BSEB Matric Adv Math (Opt) 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 the Euclidean algorithm for highest common factors.

Every integer greater than 1 can be written as a product of primes in exactly one way apart from the order of the factors. This unique factorisation justifies HCF and LCM methods.

The highest common factor of two or more numbers is the largest number 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 number divisible by each of them. For two numbers, HCF × LCM equals their product.

Writing a composite number as a product of prime numbers only, usually in ascending order or index form. It is the standard tool for finding HCF, LCM and testing perfect squares.

A real number that cannot be expressed as a ratio of two integers. Its decimal expansion is non-terminating and non-recurring, for example the square root of 2 or pi.

A number expressible as p/q where p and q are integers and q is not zero. Its decimal expansion either terminates or repeats a block of digits forever.

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.

The highest power of the variable appearing with a non-zero coefficient in a polynomial. It fixes the maximum number of zeroes and shapes the graph's behaviour.

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

A polynomial of degree two, written ax² + bx + c with a not zero. Its graph is a parabola opening upward when a is positive and downward when a is negative.

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

The rule giving roots of ax² + bx + c = 0 as x = (−b ± √(b² − 4ac)) / 2a. It solves every quadratic with real coefficients once the discriminant is known.

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

For ax² + bx + c = 0, the roots add to −b/a and multiply to c/a. These relations let you build equations or find symmetric expressions without solving.

The constant amount added to each term to obtain the next in an arithmetic progression. It may be positive, negative or zero and equals any term minus the previous one.

For an arithmetic progression with first term a and common difference d, the nth term is a + (n − 1)d. It allows any term to be found without listing earlier ones.

The total of the first n terms of an arithmetic progression, given by (n/2)[2a + (n − 1)d] or (n/2)(first term + last term).

The average of two numbers, equal to half their sum. In an arithmetic progression any term is the arithmetic mean of its immediate neighbours.

A sequence in which each term after the first is obtained by multiplying the previous term by a fixed non-zero ratio. Terms grow or shrink multiplicatively rather than additively.

The straight-line distance between points (x₁, y₁) and (x₂, y₂), equal to the square root of (x₂ − x₁)² + (y₂ − y₁)². It follows from the Pythagoras theorem.

Gives the coordinates of a point dividing the join of two points internally in a given ratio m:n as ((mx₂ + nx₁)/(m + n), (my₂ + ny₁)/(m + n)).

The midpoint of the segment joining (x₁, y₁) and (x₂, y₂) has coordinates ((x₁ + x₂)/2, (y₁ + y₂)/2). It is the section formula with ratio 1:1.

The measure of steepness of a line, equal to the rise divided by the run, or (y₂ − y₁)/(x₂ − x₁) for two points on it. Parallel lines share a slope.

Three or more points lying on one straight line. They are collinear exactly when the slopes between successive pairs are equal, or the triangle they form has zero area.

For vertices (x₁, y₁), (x₂, y₂), (x₃, y₃) the area is half the absolute value of x₁(y₂ − y₃) + x₂(y₃ − y₁) + x₃(y₁ − y₂).

A ratio of two sides of a right triangle depending on an acute angle, such as sine, cosine, tangent, cosecant, secant or cotangent. Each has a fixed value for a given angle.

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 and prove expressions.

Two angles adding to 90 degrees. Their trigonometric ratios interlock: sin(90° − θ) = cos θ, tan(90° − θ) = cot θ, and similarly for the other pairs.

The angle measured upward from the horizontal to the line of sight when an observer looks at an object higher than eye level. It appears 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.

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

A line that cuts a circle at two distinct points. The part of the secant inside the circle is a chord.

The two tangent segments drawn from a single external point to a circle are equal in length. This fact supports many circle proofs and construction problems.

A segment joining two points on a circle. A perpendicular from the centre to a chord bisects it, and equal chords are equidistant from the centre.

A quadrilateral whose four vertices lie on one circle. Its opposite angles are supplementary, and each exterior angle equals the interior opposite angle.

Also called Thales' theorem: a line drawn parallel to one side of a triangle divides the other two sides in the same ratio. Its converse tests whether a line is parallel.

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

Triangles identical in shape and size, matched by criteria such as SSS, SAS, ASA, AAS or RHS. Corresponding parts of congruent triangles are equal.

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

The area of the curved part only, equal to 2πrh for radius r and height h. It excludes the two circular ends.

The curved area plus both circular ends, equal to 2πr(h + r). It measures the material needed to make a closed cylindrical container.

The space enclosed by a cylinder, equal to πr²h, the base area multiplied by height. Capacity problems usually require this measure in cubic units.

The distance from the apex of a cone or pyramid to a point on the rim of the base. For a cone it equals the square root of r² + h².

The lateral area of a right circular cone, equal to πrl where l is the slant height. Adding the base circle gives the total surface area.

The part of a cone or pyramid left after cutting off the top with a plane parallel to the base. Its volume uses the radii of both parallel circular faces and the height.

The average of data arranged in classes, computed by the direct method, assumed-mean method or step-deviation method using class marks and frequencies.

The middle value found from the cumulative frequency table using the formula l + [(n/2 − cf)/f] × h, where l is the lower limit of the median class.

The value occurring most often, estimated for grouped data by l + [(f₁ − f₀)/(2f₁ − f₀ − f₂)] × h using the modal class frequencies.

The midpoint of a class interval, found by averaging its lower and upper limits. It represents the whole class in mean calculations.

The approximate link Mode = 3 Median − 2 Mean for moderately skewed data. It lets one measure be estimated when the other two are known.

The ratio of favourable outcomes to total equally likely outcomes in a sample space. It is a number from 0 to 1 and predicts long-run relative frequency.

The event that a given event does not occur. The probabilities of an event and its complement add to 1.

The set of all possible outcomes of a random experiment. Listing it completely is the first step in counting favourable outcomes.

The statement formed by swapping the hypothesis and conclusion of a conditional statement. A converse is not automatically true just because the original is.

The statement formed by negating and swapping the hypothesis and conclusion. It is logically equivalent to the original conditional statement.

A single case that satisfies the hypothesis but violates the conclusion, disproving a general statement. One counterexample is enough to refute a claim.

A method that assumes the statement false and derives an impossibility, thereby proving it true. Irrationality of the square root of 2 is shown this way.