for Extended candidates. All questions must be attempted. Show all working — method marks are awarded even if the final answer is wrong.</p>
<hr class="section-divider">
<h2 class="notes-h2">TOPIC 1: NUMBER</h2>
<h3 class="notes-h3">1.1 Types of Numbers</h3>
<div class="table-wrap"><table class="notes-table">
<thead>
<tr><th>Term</th><th>Definition</th><th>Examples</th></tr>
</thead><tbody>
<tr><td>Natural numbers</td><td>Positive whole numbers (and sometimes 0)</td><td>1, 2, 3, 4, …</td></tr>
<tr><td>Integers</td><td>All whole numbers (positive, negative, zero)</td><td>…-2, -1, 0, 1, 2…</td></tr>
<tr><td>Rational numbers</td><td>Can be written as a fraction p/q (q ≠ 0)</td><td>1/2, 0.75, -3, 5</td></tr>
<tr><td>Irrational numbers</td><td>Cannot be written as a fraction</td><td>√2, π, √5</td></tr>
<tr><td>Real numbers</td><td>All rational and irrational numbers</td><td>All of the above</td></tr>
<tr><td>Prime numbers</td><td>Exactly 2 factors: 1 and itself</td><td>2, 3, 5, 7, 11, 13, …</td></tr>
<tr><td>Square numbers</td><td>n² for any positive integer n</td><td>1, 4, 9, 16, 25, …</td></tr>
<tr><td>Cube numbers</td><td>n³ for any positive integer n</td><td>1, 8, 27, 64, 125, …</td></tr>
</tbody></table></div>
<p><strong>Note:</strong> 1 is NOT a prime number. 2 is the only even prime number.</p>
<h3 class="notes-h3">1.2 Place Value, Ordering, and Rounding</h3>
<p><strong>Decimal places:</strong> Count digits after the decimal point.</p>
<ul class="notes-list">
<li>Round 3.7462 to 2 decimal places → 3.75 (look at 3rd dp: 6 ≥ 5, so round up)</li>
</ul>
<p><strong>Significant figures:</strong> Count from the first non-zero digit.</p>
<ul class="notes-list">
<li>Round 0.003462 to 2 sig. figs → 0.0035 (first sig. fig is 3, second is 4; look at 6)</li>
<li>Round 47,830 to 3 sig. figs → 47,800</li>
</ul>
<p><strong>Worked Example:</strong> Write 0.04567 correct to 3 significant figures.</p>
<ul class="notes-list">
<li>First sig. fig: 4 (position 2 after decimal)</li>
<li>Second sig. fig: 5</li>
<li>Third sig. fig: 6</li>
<li>Look at 4th sig. fig: 7 ≥ 5, round up</li>
<li><strong>Answer: 0.0457</strong></li>
</ul>
<h3 class="notes-h3">1.3 Fractions, Decimals, and Percentages</h3>
<p><strong>Converting between forms:</strong></p>
<div class="table-wrap"><table class="notes-table">
<thead>
<tr><th>From</th><th>To</th><th>Method</th></tr>
</thead><tbody>
<tr><td>Fraction → Decimal</td><td>Divide numerator by denominator</td><td>3/8 = 3 ÷ 8 = 0.375</td></tr>
<tr><td>Decimal → Fraction</td><td>Write over power of 10, simplify</td><td>0.35 = 35/100 = 7/20</td></tr>
<tr><td>Percentage → Decimal</td><td>Divide by 100</td><td>35% = 0.35</td></tr>
<tr><td>Decimal → Percentage</td><td>Multiply by 100</td><td>0.35 = 35%</td></tr>
<tr><td>Fraction → Percentage</td><td>Multiply by 100</td><td>3/5 × 100 = 60%</td></tr>
</tbody></table></div>
<p><strong>Operations with fractions:</strong></p>
<ul class="notes-list">
<li><strong>Addition/Subtraction:</strong> Find LCM of denominators first</li>
<li>2/3 + 3/4 = 8/12 + 9/12 = 17/12 = 1 5/12</li>
<li><strong>Multiplication:</strong> Multiply numerators, multiply denominators</li>
<li>2/3 × 3/4 = 6/12 = 1/2</li>
<li><strong>Division:</strong> Multiply by the reciprocal (flip the second fraction)</li>
<li>2/3 ÷ 3/4 = 2/3 × 4/3 = 8/9</li>
</ul>
<p><strong>Percentage calculations:</strong></p>
<ul class="notes-list">
<li>Percentage of a quantity: 30% of 250 = 0.30 × 250 = 75</li>
<li>Percentage increase: New = Original × (1 + r/100)</li>
<li>Percentage decrease: New = Original × (1 - r/100)</li>
<li>Reverse percentage: Original = New value ÷ (1 ± r/100)</li>
</ul>
<p><strong>Worked Example:</strong> A shirt costs $45 after a 10% discount. Find the original price.</p>
<ul class="notes-list">
<li>After 10% discount: price = original × 0.90</li>
<li>Original = 45 ÷ 0.90 = <strong>$50</strong></li>
</ul>
<h3 class="notes-h3">1.4 Indices (Powers) and Roots</h3>
<p><strong>Laws of Indices:</strong></p>
<div class="table-wrap"><table class="notes-table">
<thead>
<tr><th>Law</th><th>Rule</th><th>Example</th></tr>
</thead><tbody>
<tr><td>Multiplication</td><td>aᵐ × aⁿ = aᵐ⁺ⁿ</td><td>2³ × 2⁴ = 2⁷ = 128</td></tr>
<tr><td>Division</td><td>aᵐ ÷ aⁿ = aᵐ⁻ⁿ</td><td>5⁶ ÷ 5² = 5⁴ = 625</td></tr>
<tr><td>Power of a power</td><td>(aᵐ)ⁿ = aᵐⁿ</td><td>(3²)³ = 3⁶ = 729</td></tr>
<tr><td>Zero index</td><td>a⁰ = 1</td><td>7⁰ = 1</td></tr>
<tr><td>Negative index</td><td>a⁻ⁿ = 1/aⁿ</td><td>2⁻³ = 1/8</td></tr>
<tr><td>Fractional index</td><td>a^(1/n) = ⁿ√a</td><td>8^(1/3) = ∛8 = 2</td></tr>
<tr><td>Fractional index</td><td>a^(m/n) = (ⁿ√a)ᵐ</td><td>8^(2/3) = (∛8)² = 4</td></tr>
</tbody></table></div>
<p><strong>Worked Example [E]:</strong> Simplify (2x³y²)⁴</p>
<ul class="notes-list">
<li>= 2⁴ × x³×⁴ × y²×⁴</li>
<li>= <strong>16x¹²y⁸</strong></li>
</ul>
<p><strong>Practice Questions:</strong></p>
<ol class="notes-list">
<li>Write 72 as a product of its prime factors.</li>
<li>Find the LCM and HCF of 24 and 36.</li>
<li>Calculate 27^(2/3).</li>
<li>Round 0.008473 to 2 significant figures.</li>
</ol>
<p><strong>Answers:</strong> 1. 2³ × 3² | 2. LCM=72, HCF=12 | 3. 9 | 4. 0.0085</p>
<h3 class="notes-h3">1.5 Standard Form (Scientific Notation)</h3>
<p>A number in standard form is written as: <strong>A × 10ⁿ</strong> where 1 ≤ A < 10 and n is an integer.</p>
