Maths 2 Specification

Arithmetic Sequence

• General term: aₙ = a₁ + (n − 1)d, where d is common difference
• Sum of first n terms: Sₙ = n/2 (2a₁ + (n − 1)d) or Sₙ = n/2 (a₁ + aₙ)

Geometric Sequence

• General term: aₙ = a₁ × rⁿ⁻¹, where r is common ratio
• Sum of first n terms: Sₙ = a₁(1 − rⁿ)/(1 − r) for r ≠ 1

Infinite Geometric Series

• Convergence condition: |r| < 1
• Sum to infinity: S∞ = a₁/(1 − r) when |r| < 1

Binomial Expansion

• Binomial theorem (integer n): (a + b)ⁿ = Σ(r=0 to n) C(n,r) aⁿ⁻ʳ bʳ
• Where C(n,r) = n!/(r!(n−r)!) is the binomial coefficient
• Generalised form (fractional p): (1 + x)ᵖ = 1 + px + p(p−1)x²/2! + ... (|x| < 1)

Straight Lines

• Point-slope form: y − y₁ = m(x − x₁), where m is gradient
• General form: ax + by + c = 0
• Parallel lines: m₁ = m₂ (same gradient)
• Perpendicular lines: m₁ × m₂ = −1 (product of gradients = −1)
• Distance between two points: d = √((x₂ − x₁)² + (y₂ − y₁)²)
• Midpoint: M = ((x₁ + x₂)/2, (y₁ + y₂)/2)

Circle Equations

• Centre-radius form: (x − a)² + (y − b)² = r², centre (a, b), radius r
• General expanded form: x² + y² + 2fx + 2gy + c = 0, where centre = (−f, −g) and r = √(f² + g² − c)

Circle Theorems

Sine & Cosine Rules

• Sine rule: a/sin A = b/sin B = c/sin C = 2R (where R is circumradius)
• Ambiguous case (SSA): Two possible triangles when a, b, and A are given
• Cosine rule: a² = b² + c² − 2bc cos A (and cyclic permutations for other sides)
• Area of triangle: Area = ½bc sin A (using two sides and included angle)

Radian Measure & Sector Formulas

• Radians to degrees: θ (rad) = θ (deg) × π/180; 1 radian ≈ 57.3°
• Arc length: s = rθ (θ in radians)
• Sector area: A = ½r²θ (θ in radians)

Fundamental Identities

• sin²θ + cos²θ = 1
• tan θ = sin θ / cos θ
• sec θ = 1 / cos θ
• cosec θ = 1 / sin θ
• cot θ = 1 / tan θ = cos θ / sin θ

Solving Trigonometric Equations

• Example: sin θ = 0.5 → θ = 30° or θ = 150° (within 0° to 360°)
• Method: Use identities to transform equations (e.g., rewrite tan θ in terms of sin and cos), then solve for θ in the given range
• Be aware of periodicity and all solutions in a given interval

Exponential Functions

• Form: f(x) = abˣ (with a > 0, b > 0, b ≠ 1)
• When b > 1: Exponential growth
• When 0 < b < 1: Exponential decay
• Natural exponential: f(x) = eˣ (e ≈ 2.718)

Logarithmic Laws

• log(ab) = log a + log b
• log(a/b) = log a − log b
• log(aⁿ) = n log a
• log₁(a) = 1, log(1) = 0
• If a = bˣ, then x = log_b(a)
• Change of base: log_b(a) = log(a) / log(b)

Solving Exponential Equations

• General form: aˣ = b → x = log_a(b)
• Example: 2ˣ = 8 → 2ˣ = 2³ → x = 3
• Method: Take logarithms of both sides, use log laws to isolate x

Derivative Notation

• dy/dx, f'(x), ḟ, or d/dx f(x) all represent the derivative
• Physical meaning: Rate of change of y with respect to x at a point
• Geometric meaning: Gradient of the tangent to the curve at a point

Power Rule

• If f(x) = xⁿ, then f'(x) = nxⁿ⁻¹ for any rational n
• If f(x) = ax^n, then f'(x) = anxⁿ⁻¹
• Constant rule: d/dx(c) = 0
• Sum rule: d/dx(f + g) = f' + g'

Applications of Differentiation

• Increasing/decreasing intervals: f'(x) > 0 (increasing), f'(x) < 0 (decreasing)
• Stationary points: Solve f'(x) = 0; use sign of f' or f''(x) to classify as maxima/minima
• Rate of change: dy/dx gives rate of change of quantity with respect to another

Indefinite Integral

• Antiderivative: ∫f(x)dx = F(x) + C such that F'(x) = f(x)
• Constant of integration: C is arbitrary constant (family of solutions)
• Power rule: ∫xⁿ dx = xⁿ⁺¹/(n+1) + C (n ≠ −1)

Definite Integral

• Area under curve: ∫ₐᵇ f(x)dx = [F(x)]ₐᵇ = F(b) − F(a), where F is an antiderivative
• Fundamental Theorem of Calculus: Links differentiation and integration as inverses

Integration Techniques

• Combining integrals: ∫(f + g)dx = ∫f dx + ∫g dx
• Trapezium rule: Approximate area under curve using n equally spaced points: A ≈ (h/2)(y₀ + 2y₁ + 2y₂ + ... + 2yₙ₋₁ + yₙ), where h = (b − a)/n
• Solving differential equations: For dy/dx = f(x), integrate both sides to obtain y = ∫f(x)dx + C

Common Function Families

• Linear: f(x) = mx + c – slope m, intercept c
• Quadratic: f(x) = ax² + bx + c – vertex at x = −b/2a, parabola opening upward (a > 0) or downward (a < 0)
• Cubic: f(x) = ax³ + ... – 0, 1, or 2 turning points; shape determined by leading coefficient a
• Quartic: f(x) = ax⁴ + ... – up to 3 turning points; even-degree polynomial
• Trigonometric: sin x, cos x, tan x – periodicity 360° (or 2π rad), amplitude, phase shift
• Exponential: f(x) = aˣ (a > 1) – rapid growth; asymptote at y = 0
• Logarithmic: f(x) = log_a(x) – slow increase, vertical asymptote at x = 0
• Modulus: f(x) = |x| – V-shape, corner at origin

Transformations

Determining Graph Features

• Stationary points: Solve f'(x) = 0; use f''(x) or sign analysis of f' to classify as maxima/minima
• x-intercepts: Set f(x) = 0 and solve
• y-intercept: Evaluate f(0)
• Number of real roots: For polynomial, maximum number equals its degree; use discriminant and sign changes to find actual count
• Asymptotes: Horizontal (end behaviour), vertical (undefined points), oblique (for rational functions)