GCE A/L Combined Mathematics key terms

For the quadratic ax² + bx + c, the quantity b² − 4ac. Its sign tells you whether the roots are real and distinct, real and equal, or a complex conjugate pair.

The values of x satisfying ax² + bx + c = 0, found by factorising or by the formula x = (−b ± √(b² − 4ac)) / 2a. Their sum is −b/a and their product is c/a.

A polynomial P(x) has (x − a) as a factor exactly when P(a) = 0. It converts the search for factors into evaluating the polynomial at suspected values.

When a polynomial P(x) is divided by (x − a), the remainder equals P(a). Repeated application with synthetic division allows full factorisation of higher-degree polynomials.

Rewriting a single rational expression as a sum of simpler fractions whose denominators are the factors of the original denominator. Used before integrating rational functions or expanding series.

For a complex number z = x + iy, the modulus |z| = √(x² + y²) is its distance from the origin, and the argument arg z is the angle its position vector makes with the positive real axis.

For z = x + iy, the conjugate is z̄ = x − iy, obtained by reflecting z in the real axis. Then z z̄ = |z|², which is real and non-negative.

Writing z as r(cos θ + i sin θ), where r is the modulus and θ the argument. It makes multiplication and division easy: moduli multiply or divide and arguments add or subtract.

The statement that [r(cos θ + i sin θ)]ⁿ = rⁿ(cos nθ + i sin nθ) for integer n. It gives powers and nth roots of complex numbers and underlies trigonometric multiple-angle identities.

The n distinct complex solutions of zⁿ = 1, namely cis(2kπ/n) for k = 0, 1, …, n − 1. They lie equally spaced on the unit circle and sum to zero.

Identities such as sin(A ± B) = sin A cos B ± cos A sin B and cos(A ± B) = cos A cos B ∓ sin A sin B. They express trigonometric functions of sums and differences of angles.

Special cases of the compound angle identities: sin 2A = 2 sin A cos A, cos 2A = cos²A − sin²A, and tan 2A = 2 tan A / (1 − tan²A). Used to simplify and to integrate.

An equation in trigonometric functions that holds for every value of the variable, such as sin²θ + cos²θ = 1. Proving one usually starts from one side and transforms it into the other.

The complete set of angles satisfying a trigonometric equation, expressed with an integer parameter. For example, sin θ = k gives θ = (−1)ⁿ arcsin k + nπ.

The single value returned by an inverse trigonometric function, chosen within a standard restricted range. For arcsin the range is [−π/2, π/2], for arccos [0, π], and for arctan (−π/2, π/2).

Writing a sin θ + b cos θ as R sin(θ + α), where R = √(a² + b²) and tan α = b/a. This reveals the maximum and minimum values and helps solve equations.

The measure of steepness m = (y₂ − y₁)/(x₂ − x₁) between two points. It equals the tangent of the angle the line makes with the positive x-axis.

Forms include y = mx + c, y − y₁ = m(x − x₁), and x/a + y/b = 1. Any non-vertical line can be written as ax + by + c = 0.

Two lines are perpendicular exactly when the product of their gradients is −1, so m₁m₂ = −1. A vertical line is perpendicular to a horizontal one.

The equation (x − h)² + (y − k)² = r², describing all points a distance r from centre (h, k). Expanding it gives x² + y² + 2gx + 2fy + c = 0.

For a conic, the ratio e of the distance from a point on it to the focus divided by its distance to the directrix. Circles have e = 0, ellipses 0 < e < 1, parabolas e = 1, hyperbolas e > 1.

A straight line touching the curve at a point and having the same gradient there. For a circle, the tangent at a point is perpendicular to the radius at that point.

Finding the derivative as the limit of (f(x + h) − f(x))/h as h tends to zero. It defines the gradient function and justifies all standard derivative rules.

The rule dy/dx = dy/du × du/dx for a function of a function. It lets you differentiate composite expressions such as sin(3x²) or (2x + 1)⁵.

If y = uv then dy/dx = u dv/dx + v du/dx. It differentiates products of two functions, for example x² sin x or eˣ ln x.

If y = u/v then dy/dx = (v du/dx − u dv/dx)/v². It differentiates one function divided by another, such as tan x = sin x / cos x.

Differentiating an equation relating x and y term by term, treating y as a function of x and applying the chain rule. Useful when y cannot easily be made the subject.

A point where dy/dx = 0. The second derivative decides its nature: negative means a maximum, positive a minimum, and zero requires further investigation.

The derivative of the derivative, written d²y/dx². It measures the rate of change of the gradient and classifies stationary points as maxima or minima.

Changing the variable of integration to simplify an integral, for example letting u = x² + 1 so that du = 2x dx. The limits must be converted when the integral is definite.

The rule ∫u dv = uv − ∫v du, derived from the product rule. It suits products such as x eˣ or ln x and may need repeating.

Splitting a rational integrand into simpler fractions, each of which integrates to a logarithm or a simple power. Requires the numerator degree to be lower than the denominator's.

The limit of a sum of areas, written ∫ₐᵇ f(x) dx, evaluated as F(b) − F(a) for an antiderivative F. It gives signed area, so regions below the x-axis count as negative.

The region between y = f(x), the x-axis, and two vertical lines, found by integrating f(x) between the limits. Areas between two curves are found by integrating the difference.

The solid formed by rotating a region about an axis. Its volume is π∫y² dx about the x-axis, or π∫x² dy about the y-axis, using the disc method.

For vectors a and b, the scalar a·b = |a||b| cos θ, where θ is the angle between them. It is zero for perpendicular vectors and gives the projection of one on the other.

The vector a × b with magnitude |a||b| sin θ, perpendicular to both a and b, with direction given by the right-hand rule. It is zero for parallel vectors.

A vector of length 1, often written with a hat. Dividing any non-zero vector by its magnitude gives a unit vector in the same direction.

The state in which the vector sum of all forces acting on a particle is zero, so it has no acceleration. Resolving forces into perpendicular directions gives simultaneous equations.

Replacing a force by its components along chosen perpendicular axes, usually horizontal and vertical or parallel and perpendicular to a plane. It converts vector problems into scalar equations.

The force opposing relative motion between surfaces in contact. On the point of slipping, limiting friction equals μR, where μ is the coefficient of friction and R the normal reaction.

Motion of a particle projected into the air under gravity alone. Horizontal velocity stays constant while vertical motion has constant acceleration g, so the path is a parabola.

The resultant force on a body equals the rate of change of its momentum; for constant mass this is F = ma. It links force, mass, and acceleration in dynamics problems.

The probability of event A given that B has occurred, written P(A|B) = P(A ∩ B)/P(B) where P(B) > 0. It restricts the sample space to outcomes in B.

Events A and B are independent when P(A ∩ B) = P(A)P(B), so knowing one occurred does not change the other's probability. Equivalently, P(A|B) = P(A).

Events that cannot occur together, so P(A ∩ B) = 0 and P(A ∪ B) = P(A) + P(B). They are not independent unless one has probability zero.

A rule for reversing conditional probabilities: P(A|B) = P(B|A)P(A)/P(B). It updates a prior probability in the light of new evidence.

The distribution of the number of successes in n independent trials, each with the same success probability p. Its mean is np and variance np(1 − p).

A continuous, symmetric, bell-shaped distribution fixed by its mean μ and standard deviation σ. Values are standardised as z = (x − μ)/σ for use with tables.

The square root of the variance, measuring the typical spread of values about the mean. It is in the same units as the data, unlike the variance.