<p><strong>Converting to standard form:</strong></p>
<ul class="notes-list">
<li>45,600 = 4.56 × 10⁴ (decimal point moves 4 places left)</li>
<li>0.00037 = 3.7 × 10⁻⁴ (decimal point moves 4 places right)</li>
</ul>
<p><strong>Operations in standard form:</strong></p>
<ul class="notes-list">
<li>Multiplication: (3 × 10⁴) × (2 × 10³) = 6 × 10⁷</li>
<li>Division: (6 × 10⁸) ÷ (2 × 10³) = 3 × 10⁵</li>
<li>Addition: Adjust so powers match, then add coefficients</li>
</ul>
<p><strong>Worked Example:</strong> Calculate (5.4 × 10⁶) ÷ (1.8 × 10²). Give answer in standard form.</p>
<ul class="notes-list">
<li>= (5.4 ÷ 1.8) × 10⁶⁻²</li>
<li>= 3 × 10⁴ = <strong>3.0 × 10⁴</strong></li>
</ul>
<h3 class="notes-h3">1.6 Ratio, Proportion, and Rate</h3>
<p><strong>Ratio:</strong> Compares quantities. Always simplify. 15:25 = 3:5</p>
<p><strong>Dividing in a ratio:</strong> Share 120 in the ratio 3:5.</p>
<ul class="notes-list">
<li>Total parts = 8</li>
<li>Each part = 120 ÷ 8 = 15</li>
<li>Shares: 3 × 15 = 45 and 5 × 15 = 75</li>
</ul>
<p><strong>Direct proportion:</strong> y ∝ x → y = kx (as x increases, y increases)</p>
<p><strong>Inverse proportion [E]:</strong> y ∝ 1/x → y = k/x (as x increases, y decreases)</p>
<p><strong>Speed, Distance, Time:</strong></p>
<ul class="notes-list">
<li>Distance = Speed × Time</li>
<li>Speed = Distance ÷ Time</li>
<li>Time = Distance ÷ Speed</li>
</ul>
<p><strong>Worked Example:</strong> A car travels 270 km in 3 hours. Find its average speed.</p>
<ul class="notes-list">
<li>Speed = 270 ÷ 3 = <strong>90 km/h</strong></li>
</ul>
<h3 class="notes-h3">1.7 Estimation and Bounds [E]</h3>
<p><strong>Upper and lower bounds:</strong></p>
<ul class="notes-list">
<li>A measurement of 7.4 cm (to 1 d.p.) has:</li>
<li>Lower bound: 7.35 cm</li>
<li>Upper bound: 7.45 cm</li>
</ul>
<p><strong>Bounds in calculations:</strong></p>
<ul class="notes-list">
<li>Maximum value of A + B = upper bound of A + upper bound of B</li>
<li>Minimum value of A − B = lower bound of A − upper bound of B</li>
<li>Maximum value of A ÷ B = upper bound of A ÷ lower bound of B</li>
</ul>
<hr class="section-divider">
<h2 class="notes-h2">TOPIC 2: ALGEBRA AND GRAPHS</h2>
<h3 class="notes-h3">2.1 Algebraic Manipulation</h3>
<p><strong>Expanding brackets:</strong></p>
<ul class="notes-list">
<li>Single bracket: 3(2x - 4) = 6x - 12</li>
<li>Double brackets (FOIL): (x + 3)(x - 5) = x² - 5x + 3x - 15 = x² - 2x - 15</li>
</ul>
<p><strong>Special products:</strong></p>
<ul class="notes-list">
<li>(a + b)² = a² + 2ab + b²</li>
<li>(a - b)² = a² - 2ab + b²</li>
<li>(a + b)(a - b) = a² - b² (difference of two squares)</li>
</ul>
<p><strong>Factorisation:</strong></p>
<ul class="notes-list">
<li>Common factor: 6x² + 9x = 3x(2x + 3)</li>
<li>Difference of squares: x² - 16 = (x + 4)(x - 4)</li>
<li>Quadratic trinomial: x² + 5x + 6 = (x + 2)(x + 3)</li>
<li>Find two numbers that multiply to +6 and add to +5 → 2 and 3</li>
</ul>
<p><strong>Worked Example [E]:</strong> Factorise 2x² + 7x + 3</p>
<ul class="notes-list">
<li>Multiply coefficient of x² by constant: 2 × 3 = 6</li>
<li>Find two numbers that multiply to 6 and add to 7: 1 and 6</li>
<li>Split middle term: 2x² + x + 6x + 3</li>
<li>Group: x(2x + 1) + 3(2x + 1)</li>
<li><strong>= (x + 3)(2x + 1)</strong></li>
</ul>
<h3 class="notes-h3">2.2 Solving Equations</h3>
<p><strong>Linear equations:</strong></p>
<ul class="notes-list">
<li>Collect all terms with the unknown on one side</li>
<li>3x - 7 = 2x + 5 → 3x - 2x = 5 + 7 → x = 12</li>
</ul>
<p><strong>Forming and solving equations:</strong></p>
<ul class="notes-list">
<li>"A rectangle has length (2x + 3) cm and width (x - 1) cm. The perimeter is 28 cm. Find x."</li>
<li>2(2x + 3) + 2(x - 1) = 28</li>
<li>4x + 6 + 2x - 2 = 28</li>
<li>6x + 4 = 28 → 6x = 24 → x = 4</li>
</ul>
<p><strong>Simultaneous equations [E]:</strong></p>
<p><em>Method 1: Elimination</em></p>
<ul class="notes-list">
<li>2x + 3y = 12 ... (1)</li>
<li>4x - y = 5 ... (2)</li>
<li>Multiply (2) by 3: 12x - 3y = 15 ... (3)</li>
<li>Add (1) and (3): 14x = 27 → x = 27/14</li>
<li>Substitute back to find y</li>
</ul>
<p><em>Method 2: Substitution</em></p>
<ul class="notes-list">
<li>From (2): y = 4x - 5</li>
<li>Substitute into (1): 2x + 3(4x - 5) = 12 → 14x = 27 → x = 27/14</li>
</ul>
<p><strong>Quadratic equations [E]:</strong></p>
<p><em>By factorisation:</em> x² - 5x + 6 = 0 → (x - 2)(x - 3) = 0 → x = 2 or x = 3</p>
<p><em>By the quadratic formula:</em></p>
<p>$$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$</p>
<p>For ax² + bx + c = 0</p>
<p><strong>Worked Example [E]:</strong> Solve 2x² + 3x - 5 = 0 using the formula.</p>
<ul class="notes-list">
<li>a = 2, b = 3, c = -5</li>
<li>x = (-3 ± √(9 + 40)) / 4</li>
<li>x = (-3 ± √49) / 4</li>
<li>x = (-3 ± 7) / 4</li>
<li>x = 4/4 = <strong>1</strong> or x = -10/4 = <strong>-2.5</strong></li>
</ul>
<h3 class="notes-h3">2.3 Inequalities</h3>
<p><strong>Symbols:</strong></p>
<ul class="notes-list">
<li>> greater than | < less than | ≥ greater than or equal to | ≤ less than or equal to</li>
</ul>
<p><strong>Solving:</strong></p>
<ul class="notes-list">
<li>Treat like an equation BUT: <strong>reverse the inequality sign when multiplying or dividing by a negative number</strong></li>
