BGCSE Additional Mathematics 4037 cheat sheet

Remainder Theorem: When a polynomial f(x) is divided by (x − a), the remainder is f(a).

Factor Theorem: (x − a) is a factor of f(x) if and only if f(a) = 0.

For a quadratic ax² + bx + c = 0, the discriminant Δ = b² − 4ac determines the nature of roots: Δ > 0 two distinct real roots; Δ = 0 one repeated real root; Δ < 0 no real roots (complex conjugate pair).

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

A function is one-to-one if each output corresponds to exactly one input; its inverse f⁻¹ exists only for one-to-one functions. The graphs of f and f⁻¹ are reflections in the line y = x.

Composite function: (f ∘ g)(x) = f(g(x)); the domain of f ∘ g is the set of x in the domain of g such that g(x) lies in the domain of f.

Completing the square: ax² + bx + c = a(x + b/(2a))² + (c − b²/(4a)). The vertex of the parabola y = ax² + bx + c is at (−b/(2a), c − b²/(4a)).

Partial fractions: A rational function with distinct linear factors in the denominator decomposes as A/(x − a) + B/(x − b); for a repeated factor (x − a)² use A/(x − a) + B/(x − a)².

To solve |ax + b| = c (c ≥ 0), set ax + b = c or ax + b = −c; to solve |ax + b| < c, write −c < ax + b < c.

For |ax + b| > c, split into ax + b > c or ax + b < −c.

Solving simultaneous equations: substitution or elimination; for a linear and a quadratic, substitute the linear expression into the quadratic to obtain a quadratic in one variable.

A quadratic inequality ax² + bx + c > 0 is solved by finding the critical values (roots) and testing intervals; the solution set is a union of intervals.

Graph transformations: y = f(x) + a translates the graph of y = f(x) by a units vertically; y = f(x + a) translates it by −a units horizontally; y = af(x) stretches vertically by factor a; y = f(ax) stretches horizontally by factor 1/a.

The graph of y = |f(x)| is obtained by reflecting any part of y = f(x) that lies below the x-axis into the upper half-plane.

To solve an inequality involving a rational expression, bring all terms to one side, factor the numerator and denominator, and use a sign chart with the critical values where numerator or denominator is zero.

Arithmetic progression: nth term uₙ = a + (n − 1)d; sum of first n terms Sₙ = n/2 [2a + (n − 1)d] = n/2 (a + l), where l is the last term.

Geometric progression: nth term uₙ = arⁿ⁻¹; sum of first n terms Sₙ = a(1 − rⁿ)/(1 − r) for r ≠ 1; sum to infinity S∞ = a/(1 − r) provided |r| < 1.

Binomial theorem for positive integer n: (a + b)ⁿ = Σ_{r=0}^{n} C(n, r) aⁿ⁻ʳ bʳ, where C(n, r) = n! / [r!(n − r)!].

General term in the expansion of (a + b)ⁿ is T_{r+1} = C(n, r) aⁿ⁻ʳ bʳ.

For (1 + x)ⁿ with |x| < 1 and any real n, (1 + x)ⁿ = 1 + nx + n(n−1)x²/2! + n(n−1)(n−2)x³/3! + …

The sum of the first n natural numbers is n(n + 1)/2; the sum of the first n squares is n(n + 1)(2n + 1)/6; the sum of the first n cubes is [n(n + 1)/2]².

A sequence converges if it tends to a finite limit as n → ∞; a geometric series converges if and only if |r| < 1.

Distance between points (x₁, y₁) and (x₂, y₂): √[(x₂ − x₁)² + (y₂ − y₁)²].

Midpoint of a segment joining (x₁, y₁) and (x₂, y₂): ((x₁ + x₂)/2, (y₁ + y₂)/2).

Gradient of a line through (x₁, y₁) and (x₂, y₂): m = (y₂ − y₁)/(x₂ − x₁). Two lines are perpendicular if m₁m₂ = −1.

Equation of a straight line: y − y₁ = m(x − x₁); general form ax + by + c = 0 has gradient −a/b.

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

The perpendicular distance from point (x₁, y₁) to line ax + by + c = 0 is |ax₁ + by₁ + c| / √(a² + b²).

The tangent to a circle at a point is perpendicular to the radius at that point; the perpendicular from the centre to a chord bisects the chord.

For a triangle with sides a, b, c opposite angles A, B, C: Sine Rule a/sin A = b/sin B = c/sin C; Cosine Rule a² = b² + c² − 2bc cos A.

Area of a triangle = ½ ab sin C.

In a right-angled triangle: sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse, tan θ = opposite/adjacent.

Identities: sin²θ + cos²θ = 1; tan θ = sin θ/cos θ; 1 + tan²θ = sec²θ; 1 + cot²θ = cosec²θ.

Exact values: sin 30° = ½, cos 30° = √3/2, tan 30° = 1/√3; sin 45° = cos 45° = 1/√2, tan 45° = 1; sin 60° = √3/2, cos 60° = ½, tan 60° = √3.

The general solution of sin θ = k is θ = nπ + (−1)ⁿ α, where α is the principal value; for cos θ = k, θ = 2nπ ± α; for tan θ = k, θ = nπ + α (n integer).

