BGCSE Additional Mathematics 4037 flashcards

33 cards. Read the term and the definition here. The deck at the top of this page is optional.

Remainder Theorem

When a polynomial f(x) is divided by (x − a), the remainder equals f(a). This allows finding remainders without long division.

Factor Theorem

(x − a) is a factor of polynomial f(x) if and only if f(a) = 0. It is used to factorise polynomials and find roots.

Discriminant

For ax² + bx + c = 0, Δ = b² − 4ac. If Δ > 0 two distinct real roots; Δ = 0 one repeated root; Δ < 0 no real roots.

Inverse Function

f⁻¹ exists only if f is one-to-one. It reverses f: f⁻¹(f(x)) = x. The graph of f⁻¹ is the reflection of y = f(x) in y = x.

Composite Function

(f ∘ g)(x) = f(g(x)). The domain consists of x in the domain of g for which g(x) is in the domain of f.

Completing the Square

ax² + bx + c = a(x + b/(2a))² + (c − b²/(4a)). This form reveals the vertex of the parabola and helps solve quadratic equations.

Modulus Inequality

|ax + b| < c (c > 0) is equivalent to −c < ax + b < c. |ax + b| > c splits into ax + b > c or ax + b < −c.

Graph Transformation

y = f(x) + a shifts vertically by a; y = f(x + a) shifts horizontally by −a; y = af(x) stretches vertically by factor a; y = f(ax) stretches horizontally by 1/a.

Arithmetic Progression

A sequence with constant common difference d. nth term uₙ = a + (n − 1)d; sum Sₙ = n/2 [2a + (n − 1)d].

Geometric Progression

A sequence with constant ratio r. nth term uₙ = arⁿ⁻¹; sum Sₙ = a(1 − rⁿ)/(1 − r); sum to infinity S∞ = a/(1 − r) for |r| < 1.

Binomial Theorem

(a + b)ⁿ = Σ_{r=0}^{n} C(n, r) aⁿ⁻ʳ bʳ, where C(n, r) = n!/[r!(n − r)!]. The general term is T_{r+1} = C(n, r) aⁿ⁻ʳ bʳ.

Sum of First n Natural Numbers

1 + 2 + 3 + … + n = n(n + 1)/2. This formula is useful in series summation and proof by induction.

Distance Formula

The distance between points (x₁, y₁) and (x₂, y₂) is √[(x₂ − x₁)² + (y₂ − y₁)²]. It follows from Pythagoras' theorem.

Perpendicular Lines

Two lines with gradients m₁ and m₂ are perpendicular if m₁m₂ = −1. The gradient of a normal is the negative reciprocal of the tangent's gradient.

Circle Equation

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

Sine Rule

In any triangle, a/sin A = b/sin B = c/sin C, where a, b, c are sides opposite angles A, B, C. Used when given two angles and a side or two sides and a non-included angle.

Cosine Rule

In any triangle, a² = b² + c² − 2bc cos A. Used to find a side given two sides and the included angle, or an angle given three sides.

Trigonometric Identity

sin²θ + cos²θ = 1; tan θ = sin θ/cos θ; 1 + tan²θ = sec²θ; 1 + cot²θ = cosec²θ. These identities simplify expressions and solve equations.

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).

Chain Rule

If y is a function of u and u is a function of x, then dy/dx = dy/du · du/dx. Essential for differentiating composite functions.

Product Rule

If y = uv, then dy/dx = u dv/dx + v du/dx. Used to differentiate products of functions.

Quotient Rule

If y = u/v, then dy/dx = (v du/dx − u dv/dx)/v². Used to differentiate quotients of functions.

Second Derivative Test

At a stationary point where dy/dx = 0, if d²y/dx² > 0 it is a local minimum; if d²y/dx² < 0 it is a local maximum.

Definite Integral

∫_a^b f(x) dx = F(b) − F(a), where F′(x) = f(x). It represents the signed area under the curve from x = a to x = b.

Area Between Curves

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

Volume of Revolution

When the region under y = f(x) from x = a to x = b is rotated about the x-axis, the volume is V = π ∫_a^b y² dx.

Integration by Parts

∫ u dv = uv − ∫ v du. Choose u as the function that simplifies when differentiated, often using the LIATE order.

Scalar Product

a · b = |a||b| cos θ = x₁x₂ + y₁y₂. If a · b = 0, the vectors are perpendicular. Used to find angles between vectors.

Inverse of 2×2 Matrix

For A = [[a, b], [c, d]], A⁻¹ = (1/(ad − bc)) [[d, −b], [−c, a]], provided det A = ad − bc ≠ 0.

Logarithm Laws

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. Used to evaluate logarithms with any base.

Exponential Growth and 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 Induction

To prove a statement for all positive integers: show it holds for n = 1; assume it holds for n = k; prove it holds for n = k + 1.