<li>-2x > 8 → x < -4 (sign reversed)</li>
</ul>
<p><strong>Number line:</strong></p>
<ul class="notes-list">
<li>Open circle ○ for strict inequality (< or >)</li>
<li>Closed circle ● for inclusive inequality (≤ or ≥)</li>
</ul>
<p><strong>Linear programming [E]:</strong> Shade unwanted regions. The feasible region satisfies ALL inequalities simultaneously.</p>
<h3 class="notes-h3">2.4 Formulae and Substitution</h3>
<p><strong>Substitution:</strong> Replace each letter with its value.</p>
<ul class="notes-list">
<li>Find the value of 3a² - 2b when a = 4, b = -3:</li>
<li>= 3(16) - 2(-3) = 48 + 6 = <strong>54</strong></li>
</ul>
<p><strong>Changing the subject [E]:</strong></p>
<ul class="notes-list">
<li>Make x the subject of y = 3x + 5:</li>
<li>y - 5 = 3x → x = (y - 5) / 3</li>
<li>Make r the subject of A = πr²:</li>
<li>r² = A/π → r = √(A/π)</li>
</ul>
<h3 class="notes-h3">2.5 Functions [E]</h3>
<p><strong>Function notation:</strong> f(x) means "function of x"</p>
<ul class="notes-list">
<li>If f(x) = 2x + 1, then f(3) = 2(3) + 1 = 7</li>
</ul>
<p><strong>Composite functions:</strong> fg(x) means "apply g first, then f"</p>
<ul class="notes-list">
<li>If f(x) = x + 2 and g(x) = 3x, then fg(x) = f(g(x)) = f(3x) = 3x + 2</li>
</ul>
<p><strong>Inverse functions:</strong> f⁻¹(x) reverses f(x)</p>
<ul class="notes-list">
<li>If f(x) = 2x + 1, find f⁻¹(x):</li>
<li>Let y = 2x + 1 → x = (y - 1)/2</li>
<li>f⁻¹(x) = (x - 1)/2</li>
</ul>
<h3 class="notes-h3">2.6 Graphs</h3>
<p><strong>Straight line (linear) graphs:</strong> y = mx + c</p>
<ul class="notes-list">
<li>m = gradient (slope) = rise ÷ run = (y₂ - y₁)/(x₂ - x₁)</li>
<li>c = y-intercept (where line crosses y-axis)</li>
<li>Parallel lines have <strong>equal gradients</strong></li>
<li>Perpendicular lines: m₁ × m₂ = -1 (gradients are negative reciprocals)</li>
</ul>
<p><strong>Finding the equation of a line:</strong></p>
<ul class="notes-list">
<li>Given two points (1, 3) and (4, 9):</li>
<li>Gradient m = (9-3)/(4-1) = 6/3 = 2</li>
<li>y = 2x + c → sub in (1,3): 3 = 2 + c → c = 1</li>
<li>Equation: <strong>y = 2x + 1</strong></li>
</ul>
<p><strong>Quadratic graphs:</strong> y = ax² + bx + c</p>
<ul class="notes-list">
<li>Parabola shape; opens up if a > 0, opens down if a < 0</li>
<li>Vertex (turning point) at x = -b/(2a)</li>
<li>x-intercepts: solve ax² + bx + c = 0</li>
</ul>
<p><strong>Other graphs [E]:</strong></p>
<ul class="notes-list">
<li>Cubic: y = ax³ (S-shaped curve)</li>
<li>Reciprocal: y = a/x (hyperbola, asymptotes at x=0 and y=0)</li>
<li>Exponential: y = aˣ (rapid growth or decay)</li>
</ul>
<p><strong>Gradient of a curve [E]:</strong> Draw a tangent at the point. Gradient of tangent = gradient of curve at that point.</p>
<p><strong>Distance-time and speed-time graphs:</strong></p>
<ul class="notes-list">
<li>Distance-time: gradient = speed</li>
<li>Speed-time: gradient = acceleration; area under graph = distance</li>
</ul>
<hr class="section-divider">
<h2 class="notes-h2">TOPIC 3: COORDINATE GEOMETRY</h2>
<h3 class="notes-h3">3.1 Coordinates and Midpoints</h3>
<p><strong>Midpoint of AB:</strong> M = ((x₁+x₂)/2, (y₁+y₂)/2)</p>
<p><strong>Length of AB (distance formula):</strong> d = √[(x₂-x₁)² + (y₂-y₁)²]</p>
<p><strong>Worked Example:</strong> Find the midpoint and length of the line joining A(1, -2) and B(5, 4).</p>
<ul class="notes-list">
<li>Midpoint = ((1+5)/2, (-2+4)/2) = <strong>(3, 1)</strong></li>
<li>Length = √[(5-1)² + (4-(-2))²] = √[16 + 36] = √52 = <strong>2√13</strong></li>
</ul>
<h3 class="notes-h3">3.2 Equation of a Line</h3>
<p><strong>Standard form:</strong> y = mx + c or ax + by = c</p>
<p><strong>Finding the equation given:</strong></p>
<ol class="notes-list">
<li>Gradient and one point: use y - y₁ = m(x - x₁)</li>
<li>Two points: first find m, then use above formula</li>
</ol>
<p><strong>Worked Example [E]:</strong> Find the equation of the line perpendicular to y = 3x - 1 that passes through (6, 2).</p>
<ul class="notes-list">
<li>Original gradient = 3; perpendicular gradient = -1/3</li>
<li>y - 2 = -1/3(x - 6)</li>
<li>3(y - 2) = -(x - 6)</li>
<li>3y - 6 = -x + 6</li>
<li><strong>x + 3y = 12</strong></li>
</ul>
<hr class="section-divider">
<h2 class="notes-h2">TOPIC 4: GEOMETRY</h2>
<h3 class="notes-h3">4.1 Angles</h3>
<p><strong>Angle facts (must know):</strong></p>
<ul class="notes-list">
<li>Angles on a straight line add to 180°</li>
<li>Angles at a point add to 360°</li>
<li>Vertically opposite angles are equal</li>
<li>Angles in a triangle add to 180°</li>
<li>Exterior angle of a triangle = sum of the two opposite interior angles</li>
</ul>
<p><strong>Parallel lines (transversal cutting parallel lines):</strong></p>
<ul class="notes-list">
<li>Corresponding angles: equal (F-shape)</li>
<li>Alternate angles: equal (Z-shape)</li>
<li>Co-interior (allied) angles: add to 180° (C-shape)</li>
</ul>
<h3 class="notes-h3">4.2 Polygons</h3>
<p><strong>Interior angle sum:</strong> S = (n - 2) × 180°, where n = number of sides</p>
<p><strong>Regular polygon:</strong></p>
<ul class="notes-list">
<li>Each interior angle = (n - 2) × 180° ÷ n</li>
<li>Each exterior angle = 360° ÷ n</li>
<li>Interior angle + exterior angle = 180°</li>
</ul>
<div class="table-wrap"><table class="notes-table">
