Assam HSLC Class 10 Elective Advanced Mathematics key terms

For positive integers a and b, there exist unique whole numbers q and r with a = bq + r and 0 ≤ r < b. It underlies divisibility tests and the Euclidean algorithm.

Every integer greater than 1 can be written as a product of primes, and that prime factorisation is unique apart from the order of the factors.

The highest common factor of two or more numbers is the largest positive integer dividing each of them exactly. It equals the product of the lowest powers of common primes.

The least common multiple of two or more numbers is the smallest positive integer that each number divides exactly. For two numbers, HCF × LCM equals their product.

Writing a composite number as a product of prime numbers only, usually in ascending order with powers, for example 360 = 2³ × 3² × 5.

A real number that cannot be expressed as a ratio of two integers. Its decimal expansion is non-terminating and non-recurring, as with √2, √3 and π.

A number expressible as p/q where p and q are integers and q is not zero. Its decimal expansion either terminates or recurs in a repeating block.

A decimal expansion that ends after finitely many digits. A rational number in lowest terms has one exactly when its denominator has no prime factors other than 2 and 5.

A decimal in which a digit or block of digits repeats endlessly, shown by a bar over the repeating part. Every recurring decimal represents a rational number.

The highest power of the variable appearing with a non-zero coefficient in a polynomial. It fixes the maximum number of real zeros the polynomial can have.

A value of the variable at which the polynomial evaluates to zero. Graphically these are the points where the curve cuts or touches the x-axis.

When a polynomial p(x) is divided by (x − a), the remainder equals p(a). This gives a quick way to find remainders without long division.

The linear expression (x − a) is a factor of p(x) exactly when p(a) = 0. It converts factor questions into evaluation at a point.

For the quadratic ax² + bx + c = 0, the quantity b² − 4ac. Its sign tells whether the roots are real and distinct, real and equal, or not real.

The classification of a quadratic's solutions from its discriminant: two distinct real roots when positive, equal real roots when zero, and no real roots when negative.

The roots of ax² + bx + c = 0 with a ≠ 0 are x = (−b ± √(b² − 4ac)) / 2a. It solves any quadratic, factorable or not.

Rewriting a quadratic in the form a(x + h)² + k by adding and subtracting the square of half the coefficient of x. It reveals the vertex and solves the equation.

A sequence in which each term after the first is obtained by adding a fixed constant, the common difference, to the previous term. Its nth term is a + (n − 1)d.

The constant amount added to each term of an arithmetic progression to obtain the next term. It equals the difference of any term and the one before it.

For first term a and common difference d, the total is (n/2)[2a + (n − 1)d], or (n/2)(first term + last term) when the last term is known.

A sequence in which each term after the first is obtained by multiplying the previous term by a fixed non-zero constant. Its nth term is ar^(n−1).

The fixed multiplier linking consecutive terms of a geometric progression. It is found by dividing any term by the term immediately before it.

For first term a, common ratio r ≠ 1 and n terms, the total is a(rⁿ − 1)/(r − 1). When r = 1 the sum is simply na.

The average of a set of values, found by dividing their sum by how many there are. Between two numbers it equals half their sum and lies midway between them.

For two positive numbers, the positive square root of their product. It is the middle term inserted between them so the three values form a geometric progression.

The tangent of the angle a line makes with the positive x-axis, equal to the change in y divided by the change in x between two points on it.

An algebraic relation satisfied by exactly the points of a line, commonly written as y = mx + c with slope m and y-intercept c, or in general form ax + by + c = 0.

Gives the coordinates of the point dividing the join of two points internally in a given ratio m : n, using weighted averages of the x and y coordinates.

The distance between points (x₁, y₁) and (x₂, y₂) is √[(x₂ − x₁)² + (y₂ − y₁)²]. It follows from Pythagoras' theorem applied to the coordinate differences.

A ratio of two sides of a right triangle determined by an acute angle, namely sine, cosine, tangent and their reciprocals cosecant, secant and cotangent.

An equation in trigonometric ratios that holds for every angle where both sides are defined, such as sin²θ + cos²θ = 1. Identities are used to simplify expressions.

The angle measured upward from the horizontal to the line of sight when an observer looks at an object above eye level. It sets up right triangles in height-and-distance problems.

The angle measured downward from the horizontal to the line of sight when an observer looks at an object below eye level. It equals the angle of elevation from the object.

Triangles with equal corresponding angles, whose corresponding sides are therefore in the same ratio. Their areas are in the ratio of the squares of corresponding sides.

A line drawn parallel to one side of a triangle divides the other two sides in the same ratio. Its converse is used to prove that two lines are parallel.

In a right triangle the square on the hypotenuse equals the sum of the squares on the other two sides. Its converse identifies a right angle from side lengths.

A line touching a circle at exactly one point, the point of contact. It is perpendicular to the radius drawn to that point.

The angle subtended at a point on the circle by a diameter is a right angle. More generally, angles in the same segment of a circle are equal.

A quadrilateral whose four vertices lie on one circle. Its opposite angles are supplementary, each pair adding to 180 degrees.

The average of a frequency distribution, computed as Σfx / Σf using class midpoints, or by the assumed-mean and step-deviation shortcuts on grouped data.

The middle value of data arranged in order. For grouped data it is found from the cumulative frequency table using the median class and the formula with class boundaries.

The value or class that occurs most frequently. For grouped data the modal class is located from the highest frequency and refined by a formula using neighbouring frequencies.

The positive square root of the variance, measuring how far values spread from the mean. A small value means the data cluster tightly around the average.

The mean of the squared deviations of values from their arithmetic mean. It measures dispersion and its square root is the standard deviation.

A running total of frequencies up to and including a given class. Plotting it gives an ogive, from which the median can be read graphically.

The set of all possible outcomes of a random experiment, such as the six faces when a die is thrown once. Events are subsets of this set.

For equally likely outcomes, the number of outcomes favourable to the event divided by the total number of outcomes in the sample space. It lies between 0 and 1.

The event that a given event does not occur, consisting of all outcomes in the sample space not in it. Their probabilities add to 1.

A well-defined collection of distinct objects, called its elements. Sets are described by listing elements or by a rule, and are written with braces.

A set A is a subset of B when every element of A is also an element of B. Every set is a subset of itself and the empty set is a subset of every set.

The set of all elements belonging to at least one of the given sets. Elements common to both appear only once in the result.

The set of elements common to all the given sets. It is empty when the sets share no element at all.

The set containing all elements under consideration in a particular discussion. Every other set in that discussion is a subset of it.

The set of all elements of the universal set that do not belong to the given set. It is used with union and intersection in De Morgan-type simplifications.

A rule linking elements of one set to elements of another, formally a subset of their Cartesian product. Its set of first elements is the domain.

A relation in which each element of the domain is paired with exactly one element of the codomain. Graphs of functions pass the vertical line test.

The set of all ordered pairs (a, b) with a taken from the first set and b from the second. Its size is the product of the two set sizes.

The set of all actual output values a function produces from its domain. It may be a proper subset of the codomain.