Double-angle formulas: sin 2θ = 2 sin θ cos θ; cos 2θ = cos²θ − sin²θ = 1 − 2 sin²θ = 2 cos²θ − 1; tan 2θ = 2 tan θ/(1 − tan²θ).

In radians, arc length s = rθ and sector area A = ½ r²θ, with θ in radians.

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

Power rule: d/dx (xⁿ) = n xⁿ⁻¹; d/dx (constant) = 0; d/dx (eˣ) = eˣ; d/dx (ln x) = 1/x.

Product rule: d/dx (uv) = u dv/dx + v du/dx; Quotient rule: d/dx (u/v) = (v du/dx − u dv/dx)/v²; Chain rule: dy/dx = dy/du · du/dx.

d/dx (sin x) = cos x; d/dx (cos x) = −sin x; d/dx (tan x) = sec²x.

At a stationary point, dy/dx = 0; the second derivative d²y/dx² determines its nature: > 0 local minimum, < 0 local maximum, = 0 inconclusive.

For a curve y = f(x), the gradient of the tangent at x = a is f′(a); the normal is perpendicular to the tangent, so its gradient is −1/f′(a).

The derivative gives the rate of change: if s(t) is displacement, then velocity v = ds/dt and acceleration a = dv/dt = d²s/dt².

Parametric differentiation: if x = f(t), y = g(t), then dy/dx = (dy/dt)/(dx/dt), provided dx/dt ≠ 0.

Indefinite integral: ∫ xⁿ dx = xⁿ⁺¹/(n + 1) + C for n ≠ −1; ∫ (1/x) dx = ln|x| + C; ∫ eˣ dx = eˣ + C.

∫ sin x dx = −cos x + C; ∫ cos x dx = sin x + C; ∫ sec²x dx = tan x + C.

Fundamental Theorem of Calculus: ∫_a^b f(x) dx = F(b) − F(a), where F′(x) = f(x).

The area between a curve y = f(x) and the x-axis from x = a to x = b is ∫_a^b |f(x)| dx; areas below the axis contribute negatively to the definite integral.

The area between two curves y = f(x) and y = g(x) from x = a to x = b is ∫_a^b [f(x) − g(x)] dx, where f(x) ≥ g(x) on the interval.

Volume of revolution about the x-axis: V = π ∫_a^b y² dx.

Integration by substitution: ∫ f(g(x)) g′(x) dx = ∫ f(u) du with u = g(x); for definite integrals, change the limits accordingly.

Integration by parts: ∫ u dv = uv − ∫ v du; choose u according to the order logarithmic, inverse trigonometric, algebraic, trigonometric, exponential (LIATE).

Magnitude of vector a = (x, y): |a| = √(x² + y²); unit vector in direction of a is a/|a|.

Scalar product: a · b = |a||b| cos θ = x₁x₂ + y₁y₂; perpendicular vectors have a · b = 0.

Position vector of a point dividing AB in ratio m:n is (n a + m b)/(m + n), where a and b are position vectors of A and B.

Matrix multiplication is defined when the number of columns of the first equals the number of rows of the second; in general AB ≠ BA.

The inverse of a 2×2 matrix A = [[a, b], [c, d]] is A⁻¹ = (1/(ad − bc)) [[d, −b], [−c, a]], provided det A = ad − bc ≠ 0.

A system of linear equations can be written as AX = B; if A is invertible, the unique solution is X = A⁻¹B.

The determinant of a 2×2 matrix [[a, b], [c, d]] is ad − bc; a matrix is singular if its determinant is zero.

Definition: log_a x = y ⇔ aʸ = x, for a > 0, a ≠ 1, x > 0.

Laws of logarithms: log_a (xy) = log_a x + log_a y; log_a (x/y) = log_a x − log_a y; log_a (xⁿ) = n log_a x; log_a 1 = 0; log_a a = 1.

Change of base: log_a b = log_c b / log_c a; in particular log_a b = ln b / ln a.

The natural logarithm ln x has base e; ln e = 1 and e^{ln x} = x for x > 0.

To solve aˣ = b, take logarithms of both sides: x = log b / log a.

The graph of y = eˣ passes through (0, 1) and has the x-axis as a horizontal asymptote; the graph of y = ln x passes through (1, 0) and has the y-axis as a vertical asymptote.

Exponential growth or decay: y = A e^{kt}; if k > 0 growth, if k < 0 decay; the half-life for decay is t = ln 2 / |k|.

Proof by contradiction: assume the negation of the statement, derive a logical impossibility, and conclude the original statement is true.

Proof by induction: show the statement holds for n = 1 (base case); assume it holds for n = k; use this to prove it holds for n = k + 1.

To prove a statement for all integers, consider separate cases (e.g., even and odd) and verify each case.

A counterexample is a single instance that disproves a universal statement; one counterexample is sufficient to disprove, but examples cannot prove a general statement.

When solving a problem, identify the unknown, translate the word problem into an equation or inequality, solve, and check the answer in the original context.

In problems involving rates, use the chain rule: dy/dt = dy/dx · dx/dt.

For optimisation problems, express the quantity to be optimised as a function of one variable, differentiate, set the derivative to zero, and verify the nature of the stationary point.