<thead>
<tr><th>Polygon</th><th>n</th><th>Interior angle sum</th><th>Each interior angle (regular)</th></tr>
</thead><tbody>
<tr><td>Triangle</td><td>3</td><td>180°</td><td>60°</td></tr>
<tr><td>Quadrilateral</td><td>4</td><td>360°</td><td>90°</td></tr>
<tr><td>Pentagon</td><td>5</td><td>540°</td><td>108°</td></tr>
<tr><td>Hexagon</td><td>6</td><td>720°</td><td>120°</td></tr>
<tr><td>Octagon</td><td>8</td><td>1080°</td><td>135°</td></tr>
</tbody></table></div>
<h3 class="notes-h3">4.3 Circle Theorems [E]</h3>
<p><strong>Essential theorems (must know all 8):</strong></p>
<ol class="notes-list">
<li><strong>Angle at centre:</strong> Angle at centre = 2 × angle at circumference (same arc)</li>
<li><strong>Angle in semicircle:</strong> Angle in a semicircle = 90°</li>
<li><strong>Angles in same segment:</strong> Angles in the same segment are equal</li>
<li><strong>Cyclic quadrilateral:</strong> Opposite angles add to 180°</li>
<li><strong>Tangent-radius:</strong> Tangent to a circle is perpendicular to the radius at the point of tangency</li>
<li><strong>Tangent lengths:</strong> Two tangents from an external point are equal in length</li>
<li><strong>Alternate segment theorem:</strong> Angle between tangent and chord = angle in alternate segment</li>
<li><strong>Chord bisector:</strong> Perpendicular from centre to a chord bisects the chord</li>
</ol>
<p><strong>Worked Example:</strong> O is the centre. Angle BOC = 130°. Find angle BAC.</p>
<ul class="notes-list">
<li>Angle BAC = 1/2 × angle BOC (angle at centre theorem)</li>
<li>Angle BAC = 1/2 × 130° = <strong>65°</strong></li>
</ul>
<h3 class="notes-h3">4.4 Similarity and Congruence</h3>
<p><strong>Similar shapes:</strong></p>
<ul class="notes-list">
<li>Same shape, different size (all corresponding angles equal, sides in proportion)</li>
<li>If linear scale factor = k, then:</li>
<li>Area scale factor = k²</li>
<li>Volume scale factor = k³</li>
</ul>
<p><strong>Congruent shapes:</strong></p>
<ul class="notes-list">
<li>Identical in shape AND size</li>
<li>Congruence conditions for triangles: SSS, SAS, ASA (AAS), RHS</li>
</ul>
<p><strong>Worked Example:</strong> Two similar triangles. The smaller has sides 3 cm and 5 cm. The larger has the side corresponding to 5 cm equal to 15 cm. Find the scale factor and the unknown side.</p>
<ul class="notes-list">
<li>Scale factor k = 15 ÷ 5 = 3</li>
<li>Unknown side = 3 × 3 = <strong>9 cm</strong></li>
</ul>
<h3 class="notes-h3">4.5 Symmetry</h3>
<p><strong>Line symmetry:</strong> A shape has line symmetry if it maps onto itself when reflected about a line.</p>
<p><strong>Rotational symmetry:</strong> A shape has rotational symmetry of order n if it maps onto itself n times in a full turn.</p>
<div class="table-wrap"><table class="notes-table">
<thead>
<tr><th>Shape</th><th>Lines of symmetry</th><th>Order of rotation</th></tr>
</thead><tbody>
<tr><td>Equilateral triangle</td><td>3</td><td>3</td></tr>
<tr><td>Square</td><td>4</td><td>4</td></tr>
<tr><td>Rectangle</td><td>2</td><td>2</td></tr>
<tr><td>Regular hexagon</td><td>6</td><td>6</td></tr>
<tr><td>Circle</td><td>Infinite</td><td>Infinite</td></tr>
</tbody></table></div>
<hr class="section-divider">
<h2 class="notes-h2">TOPIC 5: MENSURATION</h2>
<h3 class="notes-h3">5.1 Perimeter and Area</h3>
<div class="table-wrap"><table class="notes-table">
<thead>
<tr><th>Shape</th><th>Perimeter</th><th>Area</th></tr>
</thead><tbody>
<tr><td>Square (side a)</td><td>4a</td><td>a²</td></tr>
<tr><td>Rectangle (l × w)</td><td>2(l + w)</td><td>lw</td></tr>
<tr><td>Triangle</td><td>Sum of sides</td><td>½ × base × height</td></tr>
<tr><td>Parallelogram</td><td>Sum of sides</td><td>base × perpendicular height</td></tr>
<tr><td>Trapezium</td><td>Sum of sides</td><td>½(a + b) × h</td></tr>
<tr><td>Circle (radius r)</td><td>2πr</td><td>πr²</td></tr>
</tbody></table></div>
<p><strong>Arc and sector:</strong></p>
<ul class="notes-list">
<li>Arc length: L = (θ/360) × 2πr</li>
<li>Sector area: A = (θ/360) × πr²</li>
<li>Segment area [E]: Area of sector − Area of triangle</li>
</ul>
<p><strong>Worked Example:</strong> Find the arc length and area of a sector with radius 8 cm and angle 45°. (π = 3.142)</p>
<ul class="notes-list">
<li>Arc length = (45/360) × 2 × 3.142 × 8 = (1/8) × 50.27 = <strong>6.28 cm</strong></li>
<li>Sector area = (45/360) × 3.142 × 64 = (1/8) × 201.1 = <strong>25.1 cm²</strong></li>
</ul>
<h3 class="notes-h3">5.2 Surface Area and Volume</h3>
<div class="table-wrap"><table class="notes-table">
<thead>
<tr><th>Solid</th><th>Surface Area</th><th>Volume</th></tr>
</thead><tbody>
<tr><td>Cube (side a)</td><td>6a²</td><td>a³</td></tr>
<tr><td>Cuboid (l×w×h)</td><td>2(lw + lh + wh)</td><td>lwh</td></tr>
<tr><td>Cylinder (r, h)</td><td>2πr² + 2πrh</td><td>πr²h</td></tr>
<tr><td>Cone (r, l, h)</td><td>πr² + πrl</td><td>⅓πr²h</td></tr>
<tr><td>Sphere (r)</td><td>4πr²</td><td>(4/3)πr³</td></tr>
<tr><td>Pyramid (base B, height h)</td><td>Base + lateral faces</td><td>⅓ × B × h</td></tr>
</tbody></table></div>
<p>Where l = slant height of cone.</p>
<p><strong>Worked Example:</strong> A cylindrical tank has radius 3 m and height 5 m. Find its volume.</p>
<ul class="notes-list">
<li>V = πr²h = π × 9 × 5 = <strong>45π ≈ 141.4 m³</strong></li>
</ul>
<hr class="section-divider">
