MATSEC AM Pure Mathematics key terms

For a quadratic ax² + bx + c, the quantity b² − 4ac. Its sign tells you how many real roots exist: positive gives two, zero gives one repeated root, negative gives none.

Rewriting a quadratic in the form a(x + p)² + q. This reveals the vertex of its graph and can be used to solve the equation or prove the quadratic is always positive or negative.

The turning point of a parabola, where the curve reaches its maximum or minimum value. For y = a(x + p)² + q the vertex is at (−p, q).

The vertical line through the vertex of a parabola about which the curve is a mirror image. For y = ax² + bx + c it has equation x = −b/(2a).

Expressing a single rational expression as a sum of simpler fractions with linear or quadratic denominators. Useful for integration and for expanding expressions as series.

When a polynomial f(x) is divided by (x − a), the remainder equals f(a). Consequently (x − a) is a factor exactly when f(a) = 0.

A polynomial f(x) has (x − a) as a factor if and only if f(a) = 0. It converts the search for factors into evaluating the polynomial at candidate values.

An expression made by adding terms of the form axⁿ where n is a non-negative whole number. Its degree is the highest power of x present.

A sequence in which each term differs from the previous one by a fixed constant, called the common difference. The nth term is a + (n − 1)d.

A sequence in which each term is obtained by multiplying the previous term by a fixed constant, the common ratio. The nth term is ar^(n−1).

The limiting total of a geometric series whose common ratio r satisfies |r| < 1, given by a/(1 − r). If |r| ≥ 1 the series does not converge to a finite total.

Describing a sequence or series whose terms approach a definite finite limit, or whose partial sums approach a finite total, as more terms are taken.

Writing (a + b)ⁿ as a sum of terms using binomial coefficients. For positive integer n it terminates; for other n it gives an infinite series valid for restricted values.

A technique for summing a series by writing each term as a difference of two consecutive values, so that most terms cancel in the partial sum.

The steepness of a line, measured as the change in y divided by the change in x between two points. Parallel lines share the same gradient.

Two lines meeting at right angles. Their gradients m₁ and m₂ satisfy m₁m₂ = −1, provided neither line is vertical.

The equation of a circle with centre (a, b) and radius r, written (x − a)² + (y − b)² = r². Expanding it gives a form with x² + y² plus linear terms.

Describing a curve by giving x and y separately as functions of a third variable, such as x = t², y = 2t. The variable is called the parameter.

A unit of angle measure in which one radian is the angle subtended at the centre of a circle by an arc equal in length to the radius. A full turn is 2π radians.

The distance along a curved section of a circle. For radius r and angle θ in radians it equals rθ, and the sector area is ½r²θ.

The relation sin²θ + cos²θ = 1, true for every angle. Dividing it by cos²θ or sin²θ produces related identities involving tan²θ.

Identities expressing trigonometric functions of 2θ in terms of θ, for example sin 2θ = 2 sinθ cosθ and cos 2θ = cos²θ − sin²θ.

Writing a sinθ + b cosθ as a single sine or cosine such as R sin(θ + α), where R = √(a² + b²). Useful for solving equations and finding maximum or minimum values.

The maximum displacement of a periodic function from its central value. For y = A sin x the amplitude is A; for y = a sin x + b cos x it is √(a² + b²).

The power to which a base must be raised to produce a given number. The logarithm of N to base a is the x satisfying aˣ = N.

A function of the form aˣ in which the variable appears in the exponent. Its graph passes through (0, 1) and has the x-axis as an asymptote.

Rules that turn products into sums, quotients into differences and powers into multiples: log(xy) = log x + log y, log(x/y) = log x − log y, log(xⁿ) = n log x.

Rewriting a logarithm in another base using log_a N = log_b N ÷ log_b a. It lets you evaluate logarithms on a calculator that offers only one or two bases.

A line that a curve approaches ever more closely as it extends, without necessarily meeting it. Exponential graphs have horizontal asymptotes and rational graphs may have vertical ones.

The rule for differentiating a function of a function: dy/dx = dy/du × du/dx. It is the standard tool for differentiating composite expressions such as (3x + 1)⁵.

The rule for differentiating a product: if y = uv then dy/dx = u(dv/dx) + v(du/dx). Used when two functions of x are multiplied together.

The rule for differentiating a fraction: if y = u/v then dy/dx = (v du/dx − u dv/dx) ÷ v². It applies when both numerator and denominator depend on x.

A point on a curve where the gradient is zero, so dy/dx = 0. It may be a maximum, a minimum or a point of inflection.

The derivative of the derivative, written d²y/dx². Its sign at a stationary point distinguishes a maximum (negative) from a minimum (positive).

A straight line touching a curve at a point and having the same gradient as the curve there. Its gradient equals the derivative evaluated at that point.

The straight line through a point on a curve that is perpendicular to the tangent there. Its gradient is the negative reciprocal of the curve's gradient at that point.

A method for evaluating integrals by replacing the variable with a new one, changing dx accordingly. It reverses the chain rule and often simplifies composite integrands.

A technique based on reversing the product rule: ∫u dv = uv − ∫v du. It is used for products such as x·eˣ or x·ln x.

An integral evaluated between two limits, giving a number rather than a function. Geometrically it represents the signed area between the curve and the x-axis over that interval.

The region between a graph and the x-axis, found by integrating the function between the relevant limits. Parts below the axis count as negative unless handled separately.

A numerical method for estimating a definite integral by splitting the area into strips and treating each as a trapezium. More strips generally improve the estimate.

A numerical integration formula that fits parabolas through groups of three ordinates. It usually gives a more accurate estimate than the trapezium rule for the same number of strips.

For a complex number z = a + bi, the distance from the origin in the Argand diagram, written |z| and equal to √(a² + b²).

The angle a complex number makes with the positive real axis in the Argand diagram, measured anticlockwise. It is usually given in radians between −π and π.

The number obtained by changing the sign of the imaginary part, so a + bi becomes a − bi. Multiplying a number by its conjugate gives a real result equal to the square of its modulus.

Writing a complex number as r(cos θ + i sin θ), where r is the modulus and θ the argument. It makes multiplication and division of complex numbers straightforward.

For a polynomial with real coefficients, any non-real complex root occurs together with its conjugate. The coefficients relate to the sums and products of the roots.

A way of combining two vectors to give a number: a·b = |a||b|cos θ. It is used to find the angle between vectors and to test whether they are perpendicular.

Writing a line as r = a + λb, where a is the position vector of a point on the line and b is a direction vector. Different values of λ give different points.

Two straight lines in three dimensions that are not parallel and do not meet. Unlike lines in a plane, they have no point of intersection.

An iterative method for solving f(x) = 0 using xₙ₊₁ = xₙ − f(xₙ)/f′(xₙ). A good starting value close to the root helps the sequence converge.

Generating successive approximations to a solution by repeatedly applying a formula, starting from an initial estimate. Convergence depends on the behaviour of the formula near the root.

The probability of an event given that another event has occurred, written P(A|B) and equal to P(A ∩ B) ÷ P(B) when P(B) is not zero.

Events whose occurrence does not affect each other's probability. For such events P(A ∩ B) = P(A) × P(B).

Events that cannot happen at the same time, so P(A ∩ B) = 0. For these, the probability that either occurs is the sum of their probabilities.

The distribution of the number of successes in n independent trials, each with the same probability p of success. It is used when outcomes fall into two categories.

A continuous, symmetric bell-shaped distribution defined by its mean and variance. Probabilities are found by converting to the standard normal variable z and using tables.

A measure of how spread out data are, equal to the square root of the variance. A small value means values cluster near the mean.