Overview
This lecture covers walkthrough solutions for the IGCSE Cambridge Mathematics Paper 4 (Extended), focusing on core calculations, methods, and key concepts from percentages and algebra to geometry, statistics, vectors, and calculus.
Percentages & Finance
- Discounted price: Subtract percentage from 100, apply to original price.
- Reverse percentage: Divide new value by percentage (as decimal) for original.
- Installments: Total payment = (monthly × number) + final payment + deposit.
- Deposit as percent: (Deposit/original price) × 100.
Fuel, Distance & Depreciation
- Total distance/fuel: Add values for each segment.
- Rate: (Total fuel / total distance) × 100 = L/100 km.
- Depreciation: Apply successive percentage reductions to value, year by year.
Tables, Graphs, & Gradients
- Substitute x-values into function for missing table entries.
- Plot points and sketch curve from table.
- Tangent’s gradient = rise/run at specified point.
- Solve equations graphically by intersection points.
Algebraic Manipulation
- Combine like terms: e.g., 3m–4m+5n+8n = –m+13n.
- Powers: (ab^n)^m = a^m b^{n×m}.
- Fractions: Convert to common denominators, combine, simplify.
Geometry & Trigonometry
- Perimeter: Sum all side expressions, solve for unknown.
- Solve quadratic using formula: x = [–b±√(b²–4ac)]/(2a).
- Simultaneous equations: Equate and simplify, solve for x/y.
- Quadrilateral angles sum to 360°; apply ratios to parts.
- Circle theorems: Alternate segment, tangents, angle properties.
- Arc length: (θ/360) × 2πr.
- Sector area: (θ/360) × πr².
Statistics & Probability
- Cumulative frequency: Median at ½ total, quartiles at ¼ and ¾.
- Interquartile range = Q3–Q1.
- Probability: Favorable outcomes/total; for non-replacement adjust denominator.
- Histogram: Frequency = density × class width; mean = Σ(f×midpoint)/Σf.
Vectors
- Vector arithmetic: Add/subtract components or scaled vectors.
- Magnitude: √(x²+y²).
- Express vectors between points using position vectors and given ratios.
- Parallel vectors: If a multiple, lines are parallel.
Calculus
- Differentiate: Power rule, e.g., d/dx(ax^n) = n·a·x^{n–1}.
- Turning points: Set dy/dx = 0, solve for x.
- Second derivative test: If d²y/dx² > 0, minimum; if < 0, maximum.
Functions
- Substitution: Evaluate f(x) or g(x) at given x.
- Composite functions: g(f(x)), substitute inner into outer.
- Inverse: Rearrange to solve for x in terms of y.
- Expand and simplify for required forms.
Key Terms & Definitions
- Gradient — The slope of a line (rise/run).
- Simultaneous Equations — Equations solved together for common solutions.
- Sector — A portion of a circle bounded by two radii and an arc.
- Cumulative Frequency — Running total of frequencies up to a class.
- Histogram — Bar graph representing frequency density.
- Vector Magnitude — Length of a vector, calculated by √(x²+y²).
- Turning Point — Stationary point of a curve where dy/dx = 0.
Action Items / Next Steps
- Practice past paper problems using similar formats and methods.
- Review circle theorems, vector operations, and methods for cumulative frequency graphs.
- Revisit trigonometry and differentiation rules for exam readiness.