<h2 class="notes-h2">TOPIC 6: TRIGONOMETRY</h2>
<h3 class="notes-h3">6.1 Right-Angled Triangles (SOH-CAH-TOA)</h3>
<p>For a right-angled triangle with angle θ:</p>
<ul class="notes-list">
<li><strong>sin θ = opposite/hypotenuse</strong></li>
<li><strong>cos θ = adjacent/hypotenuse</strong></li>
<li><strong>tan θ = opposite/adjacent</strong></li>
</ul>
<p><strong>Memory aid:</strong> SOH-CAH-TOA</p>
<p><strong>Pythagoras' Theorem:</strong> a² + b² = c² (c = hypotenuse)</p>
<p><strong>Finding an unknown side:</strong></p>
<ul class="notes-list">
<li>sin 30° = opp/10 → opp = 10 sin 30° = 10 × 0.5 = 5</li>
</ul>
<p><strong>Finding an unknown angle:</strong></p>
<ul class="notes-list">
<li>tan θ = 4/3 → θ = tan⁻¹(4/3) = 53.1°</li>
</ul>
<h3 class="notes-h3">6.2 Trigonometric Ratios — Special Angles</h3>
<div class="table-wrap"><table class="notes-table">
<thead>
<tr><th>Angle</th><th>sin</th><th>cos</th><th>tan</th></tr>
</thead><tbody>
<tr><td>0°</td><td>0</td><td>1</td><td>0</td></tr>
<tr><td>30°</td><td>1/2</td><td>√3/2</td><td>1/√3</td></tr>
<tr><td>45°</td><td>1/√2</td><td>1/√2</td><td>1</td></tr>
<tr><td>60°</td><td>√3/2</td><td>1/2</td><td>√3</td></tr>
<tr><td>90°</td><td>1</td><td>0</td><td>undefined</td></tr>
</tbody></table></div>
<h3 class="notes-h3">6.3 Sine and Cosine Rules [E]</h3>
<p><strong>For any triangle with sides a, b, c and opposite angles A, B, C:</strong></p>
<p><strong>Sine Rule:</strong> a/sin A = b/sin B = c/sin C</p>
<p><em>Use when given:</em> two angles + one side OR two sides + non-included angle</p>
<p><strong>Cosine Rule:</strong> a² = b² + c² − 2bc cos A (finding a side)</p>
<p>cos A = (b² + c² − a²) / (2bc) (finding an angle)</p>
<p><em>Use when given:</em> two sides + included angle OR three sides</p>
<p><strong>Area of triangle using sine rule [E]:</strong> Area = ½ ab sin C</p>
<p><strong>Worked Example [E]:</strong> In triangle ABC, AB = 7 cm, BC = 9 cm, angle B = 55°. Find AC.</p>
<ul class="notes-list">
<li>a² = b² + c² - 2bc cos A (using cosine rule with b = 7, c = 9, A = 55°)</li>
<li>AC² = 7² + 9² - 2(7)(9) cos 55°</li>
<li>AC² = 49 + 81 - 126 × 0.5736</li>
<li>AC² = 130 - 72.27 = 57.73</li>
<li>AC = <strong>7.60 cm</strong></li>
</ul>
<h3 class="notes-h3">6.4 Bearings</h3>
<ul class="notes-list">
<li>Always measured <strong>clockwise</strong> from <strong>North</strong></li>
<li>Written as a 3-digit number: 090° (East), 180° (South), 270° (West), 000° or 360° (North)</li>
</ul>
<p><strong>Worked Example:</strong> A ship sails from A on a bearing of 120° for 50 km to B. How far South and East of A is B?</p>
<ul class="notes-list">
<li>South component = 50 cos 30° = 50 × 0.866 = 43.3 km</li>
<li>East component = 50 sin 30° = 50 × 0.5 = 25 km (Note: bearing 120° is 30° from South towards East)</li>
</ul>
<hr class="section-divider">
<h2 class="notes-h2">TOPIC 7: VECTORS [E]</h2>
<h3 class="notes-h3">7.1 Vector Notation</h3>
<p>A vector has <strong>magnitude</strong> (size) and <strong>direction</strong>.</p>
<ul class="notes-list">
<li>Denoted by bold <strong>a</strong> or with an arrow: →AB</li>
<li>Column vector: <strong>a</strong> = (x/y) where x = horizontal component, y = vertical component</li>
</ul>
<p><strong>Magnitude (length):</strong> |<strong>a</strong>| = √(x² + y²)</p>
<p><strong>Adding and subtracting vectors:</strong></p>
<ul class="notes-list">
<li>(3/2) + (1/-4) = (4/-2)</li>
<li>(3/2) - (1/-4) = (2/6)</li>
</ul>
<p><strong>Scalar multiplication:</strong></p>
<ul class="notes-list">
<li>3(2/-1) = (6/-3) (each component multiplied by the scalar)</li>
</ul>
<h3 class="notes-h3">7.2 Position Vectors</h3>
<p>If O is the origin:</p>
<ul class="notes-list">
<li>Position vector of A = →OA = <strong>a</strong></li>
<li>→AB = <strong>b</strong> - <strong>a</strong> (vector from A to B = position vector of B minus position vector of A)</li>
</ul>
<p><strong>Worked Example:</strong> O is the origin. A has position vector (3/1) and B has position vector (7/5). Find the vector →AB.</p>
<ul class="notes-list">
<li>→AB = →OB - →OA = (7/5) - (3/1) = <strong>(4/4)</strong></li>
</ul>
<h3 class="notes-h3">7.3 Parallel Vectors and Collinear Points</h3>
<ul class="notes-list">
<li><strong>a</strong> and <strong>b</strong> are parallel if <strong>a</strong> = k<strong>b</strong> for some scalar k</li>
<li>Points A, B, C are collinear if →AC = k→AB for some scalar k</li>
</ul>
<hr class="section-divider">
<h2 class="notes-h2">TOPIC 8: MATRICES [E]</h2>
<h3 class="notes-h3">8.1 Matrix Operations</h3>
<p>A matrix is a rectangular array of numbers. Size is described as "rows × columns".</p>
<p><strong>Addition/Subtraction:</strong> Matrices must have the same dimensions. Add/subtract corresponding elements.</p>
<p><strong>Scalar multiplication:</strong> Multiply every element by the scalar.</p>
<ul class="notes-list">
<li>3 × (2 4 / 1 -3) = (6 12 / 3 -9)</li>
</ul>
<p><strong>Matrix multiplication:</strong> A × B is possible only when columns of A = rows of B.</p>
<ul class="notes-list">
<li>(2×2)(2×2) → (2×2) result</li>
<li>Element at row i, column j = sum of products of row i of A with column j of B</li>
</ul>
<p><strong>Worked Example:</strong></p>
<p>(1 2 / 3 4) × (5 6 / 7 8)</p>
<p>Row 1, Col 1: (1×5) + (2×7) = 5 + 14 = 19</p>
