Odisha BSE Madhyama Mathematics flashcards

32 cards. Read the term and the definition here. The deck at the top of this page is optional.

Euclid's division lemma

For positive integers a and b, there exist unique integers q and r such that a = bq + r, where 0 ≤ r < b. Used to compute HCF by repeated division.

Fundamental Theorem of Arithmetic

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

Remainder theorem

If a polynomial p(x) is divided by the linear polynomial (x − a), then the remainder is p(a). This avoids full long division when only the remainder is needed.

Factor theorem

The polynomial (x − a) is a factor of p(x) if and only if p(a) = 0. It converts the factor question into evaluating the polynomial at a single value.

Degree of a polynomial

The highest power of the variable with a non-zero coefficient. A linear polynomial has degree 1, a quadratic degree 2, and a cubic degree 3.

Linear pair of angles

Two adjacent angles whose non-common arms form a straight line. They are supplementary, so their measures add up to exactly 180 degrees.

Exterior angle of a triangle

The angle formed between one side of a triangle and the extension of an adjacent side. It equals the sum of the two opposite interior angles.

Congruence criteria for triangles

SSS, SAS, ASA, AAS and RHS. Two triangles are congruent when the listed corresponding sides and angles match; AAA and SSA do not guarantee congruence.

Perpendicular bisector locus

The set of all points equidistant from the endpoints of a segment is the perpendicular bisector of that segment. Every point on it satisfies the equal-distance condition.

Angle sum property

The three interior angles of any triangle add up to 180 degrees. For a quadrilateral the interior angles add up to 360 degrees.

Heron's formula

Area of a triangle with sides a, b, c equals the square root of s(s − a)(s − b)(s − c), where s is the semi-perimeter (a + b + c)/2.

Distance formula

The distance between points (x₁, y₁) and (x₂, y₂) is the square root of (x₂ − x₁)² + (y₂ − y₁)², derived from the Pythagoras theorem.

Median of ungrouped data

Arrange the observations in order. For an odd count n, the median is the ((n + 1)/2)th value; for an even count it is the mean of the two middle values.

Graph of a linear equation

The solutions of ax + by + c = 0 form a straight line in the coordinate plane. Two distinct points determine the line, and every point on it is a solution.

Consistent pair of linear equations

A pair of linear equations is consistent if it has at least one solution. The graphs then intersect at one point or coincide; parallel distinct lines mean no solution.

Quadratic formula

For ax² + bx + c = 0 with a ≠ 0, the roots are x = (−b ± √(b² − 4ac)) / 2a. The formula applies whenever the discriminant is non-negative.

Discriminant of a quadratic

D = b² − 4ac for ax² + bx + c = 0. If D > 0 there are two distinct real roots, if D = 0 equal real roots, and if D < 0 no real roots.

Sum and product of roots

For ax² + bx + c = 0 with roots α and β, α + β = −b/a and αβ = c/a. These relations let a quadratic be built from its roots.

Arithmetic progression nth term

For an AP with first term a and common difference d, the nth term is aₙ = a + (n − 1)d. The difference between consecutive terms is constant.

Sum of first n terms of AP

Sₙ = (n/2)(2a + (n − 1)d), or equivalently Sₙ = (n/2)(a + l) where l is the last term of the progression.

Similar triangles area ratio

If two triangles are similar, the ratio of their areas equals the ratio of the squares of any pair of corresponding sides, medians or altitudes.

Basic proportionality theorem

If a line is drawn parallel to one side of a triangle intersecting the other two sides, it divides those two sides in the same ratio.

Pythagoras theorem

In a right-angled triangle, the square of the hypotenuse equals the sum of the squares of the other two sides. Its converse tests whether a triangle is right-angled.

Tangent-radius theorem

The tangent at any point of a circle is perpendicular to the radius drawn through the point of contact, giving a right angle at that point.

Equal tangents from external point

Tangents drawn to a circle from an external point are equal in length. The line joining the point to the centre bisects the angle between them.

Trigonometric identity

sin²θ + cos²θ = 1 for every angle θ. Dividing by cos²θ gives 1 + tan²θ = sec²θ, and dividing by sin²θ gives 1 + cot²θ = cosec²θ.

Values of sin and cos

sin 30° = 1/2, cos 30° = √3/2, sin 45° = cos 45° = 1/√2, sin 60° = √3/2, cos 60° = 1/2, and sin 90° = 1.

Angle of elevation

The angle between the horizontal line of sight and the line of sight to an object above the horizontal. Used with trigonometry to find heights and distances.

Volume of a sphere

The volume enclosed by a sphere of radius r is (4/3)πr³, and its surface area is 4πr². A hemisphere has half these values.

Surface area of a cone

For a right circular cone of radius r and slant height l, curved surface area is πrl and total surface area is πr(l + r), with l² = r² + h².

Mean by assumed-mean method

For grouped data, x̄ = a + (Σfᵢdᵢ / Σfᵢ), where a is the assumed mean and dᵢ is the deviation of each class mark from a.

Median of grouped data

Median = l + ((n/2 − cf)/f)h, where l is the lower limit of the median class, cf the cumulative frequency before it, f its frequency and h the class width.