NEB Class 12 Mathematics key terms

A number written as a + bi, where a and b are real numbers and i is the imaginary unit with i² = −1. The real part is a and the imaginary part is b.

The conjugate of a + bi is a − bi. Multiplying a complex number by its conjugate gives the real number a² + b², which is used when dividing complex numbers and when finding moduli.

The modulus |a + bi| equals the square root of a² + b². It is the distance from the origin to the point that represents the complex number in the Argand plane.

For any real angle θ and any integer n, (cos θ + i sin θ)^n = cos nθ + i sin nθ. It turns powers and roots of complex numbers in polar form into simple multiples of angles.

An ordered arrangement of some or all members of a set. Because order matters, ABC and CBA count as different permutations; the number of r-permutations of n objects is n!/(n − r)!.

A selection of objects in which order does not matter, so choosing A then B is the same as choosing B then A. The number of r-combinations of n objects is C(n, r) = n!/(r!(n − r)!).

A formula that expands (a + b)^n, for a positive integer n, as a sum of n + 1 terms built from binomial coefficients, with powers of a decreasing and powers of b increasing.

In the expansion of (a + b)^n, the term containing b^r is T(r+1) = C(n, r)·a^(n−r)·b^r. It lets you find one particular term without writing out the whole expansion.

The angle subtended at the centre of a circle by an arc whose length equals the radius. One radian is 180/π degrees, about 57.3°, and π radians equals 180°.

The single angle chosen from the infinitely many angles satisfying an inverse trigonometric relation. It lies in a restricted range, such as −π/2 to π/2 for sin⁻¹, so the inverse is a function.

An expression listing every angle that satisfies a trigonometric equation, using an integer n to cover all repetitions, for example θ = nπ + (−1)^n α when sin θ = sin α.

An identity that expresses a trigonometric ratio of the sum or difference of two angles through ratios of the separate angles, such as sin(A + B) = sin A cos B + cos A sin B.

An identity that is true only when the angles satisfy a stated relation, usually A + B + C = π for a triangle, which allows results such as sin 2A + sin 2B + sin 2C = 4 sin A sin B sin C.

The smallest positive value p for which f(x + p) = f(x) for every x. Sine and cosine have period 2π, while tangent and cotangent have period π.

The gradient of a line, equal to the tangent of its angle of inclination and to (y₂ − y₁)/(x₂ − x₁) for two points on it. Parallel lines have equal slopes.

The shortest distance from a point to a line, measured along a segment meeting the line at 90°. For ax + by + c = 0 and the point (x₁, y₁) it equals |ax₁ + by₁ + c|/√(a² + b²).

A curve produced by cutting a double cone with a plane. Depending on the angle of the cut it is a circle, parabola, ellipse or hyperbola, each described by a second-degree equation in x and y.

A number e that measures how far a conic departs from a circle. An ellipse has e < 1, a parabola has e = 1, and a hyperbola has e > 1.

The set of points equidistant from a fixed point, the focus, and a fixed line, the directrix. The standard form y² = 4ax has vertex at the origin, focus (a, 0) and directrix x = −a.

The set of points whose distances from two fixed foci add up to a constant. Standard form x²/a² + y²/b² = 1 with a > b has foci (±ae, 0) and eccentricity e = √(1 − b²/a²).

The set of points for which the difference of the distances from two fixed foci is constant. Standard form x²/a² − y²/b² = 1 opens left and right and has eccentricity e = √(1 + b²/a²).

A vector of length one, used to state direction. Any nonzero vector a becomes a unit vector when divided by its magnitude, giving â = a/|a|.

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

For vectors a and b in space, a × b is a vector perpendicular to both, with magnitude |a||b| sin θ and direction fixed by the right-hand rule. Its magnitude equals the area of the parallelogram they span.

The number a·(b × c) formed from three vectors. Its absolute value is the volume of the parallelepiped with those vectors as edges, and it is zero exactly when the vectors are coplanar.

The cosines of the angles that a line or vector makes with the positive x, y and z axes, usually written l, m and n. They always satisfy l² + m² + n² = 1.

Vectors that lie in, or are parallel to, one common plane. Three vectors are coplanar precisely when their scalar triple product is zero, which makes them linearly dependent.

The value that f(x) approaches as x approaches a chosen point, whether or not f is defined at that point. Limits underpin the definitions of continuity and the derivative.

A function is continuous at a point when its limit there exists, the function is defined there, and those two values are equal. Graphically the curve has no break, jump or hole at that point.

The instantaneous rate of change of a function, defined as the limit of [f(x + h) − f(x)]/h as h tends to zero. Geometrically it is the slope of the tangent to the curve.

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

The integral of a function between two limits a and b, written ∫ₐᵇ f(x) dx. It is evaluated as F(b) − F(a), where F is any antiderivative of f.

A technique derived from the product rule, ∫u dv = uv − ∫v du. It suits integrands that are products, with u chosen (often by the LIATE order) so the remaining integral is simpler.

The region bounded by the curve y = f(x), the x-axis and the vertical lines x = a and x = b, found as the definite integral of f from a to b. Parts below the axis count as negative.

The square root of the mean of the squared deviations from the mean. It measures spread in the same units as the data, and a small value means observations cluster near the mean.

A number r between −1 and 1 that measures the strength and direction of the linear relationship between two variables. A value near zero indicates no linear relationship.

The straight line of best fit for paired data, obtained by the least-squares method and used to predict one variable from the other. Its slope equals r times the ratio of the two standard deviations.

The probability that event A occurs given that event B has already occurred, written P(A|B) = P(A ∩ B)/P(B) with P(B) > 0. It effectively shrinks the sample space to B.

A rule for reversing a conditional probability: P(A|B) = P(B|A)·P(A)/P(B). It updates a prior probability in the light of new evidence and is often applied to a partition of outcomes.

The probability distribution of the number of successes in n independent trials that each have the same success probability p. Its probabilities are P(X = r) = C(n, r)·p^r·(1 − p)^(n−r).