<p>Row 1, Col 2: (1×6) + (2×8) = 6 + 16 = 22</p>
<p>Row 2, Col 1: (3×5) + (4×7) = 15 + 28 = 43</p>
<p>Row 2, Col 2: (3×6) + (4×8) = 18 + 32 = 50</p>
<p>Result: <strong>(19 22 / 43 50)</strong></p>
<h3 class="notes-h3">8.2 Determinant and Inverse</h3>
<p>For a 2×2 matrix M = (a b / c d):</p>
<ul class="notes-list">
<li><strong>Determinant:</strong> det(M) = ad - bc</li>
<li><strong>Inverse:</strong> M⁻¹ = 1/det(M) × (d -b / -c a) — (swap diagonal, negate off-diagonal)</li>
<li>Inverse exists only when det(M) ≠ 0</li>
</ul>
<p><strong>Solving simultaneous equations using matrices:</strong></p>
<ul class="notes-list">
<li>System: 2x + y = 7, 3x - 2y = 0</li>
<li>Matrix form: (2 1 / 3 -2)(x / y) = (7 / 0)</li>
<li>x/y = M⁻¹ × (7 / 0)</li>
</ul>
<hr class="section-divider">
<h2 class="notes-h2">TOPIC 9: TRANSFORMATIONS</h2>
<h3 class="notes-h3">9.1 Types of Transformations</h3>
<div class="table-wrap"><table class="notes-table">
<thead>
<tr><th>Transformation</th><th>What changes</th><th>What stays the same</th></tr>
</thead><tbody>
<tr><td>Reflection</td><td>Position/orientation</td><td>Shape, size</td></tr>
<tr><td>Rotation</td><td>Position/orientation</td><td>Shape, size</td></tr>
<tr><td>Translation</td><td>Position</td><td>Shape, size, orientation</td></tr>
<tr><td>Enlargement</td><td>Size, position</td><td>Shape, angles</td></tr>
</tbody></table></div>
<h3 class="notes-h3">9.2 Describing Transformations</h3>
<p><strong>Reflection:</strong> State the mirror line (e.g., reflection in the line y = x)</p>
<p><strong>Rotation:</strong> State centre, angle, direction (clockwise or anticlockwise)</p>
<p><strong>Translation:</strong> State the vector (x/y)</p>
<p><strong>Enlargement:</strong> State the centre of enlargement and the scale factor k</p>
<ul class="notes-list">
<li>If k > 0: object and image on same side of centre</li>
<li>If k < 0: object and image on opposite sides (inverted) [E]</li>
<li>If |k| > 1: enlargement; if |k| < 1: reduction</li>
</ul>
<h3 class="notes-h3">9.3 Transformations Using Matrices [E]</h3>
<div class="table-wrap"><table class="notes-table">
<thead>
<tr><th>Transformation</th><th>Matrix</th></tr>
</thead><tbody>
<tr><td>Reflection in x-axis</td><td>(1 0 / 0 -1)</td></tr>
<tr><td>Reflection in y-axis</td><td>(-1 0 / 0 1)</td></tr>
<tr><td>Reflection in y = x</td><td>(0 1 / 1 0)</td></tr>
<tr><td>Rotation 90° anticlockwise</td><td>(0 -1 / 1 0)</td></tr>
<tr><td>Rotation 90° clockwise</td><td>(0 1 / -1 0)</td></tr>
<tr><td>Rotation 180°</td><td>(-1 0 / 0 -1)</td></tr>
<tr><td>Enlargement scale factor k</td><td>(k 0 / 0 k)</td></tr>
</tbody></table></div>
<hr class="section-divider">
<h2 class="notes-h2">TOPIC 10: STATISTICS</h2>
<h3 class="notes-h3">10.1 Averages and Spread</h3>
<div class="table-wrap"><table class="notes-table">
<thead>
<tr><th>Measure</th><th>Definition</th><th>When to use</th></tr>
</thead><tbody>
<tr><td>Mean</td><td>Sum of all values ÷ number of values</td><td>Balanced, no extreme values</td></tr>
<tr><td>Median</td><td>Middle value when ordered</td><td>Skewed data or extreme values</td></tr>
<tr><td>Mode</td><td>Most frequently occurring value</td><td>Categorical data</td></tr>
<tr><td>Range</td><td>Largest − Smallest</td><td>Simple measure of spread</td></tr>
<tr><td>Interquartile Range (IQR)</td><td>UQ − LQ</td><td>Better spread measure, ignores extremes</td></tr>
</tbody></table></div>
<p><strong>Calculating mean from frequency table:</strong></p>
<ul class="notes-list">
<li>Mean = Σ(fx) / Σf</li>
<li>Where f = frequency, x = value (use midpoint for grouped data)</li>
</ul>
<p><strong>Worked Example:</strong> Calculate the mean of: 3, 7, 4, 8, 3, 6, 5</p>
<ul class="notes-list">
<li>Sum = 3+7+4+8+3+6+5 = 36</li>
<li>Mean = 36 ÷ 7 = <strong>5.14 (3 s.f.)</strong></li>
</ul>
<h3 class="notes-h3">10.2 Histograms [E]</h3>
<p>For grouped data where class widths are <strong>unequal</strong>, use <strong>frequency density</strong>:</p>
<ul class="notes-list">
<li>Frequency density = Frequency ÷ Class width</li>
<li>Area of bar = Frequency (not height!)</li>
</ul>
<h3 class="notes-h3">10.3 Cumulative Frequency</h3>
<ul class="notes-list">
<li>Plot cumulative frequency against <strong>upper class boundary</strong></li>
<li>Draw smooth S-shaped curve</li>
<li>From the curve: read off Median, Lower Quartile (Q1), Upper Quartile (Q3)</li>
<li>Median: at cumulative frequency = n/2</li>
<li>Q1: at cf = n/4</li>
<li>Q3: at cf = 3n/4</li>
<li>IQR = Q3 − Q1</li>
</ul>
<p><strong>Box and whisker diagram:</strong> Shows Min, Q1, Median, Q3, Max on a number line.</p>
<h3 class="notes-h3">10.4 Scatter Diagrams</h3>
<ul class="notes-list">
<li><strong>Positive correlation:</strong> Both variables increase together</li>
<li><strong>Negative correlation:</strong> One increases as the other decreases</li>
<li><strong>No correlation:</strong> No clear pattern</li>
</ul>
<p><strong>Line of best fit:</strong> Drawn through the data, passing through the mean point (x̄, ȳ).</p>
<hr class="section-divider">
<h2 class="notes-h2">TOPIC 11: PROBABILITY</h2>
<h3 class="notes-h3">11.1 Basic Probability</h3>
<p><strong>Probability</strong> of event A: P(A) = (number of favourable outcomes) / (total number of possible outcomes)</p>
<ul class="notes-list">
<li>0 ≤ P(A) ≤ 1</li>
<li>P(A does not happen) = 1 - P(A)</li>
