EJU Mathematics Course 2 key terms

Any value on the continuous number line, including rationals such as 3/4 and irrationals such as √2; it can be positive, negative, or zero and is ordered by size.

The distance of a number from zero on the number line, written |x|; it equals x when x is nonnegative and −x when x is negative.

Rewriting a fraction so no radical remains in the denominator, typically by multiplying numerator and denominator by a conjugate such as √a − √b.

Writing a polynomial as a product of simpler polynomials, for example x² − 5x + 6 = (x − 2)(x − 3); it is the main tool for solving polynomial equations and simplifying fractions.

A quotient of two polynomials, defined only where the denominator is nonzero; it is simplified by factoring and cancelling common factors.

A function of the form y = ax² + bx + c with a ≠ 0, whose graph is a parabola opening upward when a > 0 and downward when a < 0.

The turning point of a parabola, located at x = −b/(2a) for y = ax² + bx + c; it gives the minimum value when a > 0 and the maximum when a < 0.

Rewriting ax² + bx + c in the form a(x − p)² + q, which reveals the vertex (p, q) and the range of the quadratic function at once.

The quantity D = b² − 4ac for ax² + bx + c = 0; D > 0 gives two distinct real roots, D = 0 one repeated root, and D < 0 no real roots.

An inequality such as ax² + bx + c > 0, solved by finding the real roots of the associated equation and testing the sign of the parabola on each interval between them.

Union, intersection, and complement combine sets: A ∪ B contains elements in either, A ∩ B those in both, and the complement of A contains elements of the universal set not in A.

A set A is a subset of B when every element of A also belongs to B, written A ⊆ B; the empty set is a subset of every set.

Rules stating that the complement of a union is the intersection of the complements, and the complement of an intersection is the union of the complements.

The statement obtained from "if P then Q" by negating and swapping, giving "if not Q then not P"; it is logically equivalent to the original implication.

The statement formed by swapping hypothesis and conclusion of "if P then Q", giving "if Q then P"; it is not automatically equivalent to the original.

P is necessary and sufficient for Q when each implies the other, so P holds exactly when Q holds; this biconditional is written P ⇔ Q.

An ordered arrangement of objects; the number of ways to choose and order r objects from n distinct ones is nPr = n!/(n − r)!.

A selection of objects where order does not matter; choosing r from n distinct objects can be done in nCr = n!/(r!(n − r)!) ways.

The number nCr appearing in the expansion of (a + b)ⁿ; the term containing aⁿ⁻ʳbʳ has coefficient nCr, and the coefficients form Pascal's triangle.

The identity (a + b)ⁿ = Σ nCr aⁿ⁻ʳbʳ for r from 0 to n, used to expand powers of binomials and to find a specific term without full expansion.

When all outcomes are equally likely, the probability of an event equals the number of favourable outcomes divided by the total number of outcomes.

The event that a given event does not occur; its probability is 1 minus the probability of the event, which often simplifies "at least one" problems.

Two events whose occurrence does not affect each other, so the probability of both equals the product of their individual probabilities.

Events where the occurrence of one changes the probability of the other, as when drawing cards without replacement from a deck.

The probability that event A occurs given that B has occurred, written P(A|B) and computed as P(A ∩ B)/P(B) when P(B) > 0.

The probability-weighted average of a random variable, found by multiplying each possible value by its probability and summing the results.

A numerical quantity whose value depends on the outcome of a random trial, such as the number of heads in three coin tosses.

A repeated division procedure for finding the greatest common divisor of two integers: replace the pair by the divisor and the remainder until the remainder is zero.

The largest positive integer dividing both of two given integers; it can be expressed as an integer combination of them, which underpins solving linear equations in integers.

An equation such as ax + by = c to be solved in integers; it has solutions exactly when the greatest common divisor of a and b divides c.

Writing an integer as a sum of powers of a base b with digits from 0 to b − 1, for example 1011 in base 2 equals 11 in base 10.

For a point P and a circle, the product PA·PB of the distances along any secant through P is constant; it equals the square of the tangent length when P is outside.

An angle inscribed in a circle equals half the central angle standing on the same arc, so angles subtending the same arc are equal.

A quadrilateral whose four vertices lie on one circle; its opposite angles sum to 180°, which is a standard test for concyclicity.

Triangles with equal corresponding angles and proportional corresponding sides; their areas are in the ratio of the squares of corresponding lengths.

In a triangle, a/sin A = b/sin B = c/sin C = 2R, and c² = a² + b² − 2ab cos C; these relate sides, angles, and the circumradius.

An equation true for all angles, such as sin²θ + cos²θ = 1 or the double-angle formulas; identities are used to simplify expressions and solve equations.

An equation involving trigonometric functions of an unknown angle, solved by reducing to a basic form and listing all solutions in the required interval using periodicity.

The area of a triangle equals half the product of two sides and the sine of the included angle, S = ½ab sin C, and also equals abc/(4R).

The exponent to which a base must be raised to produce a given number: log_a b = x means aˣ = b, defined for a > 0, a ≠ 1, and b > 0.

Rules turning 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.

The formula log_a b = log_c b / log_c a, allowing logarithms in any base to be rewritten using common logarithms or natural logarithms.

The instantaneous rate of change of a function, defined as the limit of the difference quotient; geometrically it is the slope of the tangent line at a point.

The straight line touching a curve at a point with slope equal to the derivative there; its equation at x = a is y − f(a) = f′(a)(x − a).

Whether a function increases or decreases on an interval, decided by the sign of its derivative: positive means increasing, negative means decreasing.

A local maximum or minimum of a function, found where the derivative is zero or undefined and changes sign; endpoints of the interval must also be checked.

The limit of a sum of signed rectangle areas, written ∫ from a to b of f(x) dx; it equals F(b) − F(a) for any antiderivative F of f.

The area enclosed by two graphs, computed as the definite integral of the upper function minus the lower function over the interval between their intersection points.

A sequence in which each term after the first is obtained by adding a fixed difference d; the nth term is a + (n − 1)d and the sum of n terms is n(a + l)/2.

A sequence in which each term is obtained by multiplying the previous term by a fixed ratio r; the nth term is a·rⁿ⁻¹ and the sum of n terms is a(rⁿ − 1)/(r − 1) for r ≠ 1.

A rule expressing each term of a sequence in terms of earlier terms, such as aₙ₊₁ = 2aₙ + 1; solving it means finding an explicit formula for the nth term.

A proof technique for statements about all positive integers: prove the base case, assume the statement for n, then derive it for n + 1.

A quantity with both magnitude and direction, represented by an arrow or by components; vectors add componentwise and can be multiplied by scalars.

For vectors a and b, the scalar a·b = |a||b| cos θ; it is zero exactly when the vectors are perpendicular and gives the angle between them.

The vector from a fixed origin to a point, used to describe points and to express other vectors as differences of position vectors.

The line through point A with direction vector d is written p = a + t d for real t; two lines are parallel when their direction vectors are scalar multiples.