CSEC Additional Mathematics cheat sheet

Remainder Theorem: the remainder when f(x) is divided by (x − a) is f(a); by (ax − b) it is f(b/a).

Factor Theorem: (x − a) is a factor of f(x) exactly when f(a) = 0; use it to find unknown coefficients.

Completing the square: ax² + bx + c = a(x + b/2a)² + (c − b²/4a); the turning point of y = f(x) is (−b/2a, c − b²/4a).

Quadratic formula x = (−b ± √(b² − 4ac))/2a; discriminant b² − 4ac > 0 gives two distinct real roots, = 0 equal roots, < 0 no real roots.

Sum and product of roots of ax² + bx + c = 0: α + β = −b/a and αβ = c/a.

For real x, the quadratic ax² + bx + c has constant sign iff its discriminant is negative; sign follows the sign of a.

Function notation: fg(x) means f(g(x)), applied right to left; the inverse f⁻¹ reverses f, so f(f⁻¹(x)) = x on the stated domain.

Log laws: log(xy) = log x + log y, log(x/y) = log x − log y, log xⁿ = n log x, log_a a = 1, log_a 1 = 0; solve aˣ = b by taking logs.

Indices: a^(m/n) = ⁿ√(aᵐ); a⁰ = 1 and a^(−n) = 1/aⁿ for a ≠ 0.

Distance between points: √((x₂ − x₁)² + (y₂ − y₁)²); midpoint is ((x₁ + x₂)/2, (y₁ + y₂)/2).

Gradient m = (y₂ − y₁)/(x₂ − x₁); parallel lines have equal gradients, perpendicular lines have gradients whose product is −1.

Line forms: y = mx + c, y − y₁ = m(x − x₁), and x/a + y/b = 1 for intercepts a and b.

Circle with centre (h, k) and radius r: (x − h)² + (y − k)² = r²; general form x² + y² + 2gx + 2fy + c = 0 has centre (−g, −f) and radius √(g² + f² − c).

Exact values: sin 30° = 1/2, cos 30° = √3/2, tan 30° = 1/√3; sin 45° = cos 45° = 1/√2; sin 60° = √3/2, cos 60° = 1/2, tan 60° = √3.

Identities: sin²θ + cos²θ = 1, tan θ = sin θ/cos θ, and 1 + tan²θ = sec²θ where sec is used.

Sine rule a/sin A = b/sin B = c/sin C; cosine rule a² = b² + c² − 2bc cos A; area of triangle = ½ab sin C.

For y = a sin(bx) + c: amplitude = |a|, period = 360°/b (or 2π/b in radians), vertical shift c.

Radians: π radians = 180°; arc length s = rθ and sector area A = ½r²θ with θ in radians.

Derivative from first principles: f′(x) = lim(h→0) [f(x + h) − f(x)]/h.

Power rule: d/dx (axⁿ) = anxⁿ⁻¹; derivative of a constant is 0; differentiate term by term for polynomials.

Product rule (uv)′ = u′v + uv′; quotient rule (u/v)′ = (u′v − uv′)/v².

Chain rule: d/dx [f(g(x))] = f′(g(x))·g′(x); for y = (ax + b)ⁿ, dy/dx = an(ax + b)ⁿ⁻¹.

Stationary points occur where dy/dx = 0; use the sign of d²y/dx² (negative = maximum, positive = minimum) or a sign table to classify.

Tangents and normals: gradient of tangent at x = a is f′(a); normal gradient is −1/f′(a); use y − y₁ = m(x − x₁).

Integration reverses differentiation: ∫axⁿ dx = axⁿ⁺¹/(n + 1) + C for n ≠ −1.

Definite integrals: ∫ₐᵇ f(x) dx = F(b) − F(a); this gives the area between the curve and the x-axis, with regions below the axis contributing negative values.

Rates of change: if y = f(t), then dy/dt is the rate of change of y; connected rates use dy/dt = dy/dx · dx/dt.

Simple interest I = PRT/100; compound amount A = P(1 + r/100)ⁿ, with n compounding periods.

Depreciation: value after n years = P(1 − r/100)ⁿ; appreciation uses (1 + r/100)ⁿ.

For growth/decay modelled by y = Ae^(kt), k > 0 gives growth and k < 0 gives decay; use logs to solve for t.

Arithmetic progression: nth term a + (n − 1)d, sum Sₙ = (n/2)[2a + (n − 1)d].

Geometric progression: nth term arⁿ⁻¹, sum Sₙ = a(1 − rⁿ)/(1 − r) for r ≠ 1; sum to infinity S∞ = a/(1 − r) only when |r| < 1.

Statistics: mean = Σfx/Σf for grouped data; median position is the (n + 1)/2 th value; mode is the most frequent value.

Probability: P(A or B) = P(A) + P(B) − P(A and B); for independent events P(A and B) = P(A)P(B); P(not A) = 1 − P(A).

Vectors in 2D: magnitude of (x, y) is √(x² + y²); add components separately; scalar multiplication scales each component.

Linear programming: graph each inequality, shade the feasible region, then evaluate the objective function at the vertices to find the optimum.