UPMSP High School Mathematics key terms

For any positive integers a and b, there exist unique whole numbers q and r such that a = bq + r, where 0 ≤ r < b. It is the basis for computing HCF.

Every composite number can be expressed as a product of primes, and this factorisation is unique apart from the order of the factors.

The highest common factor of two or more numbers is the largest number that divides each of them exactly, found by prime factorisation or Euclid's algorithm.

The least common multiple of two or more numbers is the smallest number that is a multiple of each of them; for two numbers, HCF × LCM equals their product.

Writing a composite number as a product of prime numbers only, for example 72 = 2³ × 3². It is used to find HCF, LCM and to test whether a fraction terminates.

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

A number expressible as p/q where p and q are integers and q ≠ 0. Its decimal expansion either terminates or repeats a block of digits forever.

A decimal expansion that ends after finitely many digits. A fraction in lowest terms gives a terminating decimal exactly when its denominator has only 2 and 5 as prime factors.

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

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

For a quadratic ax² + bx + c with zeroes α and β, the sum α + β equals −b/a and the product αβ equals c/a. These relations let you build a quadratic from its zeroes.

A polynomial of degree two, usually written ax² + bx + c with a ≠ 0. Its graph is a parabola and it has at most two real zeroes.

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

The roots of ax² + bx + c = 0 are given by x = (−b ± √(b² − 4ac)) / 2a. It solves every quadratic with a ≠ 0, provided the discriminant is not negative.

The classification of a quadratic's solutions using the discriminant: two distinct real roots when D > 0, equal real roots when D = 0, and no real roots when D < 0.

Rewriting a quadratic as a perfect square plus a constant, for example x² + 6x + 2 = (x + 3)² − 7. It derives the quadratic formula and locates the vertex.

An equation of the form ax + by + c = 0 with a and b not both zero. Its solutions are ordered pairs forming a straight line in the coordinate plane.

Two linear equations that have at least one common solution. Their graphs either intersect at one point or coincide, giving one solution or infinitely many.

Solving a pair of linear equations by drawing both lines on one grid and reading off the coordinates of their point of intersection, if one exists.

A list of numbers in which each term after the first is obtained by adding a fixed constant to the previous term, such as 5, 8, 11, 14.

The fixed number added to each term of an arithmetic progression to get the next term, written d and equal to any term minus the one before it.

The general term of an arithmetic progression, given by aₙ = a + (n − 1)d, where a is the first term and d the common difference.

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

The distance between points (x₁, y₁) and (x₂, y₂) is √[(x₂ − x₁)² + (y₂ − y₁)²]. It follows from the Pythagorean theorem applied to the horizontal and vertical gaps.

The point dividing the join of (x₁, y₁) and (x₂, y₂) internally in the ratio m : n has coordinates ((mx₂ + nx₁)/(m + n), (my₂ + ny₁)/(m + n)).

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

If a line is drawn parallel to one side of a triangle and cuts the other two sides, it divides them in the same ratio. Its converse is used to prove lines parallel.

Triangles whose corresponding angles are equal and whose corresponding sides are in the same ratio. Similarity preserves shape but not necessarily size.

If two angles of one triangle equal two angles of another, the triangles are similar. Since the third angles then also match, only two need checking.

A line that touches a circle at exactly one point, called the point of contact. At that point the tangent is perpendicular to the radius drawn to it.

The distance from an external point to the point of contact. Tangents drawn from the same external point to a circle are equal in length.

A line that cuts a circle at two distinct points. A tangent may be viewed as the limiting position of a secant whose two intersection points merge.

A ratio of two sides of a right triangle defined by an acute angle, such as sine, cosine or tangent. The value depends only on the angle, not on the triangle's size.

An equation true for every angle where both sides are defined, for example sin²θ + cos²θ = 1. Identities are used to simplify expressions and prove equalities.

Two angles whose sum is 90°. For such angles sin(90° − θ) = cos θ, tan(90° − θ) = cot θ and sec(90° − θ) = cosec θ.

The angle measured upward from the horizontal to the line of sight when an observer looks at an object above 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.

The region enclosed by two radii and the arc between them. Its area is (θ/360) × πr² and its arc length is (θ/360) × 2πr for central angle θ in degrees.

The region between a chord and the arc it cuts off. Its area is found by subtracting the triangle formed by the chord and two radii from the corresponding sector.

The distance measured along a circle between two points on it. For central angle θ in degrees it equals (θ/360) × 2πr.

The portion 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 circular ends and the height.

The total exposed area of a shape built by joining two or more standard solids. Only surfaces visible from outside are counted; hidden joining faces are excluded.

The total space occupied by a shape formed from joined standard solids, obtained by adding the volumes of the individual parts since volume is additive.

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 of ordered data, found for grouped data by locating the median class and applying median = l + [(n/2 − cf)/f] × h.

The value that occurs most often, found for grouped data from the modal class using mode = l + [(f₁ − f₀)/(2f₁ − f₀ − f₂)] × h.

For moderately skewed distributions, mode ≈ 3 × median − 2 × mean. It gives a quick estimate of one average when the other two are known.

For equally likely outcomes, the number of outcomes favourable to the event divided by the total number of outcomes. The value always lies between 0 and 1.

The event that a given event does not happen. Its probability equals 1 minus the probability of the event, so the two probabilities add to 1.

An event that must occur in every trial of an experiment, so its probability is 1. The impossible event, by contrast, has probability 0.