<li>P(certain event) = 1</li>
</ul>
<h3 class="notes-h3">11.2 Combined Events</h3>
<p><strong>Mutually exclusive events (cannot happen at same time):</strong></p>
<ul class="notes-list">
<li>P(A or B) = P(A) + P(B)</li>
</ul>
<p><strong>Independent events (one does not affect the other):</strong></p>
<ul class="notes-list">
<li>P(A and B) = P(A) × P(B)</li>
</ul>
<p><strong>Tree diagrams:</strong> Used to show all possible outcomes of two or more events.</p>
<ul class="notes-list">
<li>Multiply along branches to find probability of combined outcome</li>
<li>Add probabilities of different routes to the same outcome</li>
</ul>
<p><strong>Worked Example:</strong> A bag has 3 red and 5 blue balls. Two balls are drawn without replacement. Find P(both red).</p>
<ul class="notes-list">
<li>P(first red) = 3/8</li>
<li>P(second red | first was red) = 2/7</li>
<li>P(both red) = 3/8 × 2/7 = <strong>6/56 = 3/28</strong></li>
</ul>
<h3 class="notes-h3">11.3 Conditional Probability [E]</h3>
<p>P(B|A) = P(A and B) / P(A)</p>
<p>This is the probability of B given that A has already happened.</p>
<p><strong>Venn diagrams:</strong> Use for events with overlapping outcomes.</p>
<ul class="notes-list">
<li>P(A or B) = P(A) + P(B) - P(A and B)</li>
<li>P(only A) = P(A) - P(A and B)</li>
</ul>
<hr class="section-divider">
<h2 class="notes-h2">TOPIC 12: SEQUENCES</h2>
<h3 class="notes-h3">12.1 Number Sequences</h3>
<p><strong>Finding the nth term of a linear sequence (arithmetic):</strong></p>
<ul class="notes-list">
<li>nth term = a + (n - 1)d</li>
<li>where a = first term, d = common difference</li>
</ul>
<p><strong>Worked Example:</strong> Find the nth term of 5, 8, 11, 14, ...</p>
<ul class="notes-list">
<li>d = 3, a = 5</li>
<li>nth term = 5 + (n-1)3 = 5 + 3n - 3 = <strong>3n + 2</strong></li>
</ul>
<p><strong>Geometric sequences [E]:</strong> Each term is multiplied by a constant ratio r.</p>
<ul class="notes-list">
<li>nth term = a × rⁿ⁻¹</li>
</ul>
<h3 class="notes-h3">12.2 Recognising Sequences</h3>
<div class="table-wrap"><table class="notes-table">
<thead>
<tr><th>Sequence</th><th>Type</th><th>nth term</th></tr>
</thead><tbody>
<tr><td>2, 4, 6, 8, ...</td><td>Even numbers</td><td>2n</td></tr>
<tr><td>1, 3, 5, 7, ...</td><td>Odd numbers</td><td>2n - 1</td></tr>
<tr><td>1, 4, 9, 16, ...</td><td>Square numbers</td><td>n²</td></tr>
<tr><td>1, 8, 27, 64, ...</td><td>Cube numbers</td><td>n³</td></tr>
<tr><td>1, 1, 2, 3, 5, 8, ...</td><td>Fibonacci</td><td>(each term = sum of previous two)</td></tr>
<tr><td>1, 2, 4, 8, 16, ...</td><td>Powers of 2</td><td>2ⁿ⁻¹</td></tr>
</tbody></table></div>
<hr class="section-divider">
<h2 class="notes-h2">EXAM TECHNIQUE GUIDE</h2>
<h3 class="notes-h3">Paper Strategy</h3>
<p><strong>Paper 1 / Paper 2 (Non-calculator):</strong></p>
<ul class="notes-list">
<li>Show all arithmetic working — no calculator to verify</li>
<li>Know exact values: sin 30° = 0.5, cos 60° = 0.5, sin 45° = 1/√2</li>
<li>Time: 1 mark per minute as a guide</li>
</ul>
<p><strong>Paper 3 / Paper 4 (Calculator):</strong></p>
<ul class="notes-list">
<li>Give answers to 3 significant figures unless the question specifies otherwise</li>
<li>Write down intermediate values — don't round until the final answer</li>
<li>If an answer seems unreasonably large or small, recheck</li>
</ul>
<h3 class="notes-h3">Common Mistakes to Avoid</h3>
<ol class="notes-list">
<li><strong>Dividing by a negative:</strong> When solving inequalities, ALWAYS reverse the sign</li>
<li><strong>Standard form:</strong> Ensure 1 ≤ A < 10. Example: 23 × 10³ is NOT standard form — it should be 2.3 × 10⁴</li>
<li><strong>Rounding too early:</strong> Keep full calculator values until the final step</li>
<li><strong>Circle theorems:</strong> Always state which theorem you used — marks are awarded for reasoning</li>
<li><strong>Transformation descriptions:</strong> Always give ALL required information (e.g., rotation needs centre + angle + direction)</li>
<li><strong>Probability:</strong> Must be between 0 and 1. If you get > 1 or < 0, recheck</li>
<li><strong>Cumulative frequency:</strong> Plot against UPPER class boundary, not midpoint</li>
<li><strong>Cosine rule:</strong> Remember the formula is a² = b² + c² <strong>−</strong> 2bc cos A (minus sign!)</li>
</ol>
<h3 class="notes-h3">Command Words</h3>
<div class="table-wrap"><table class="notes-table">
<thead>
<tr><th>Word</th><th>What to do</th></tr>
</thead><tbody>
<tr><td>Calculate</td><td>Work out a numerical answer. Show working.</td></tr>
<tr><td>Simplify</td><td>Reduce to simplest form (algebra)</td></tr>
<tr><td>Solve</td><td>Find the value(s) of the unknown</td></tr>
<tr><td>Factorise</td><td>Write as a product of factors</td></tr>
<tr><td>Expand</td><td>Remove brackets</td></tr>
<tr><td>Sketch</td><td>Show general shape — accuracy less critical. Label key features.</td></tr>
<tr><td>Draw</td><td>Accurate graph/diagram required</td></tr>
<tr><td>Show that</td><td>Prove the given result. Start from one side, work to the other.</td></tr>
<tr><td>Explain</td><td>Give a reason in words or with reference to a theorem</td></tr>
<tr><td>Write down</td><td>Answer required — little or no working needed</td></tr>
<tr><td>Estimate</td><td>Use rounded values to give approximate answer</td></tr>
