IGCSE Mathematics Revision Notes (Cambridge CIE)

Number and Algebra

1. Number Basics

• Integers: Whole numbers, positive, negative or zero.
• Primes: Numbers with exactly two factors (1 and itself), e.g., 2, 3, 5, 7.
• LCM & HCF: Lowest Common Multiple & Highest Common Factor.

Q1: Find the HCF of 24 and 36.

2. Standard Form

A x 10ⁿ, where 1 ≤ A < 10 and n is an integer.

Q2: Write 0.000045 in standard form.

3. Approximation and Estimation

• Significant Figures (SF): Non-zero digits are significant. Zeros between non-zero digits are significant. Leading zeros are not. Trailing zeros after a decimal point are significant.
• Decimal Places (DP): Count digits after the decimal point.

Q3: Round 12.047 to 2 significant figures.

4. Ratio, Proportion and Percentage

• Ratio: Comparing quantities, e.g., 2:3.
• Direct Proportion: y = kx.
• Inverse Proportion: y = k/x.
• Percentage Change: (Change / Original) x 100%.

Q4: Share $120 in the ratio 3:5.

5. Algebraic Manipulation

• Expanding: Removing brackets, e.g., a(b+c) = ab + ac.
• Factorising: Putting into brackets, common factors, difference of two squares (a²-b²=(a-b)(a+b)), quadratic factorisation.
• Simplifying: Collecting like terms.

Q5: Factorise 3x² - 12.

6. Equations and Inequalities

• Linear Equations: Isolate the variable.
• Quadratic Equations: Factorising, Quadratic Formula (x = [-b ± √(b²-4ac)] / 2a).
• Simultaneous Equations: Substitution or Elimination.
• Inequalities: Treat like equations, but reverse sign when multiplying/dividing by a negative number.

Q6: Solve 2x - 5 < 7.

7. Sequences

Finding the nth term for arithmetic sequences (a + (n-1)d).

Q7: Find the nth term of the sequence 3, 7, 11, 15, ...

8. Functions

f(x) notation, composite functions (fg(x)), inverse functions (f¹(x)).

Q8: If f(x) = 3x - 1, find f¹(x).

Geometry and Measure

9. Angles and Polygons

• Angles on a straight line: Sum to 180°.
• Angles at a point: Sum to 360°.
• Parallel Lines: Alternate angles are equal, Corresponding angles are equal, Interior angles sum to 180°.
• Polygon Interior Angle Sum: (n-2) x 180°.
• Each Interior Angle of Regular Polygon: [(n-2) x 180°] / n.
• Each Exterior Angle of Regular Polygon: 360° / n.

Q9: A regular hexagon has an exterior angle of what measure?

10. Mensuration (Area, Volume, Surface Area)

Q10: Calculate the area of a circle with radius 5 cm (use π = 3.14).

11. Trigonometry (Right-Angled Triangles)

• SOH CAH TOA: Sin θ = Opp/Hyp, Cos θ = Adj/Hyp, Tan θ = Opp/Adj.
• Pythagoras Theorem: a² + b² = c².

Q11: In a right-angled triangle, the opposite side is 8 cm and the hypotenuse is 10 cm. Find sin θ.

12. Trigonometry (Non-Right-Angled Triangles)

• Sine Rule: a/sin A = b/sin B = c/sin C.
• Cosine Rule: a² = b² + c² - 2bc cos A.
• Area: ½ab sin C.

Q12: Find the area of a triangle with sides 6 cm and 8 cm and the included angle 30°.

13. Vectors and Transformations

• Vectors: Magnitude and Direction. Addition/Subtraction.
• Transformations: Translation, Reflection, Rotation, Enlargement.

Q13: A vector v = (2, -3). Calculate 2v.

Coordinate Geometry

14. Lines and Gradients

• Gradient (m): (y2-y1) / (x2-x1).
• Equation of a Line: y = mx + c or y - y1 = m(x - x1).
• Parallel Lines: Same gradient.
• Perpendicular Lines: Product of gradients = -1 (m1m2 = -1).

Q14: Find the gradient of the line passing through (1, 2) and (3, 8).

Q15: Write the equation of a line with gradient 2 and y-intercept 3.

Probability

15. Basic Probability

• P(Event) = (Number of favourable outcomes) / (Total number of outcomes).
• P(A and B) = P(A) x P(B) (for independent events).
• P(A or B) = P(A) + P(B) (for mutually exclusive events).

Q16: A bag contains 3 red and 5 blue balls. What is the probability of picking a red ball?

Q17: Two dice are rolled. What is the probability of getting two sixes?

Statistics

16. Data Handling

• Mean: Sum of values / Number of values.
• Median: Middle value when ordered.
• Mode: Most frequent value.
• Range: Highest - Lowest.
• Interquartile Range (IQR): Q3 - Q1.

Q18: Find the median of the numbers: 7, 2, 5, 8, 3.

17. Representing Data

• Bar charts, Pie charts, Histograms, Cumulative frequency curves, Box-and-whisker plots.
• Histograms: Area of bar proportional to frequency. Frequency density = Frequency / Class Width.

Q19: The frequency density for a class interval 10-20 is 2. What is the frequency?

Q20: A student scores 60, 75, 80, 65, 70 in five tests. Calculate the mean score.

Answer Key

• A1: 12
• A2: 4.5 x 10^-5
• A3: 12
• A4: $45 and $75
• A5: 3(x-2)(x+2)
• A6: x < 6
• A7: 4n - 1
• A8: f¹(x) = (x+1)/3
• A9: 60°
• A10: 78.5 cm²
• A11: 0.8
• A12: 12 cm²
• A13: (4, -6)
• A14: 3
• A15: y = 2x + 3
• A16: 3/8
• A17: 1/36
• A18: 5
• A19: 20
• A20: 70