NSSCAS Mathematics (8227) key terms

A number written as a root that cannot be reduced to a rational value, such as √5 or 2 + √3. Surds are kept in exact form because their decimal expansions never terminate or repeat.

Rewriting a fraction so that no surd remains below the line, usually by multiplying numerator and denominator by a suitable surd or by the conjugate of a binomial denominator.

The expression b² − 4ac taken from the quadratic ax² + bx + c. Its sign shows how many real roots the equation has: positive gives two, zero gives one repeated root, negative gives none.

The rule that (x − a) is a factor of a polynomial exactly when substituting x = a gives zero. It converts factor questions into evaluations and helps find unknown coefficients.

The number of ways of choosing r items from n, written nCr or as the coefficient in a binomial expansion. In the expansion of (a + b)ⁿ it multiplies each term of the form aⁿ⁻ʳbʳ.

The power to which a chosen base must be raised to produce a given number: logₐN = x means aˣ = N. Logarithms turn multiplication into addition and solve equations with the unknown in an exponent.

The complete set of input values for which a function is defined. Square roots need non-negative inputs under them, denominators must not be zero, and logarithms need positive arguments.

The set of all output values a function actually produces as its input runs through the whole domain. It is found by combining the domain with the shape and turning points of the graph.

A function formed by applying one function and then another, written fg(x) for f(g(x)). Its domain contains only inputs that g accepts and whose outputs f then accepts.

The function that undoes another, written f⁻¹, so that f⁻¹(f(x)) = x for every x in the domain. It exists only for one-to-one functions, and its graph is the reflection in y = x.

The function |x| giving the distance of x from zero, so |x| equals x when x is non-negative and −x when x is negative. Its graph is V-shaped and it expresses distances and inequalities.

Changing a curve's position or shape by altering its equation: f(x) + a shifts it vertically, f(x + a) shifts it horizontally, af(x) stretches it vertically, and −f(x) reflects it in the x-axis.

A measure of how steeply a straight line rises, equal to the change in y divided by the change in x between two points. Parallel lines share a gradient; perpendicular gradients multiply to −1.

The line that cuts a given segment in half at right angles. Every point on it is equidistant from the two endpoints, which makes it the standard tool for locating a circle's centre.

The point exactly halfway between two given points, found by averaging their x-coordinates and averaging their y-coordinates separately. It lies on the perpendicular bisector of the segment joining them.

The angle subtended at a circle's centre by an arc equal in length to the radius. One full turn measures 2π radians, so π radians corresponds to 180 degrees.

The distance measured along a curved part of a circle. With the angle θ given in radians and radius r, the arc length is rθ.

The region of a circle enclosed by two radii and the arc between them. With angle θ in radians, its area is ½r²θ, a formula that only works in radian measure.

Half the difference between the greatest and least values of a periodic graph. For y = a sin x or y = a cos x the amplitude is the modulus of a.

The smallest positive interval in the input after which a periodic function repeats its values exactly. For sin kx and cos kx the period is 2π/k; for tan kx it is π/k.

A sequence in which each term after the first comes from adding a fixed constant. The nth term is a + (n − 1)d and the sum of n terms is ½n(2a + (n − 1)d).

A sequence in which each term is a fixed multiple of the term before it. The nth term is arⁿ⁻¹, and the terms shrink towards zero whenever the common ratio is between −1 and 1.

The constant multiplier r linking consecutive terms of a geometric progression, found by dividing any term by the previous one. Its size determines whether the corresponding series converges.

The limiting value approached by the partial sums of an infinite geometric series. It exists only when the common ratio satisfies |r| < 1, and then it equals a/(1 − r).

The gradient function of a curve, giving the instantaneous rate of change of y with respect to x. It is defined as the limit of a chord's gradient as the interval between the points shrinks to zero.

A point on a curve where the derivative equals zero, so the tangent there is horizontal. It may be a local maximum, a local minimum, or a point of inflection.

The result of differentiating a function twice, written d²y/dx². At a stationary point a negative value indicates a maximum, a positive value a minimum, and zero is inconclusive.

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

The rule for differentiating a function of a function: dy/dx equals dy/du multiplied by du/dx. It handles composite expressions such as (3x + 1)⁵ or e^(x²).

A function whose derivative is the one you started with. Any two antiderivatives of the same function differ only by a constant, so an indefinite integral carries a constant of integration.

The integral of a function between two limits, evaluated by subtracting the antiderivative's value at the lower limit from its value at the upper limit. It represents signed area between the curve and the axis.

The region between a graph and the x-axis over a stated interval, found by integrating the function between the two x-values. Parts below the axis contribute negative amounts.

The volume swept out when a region bounded by a curve is rotated one full turn about an axis. For rotation about the x-axis it is π times the integral of y² between the limits.

An arrangement of items in which the order matters. The number of ordered arrangements of r items chosen from n distinct items is n! divided by (n − r)!.

A selection of items in which order is irrelevant, so choosing A then B counts the same as B then A. The number of ways of choosing r from n distinct items is n!/(r!(n − r)!).

A variable that takes only separate, countable values, each carrying a probability. Its probability distribution must give non-negative values that add up to exactly one.

The mean value of a random variable over many trials, found by multiplying each possible value by its probability and adding the products. It is written E(X).

Describing events that cannot occur on the same trial, so their overlap has probability zero. For such events the probability that either happens is the sum of their separate probabilities.

Events whose outcomes do not influence one another, so the probability of both occurring equals the product of their individual probabilities. Knowing one happened gives no information about the other.

The normal distribution with mean zero and standard deviation one, used as the reference for tables. Any normal variable is converted to it by subtracting the mean and dividing by the standard deviation.

A vector drawn from a fixed origin to a point, recording that point's location. It lets points be handled algebraically, so distances and midpoints become vector calculations.

The product of two vectors defined as the product of their magnitudes times the cosine of the angle between them, giving a single number. It also equals the sum of corresponding component products, and is zero for perpendicular vectors.

A vector of length exactly one, usually pointing in some specified direction. Any non-zero vector becomes a unit vector when divided by its own magnitude.