Mathematics — Core Revision Notes

1. Number & Fractions

Number types: Natural numbers (counting numbers: 1, 2, 3…), Integers (whole numbers including negatives), Rational numbers (can be expressed as a fraction p/q), Irrational numbers (cannot be expressed as a fraction: π, √2). BIDMAS/BODMAS determines order of operations: Brackets → Indices → Division/Multiplication (left to right) → Addition/Subtraction (left to right). Standard form (scientific notation): A × 10ⁿ where 1 ≤ A < 10. To convert: count the number of places the decimal point moves — right = negative n, left = positive n. Percentage change = (change / original) × 100%. Compound interest: A = P(1 + r/100)ⁿ. Reverse percentages: if a price AFTER a 20% increase is £120, original = 120 ÷ 1.20 = £100. Upper and lower bounds: for a measurement of 4.7 cm (to 1 d.p.), lower bound = 4.65 cm, upper bound = 4.75 cm.

Ratio, proportion & fractions — exam essentials

• To share in ratio a:b:c — find total parts, divide amount by total, multiply by each part
• Direct proportion: y = kx — graph is straight line through origin
• Inverse proportion: y = k/x — graph is a hyperbola; as x doubles, y halves
• Adding fractions: find common denominator, add numerators
• Multiplying fractions: multiply numerators, multiply denominators, simplify
• Dividing fractions: multiply by the reciprocal of the second fraction (flip and multiply)
• Recurring decimals to fractions: multiply by 10ⁿ (where n = length of repeating block), subtract to eliminate the repeat

2. Algebra

Expanding brackets: a(b + c) = ab + ac. For double brackets: (x + a)(x + b) = x² + (a+b)x + ab. Factorising a quadratic: find two numbers that multiply to 'c' and add to 'b' in x² + bx + c. For ax² + bx + c: use the AC method (find two numbers that multiply to ac and add to b). Completing the square: x² + bx = (x + b/2)² − (b/2)². Quadratic formula: x = [−b ± √(b² − 4ac)] / 2a. The discriminant b² − 4ac tells you: > 0 means two distinct real roots, = 0 means one repeated root, < 0 means no real roots. Simultaneous equations: elimination (multiply to make coefficients equal, then add or subtract) or substitution (substitute one equation into the other). Inequalities: solve like equations BUT flip the inequality sign when multiplying or dividing by a negative number.

Indices and surds — the rules

• aᵐ × aⁿ = aᵐ⁺ⁿ (same base, multiply: add powers)
• aᵐ ÷ aⁿ = aᵐ⁻ⁿ (same base, divide: subtract powers)
• (aᵐ)ⁿ = aᵐⁿ (power of a power: multiply indices)
• a⁰ = 1 (any non-zero number to the power 0 equals 1)
• a⁻ⁿ = 1/aⁿ (negative power = reciprocal)
• a^(1/n) = ⁿ√a (fractional power = nth root)
• Surd rules: √a × √b = √(ab); √a / √b = √(a/b); to rationalise denominator, multiply by conjugate

3. Geometry & Circle Theorems

Interior angles of a polygon: sum = (n − 2) × 180°. Each interior angle of a regular polygon = (n − 2) × 180° / n. Each exterior angle of a regular polygon = 360° / n. Circle theorems: (1) Angle at centre = 2 × angle at circumference (same arc). (2) Angles in the same segment are equal. (3) Angle in a semicircle = 90°. (4) Opposite angles in a cyclic quadrilateral add up to 180°. (5) Tangent is perpendicular to the radius at point of contact. (6) Two tangents from an external point are equal in length. (7) Alternate segment theorem: angle between tangent and chord = angle in alternate segment. Arc length = (θ/360) × 2πr. Area of sector = (θ/360) × πr². Always give a reason for each step in circle theorem proofs.

Circle components: radius (R), diameter (D = 2R), circumference (C = 2πR = πD), chord, arc, sector, and segment. All circle theorem proofs require you to name the theorem used.

Trigonometry reference

• SOH: sin θ = Opposite / Hypotenuse
• CAH: cos θ = Adjacent / Hypotenuse
• TOA: tan θ = Opposite / Adjacent
• Sine rule: a/sin A = b/sin B = c/sin C (use when two angles + one side, or two sides + non-included angle)
• Cosine rule: a² = b² + c² − 2bc cos A (use when two sides + included angle, or three sides given)
• Area of triangle = ½ab sin C
• 3D trigonometry: identify the right-angled triangle in 3D, label it carefully, then apply SOHCAHTOA

4. Statistics & Probability

Mean = Σfx / Σf (for frequency tables). Median = middle value when data is ordered (use (n+1)/2 position). Mode = most frequent value. Range = max − min. Interquartile range (IQR) = Q3 − Q1 (measures spread of middle 50%). Standard deviation measures spread around the mean — a larger value means more spread. Cumulative frequency graphs: S-shaped curve; read off median at Σf/2, Q1 at Σf/4, Q3 at 3Σf/4. Histograms: frequency DENSITY = frequency ÷ class width (NOT frequency on y-axis). Box plots show min, Q1, median, Q3, max. Probability: P(A) = favourable outcomes / total outcomes. P(A or B) = P(A) + P(B) for mutually exclusive events. P(A and B) = P(A) × P(B) for independent events. Conditional probability: P(A|B) = P(A and B) / P(B). Tree diagrams multiply along branches and add between branches.

5. Sequences, Functions & Graphs

Arithmetic sequence: add/subtract a constant difference d. nth term = a + (n − 1)d where a is the first term. Geometric sequence: multiply by a constant ratio r. nth term = arⁿ⁻¹. Quadratic sequences: second difference is constant — the nth term contains n². To find the nth term of a quadratic sequence: find ½ × second difference (this gives the n² coefficient), then form a linear adjustment. Functions: f(x) notation means the output for input x. Composite function fg(x) means 'apply g first, then f'. Inverse function f⁻¹(x): swap x and y, then rearrange for y. For graph transformations: y = f(x + a) shifts left by a; y = f(x) + a shifts up by a; y = f(ax) stretches horizontally by factor 1/a; y = af(x) stretches vertically by factor a; y = −f(x) reflects in x-axis; y = f(−x) reflects in y-axis.

Vectors & Transformations

• Vector addition: add components separately: (a,b) + (c,d) = (a+c, b+d)
• Scalar multiplication: k(a,b) = (ka, kb) — scales the length, preserves/reverses direction
• Magnitude of vector (a,b) = √(a² + b²)
• Translation: described by a column vector — every point moves by the same amount
• Reflection: requires the equation of the mirror line — use coordinate geometry to find image points
• Rotation: requires centre, angle, and direction (clockwise/anticlockwise)
• Enlargement: requires centre and scale factor — negative scale factor rotates 180° through centre
• For similarity: corresponding lengths are in ratio k; areas in ratio k²; volumes in ratio k³