<tr><td>Find</td><td>Work out, showing appropriate method</td></tr>
<tr><td>Prove</td><td>Formal argument required — all steps must be justified</td></tr>
</tbody></table></div>
<h3 class="notes-h3">Mark Types</h3>
<ul class="notes-list">
<li><strong>M marks</strong> (Method marks): Awarded for correct method, even if final answer is wrong</li>
<li><strong>A marks</strong> (Accuracy marks): Awarded only when method is correct</li>
<li><strong>B marks</strong> (Independent marks): Awarded regardless of method</li>
<li><strong>FT marks</strong> (Follow through): Marks awarded for correct work based on an earlier wrong answer</li>
</ul>
<h3 class="notes-h3">Checklist Before Submitting</h3>
<ul class="notes-list">
<li>[ ] Have I answered every question? (No blanks — attempt everything)</li>
<li>[ ] Have I shown all working? (Especially for multi-step questions)</li>
<li>[ ] Are my answers in the correct units? (cm vs m vs km; cm² vs m², etc.)</li>
<li>[ ] Have I rounded correctly? (3 s.f. unless specified)</li>
<li>[ ] Have I re-read questions I found difficult? (Often a re-read reveals something missed)</li>
<li>[ ] Are my diagrams labelled? (Sides, angles, coordinates)</li>
</ul>
<hr class="section-divider">
<h2 class="notes-h2">PRACTICE PAPER — EXAM-STYLE QUESTIONS</h2>
<p><strong>Section A (Non-calculator)</strong></p>
<ol class="notes-list">
<li>Write 2.7 × 10⁻³ as an ordinary number. <em>(1 mark)</em></li>
</ol>
<ol class="notes-list">
<li>Factorise completely: 12x²y - 8xy². <em>(2 marks)</em></li>
</ol>
<ol class="notes-list">
<li>Solve the inequality -3x + 4 > 13. <em>(2 marks)</em></li>
</ol>
<ol class="notes-list">
<li>Find the nth term of the sequence 7, 13, 19, 25, ... <em>(2 marks)</em></li>
</ol>
<p><strong>Section B (Calculator)</strong></p>
<ol class="notes-list">
<li>A sphere has radius 5 cm. Calculate its volume. Give your answer in terms of π. <em>(2 marks)</em></li>
</ol>
<ol class="notes-list">
<li>[E] The line through A(2, -1) is perpendicular to y = 2x + 3. Find the equation of this line. <em>(3 marks)</em></li>
</ol>
<ol class="notes-list">
<li>In a class, 15 students study French, 10 study German, and 4 study both. How many students study neither? (Total class = 25 students.) <em>(3 marks)</em></li>
</ol>
<ol class="notes-list">
<li>[E] Solve 3x² - 5x - 2 = 0. Give exact answers. <em>(3 marks)</em></li>
</ol>
<p><strong>Answers:</strong></p>
<ol class="notes-list">
<li>0.0027</li>
<li>4xy(3x - 2y)</li>
<li>x < -3</li>
<li>6n + 1</li>
<li>(500/3)π cm³</li>
<li>y = -1/2 x [perpendicular gradient = -1/2; y - (-1) = -1/2(x - 2) → y = -x/2]</li>
<li>25 - (15 + 10 - 4) = 4 students</li>
<li>(3x + 1)(x - 2) = 0 → x = -1/3 or x = 2</li>
</ol>
<hr class="section-divider">
<h2 class="notes-h2">KEY FORMULAE SUMMARY SHEET</h2>
<p><strong>Cut out or save this section for quick reference.</strong></p>
<pre class="code-block"><code>
NUMBER
Percentage change: (change ÷ original) × 100
Reverse %: original = new ÷ (1 ± r/100)
Bounds: ± half of the stated precision

ALGEBRA
Quadratic formula: x = [-b ± √(b²-4ac)] / 2a
Gradient: m = (y₂ - y₁) / (x₂ - x₁)

GEOMETRY
Interior angle sum: (n-2) × 180°
Arc length: (θ/360) × 2πr
Sector area: (θ/360) × πr²

TRIGONOMETRY (right-angled)
sin = opp/hyp cos = adj/hyp tan = opp/adj
a² + b² = c² (Pythagoras)

TRIGONOMETRY (any triangle)
Sine rule: a/sin A = b/sin B = c/sin C
Cosine rule: a² = b² + c² - 2bc cos A
Area: ½ ab sin C

MENSURATION
Circle area: πr² Circumference: 2πr
Cylinder vol: πr²h SA: 2πr² + 2πrh
Cone vol: ⅓πr²h SA: πrl + πr²
Sphere vol: (4/3)πr³ SA: 4πr²

STATISTICS
Mean (freq. table): Σfx / Σf
IQR = Q3 - Q1
</code></pre>
<hr class="section-divider">
<h2 class="notes-h2">FREQUENTLY TESTED PAST PAPER TOPICS</h2>
<p>Based on analysis of May/June and Oct/Nov papers (2019–2024):</p>
<div class="table-wrap"><table class="notes-table">
<thead>
<tr><th>Topic</th><th>Frequency</th><th>Paper(s)</th></tr>
</thead><tbody>
<tr><td>Percentages / reverse percentage</td><td>Every paper</td><td>1, 2, 3, 4</td></tr>
<tr><td>Quadratic equations / factorisation</td><td>Every paper</td><td>2, 4</td></tr>
<tr><td>Circle theorems</td><td>Every paper</td><td>4</td></tr>
<tr><td>Cumulative frequency / box plot</td><td>Every paper</td><td>3, 4</td></tr>
<tr><td>Trigonometry (right-angled)</td><td>Every paper</td><td>1, 2, 3, 4</td></tr>
<tr><td>Sine/cosine rules</td><td>Every paper</td><td>4</td></tr>
<tr><td>Vectors</td><td>Every Extended paper</td><td>4</td></tr>
<tr><td>Transformations</td><td>Every paper</td><td>3, 4</td></tr>
<tr><td>Sequences (nth term)</td><td>Every paper</td><td>1, 2, 3, 4</td></tr>
<tr><td>Similarity / scale factors</td><td>Every paper</td><td>3, 4</td></tr>
<tr><td>Probability tree diagrams</td><td>Every paper</td><td>3, 4</td></tr>
<tr><td>Standard form</td><td>Every paper</td><td>1, 2</td></tr>
<tr><td>Functions and inverse functions</td><td>Every Extended paper</td><td>2, 4</td></tr>
</tbody></table></div>
<hr class="section-divider">
<p><em>These notes cover the complete Cambridge IGCSE Mathematics (0580) syllabus for Core and Extended candidates.</em></p>
<p><em>For official past papers, mark schemes, and examiner reports, visit cambridgeinternational.org</em></p>
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