Odisha BSE HSC 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 and 0 ≤ r < b.

Fundamental theorem of arithmetic

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

HCF and LCM relation

For any two positive integers a and b, HCF(a, b) multiplied by LCM(a, b) equals the product a × b.

Terminating decimal condition

A rational number in lowest terms has a terminating decimal expansion exactly when its denominator has no prime factors other than 2 and 5.

Irrational number

A real number that cannot be expressed as a fraction of two integers; examples include √2, √3, and π.

Remainder theorem

If a polynomial p(x) is divided by x − a, then the remainder equals p(a), the value of the polynomial at x = a.

Factor theorem

A polynomial p(x) has x − a as a factor if and only if p(a) = 0; this statement is the factor theorem.

Quadratic discriminant

For the quadratic ax² + bx + c = 0, the discriminant D = b² − 4ac tells whether the roots are distinct real, equal real, or non-real.

Quadratic formula

The roots of ax² + bx + c = 0 with a ≠ 0 are x = (−b ± √(b² − 4ac))/(2a), provided b² − 4ac is not negative.

nth term of AP

For an arithmetic progression with first term a and common difference d, the nth term is a_n = a + (n − 1)d.

Sum of AP terms

For an arithmetic progression, the sum of the first n terms is S_n = n/2[2a + (n − 1)d], or S_n = n/2(a + l) when the last term l is known.

Rationalising a surd denominator

To rationalise a denominator such as √a + √b, multiply numerator and denominator by the conjugate √a − √b, because (√a + √b)(√a − √b) = a − b.

Basic proportionality theorem

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

Similar triangle tests

Two triangles are similar when corresponding angles are equal and corresponding sides are proportional; the standard tests are AA, SSS, and SAS similarity.

Area ratio of similar triangles

The ratio of the areas of two similar triangles is equal to the square of the ratio of any pair of corresponding sides.

Tangent–radius theorem

A tangent to a circle is perpendicular to the radius drawn to the point of contact, so the angle between the tangent and radius is 90°.

Equal tangents theorem

Tangents drawn from an external point to a circle are equal in length, and they make equal angles with the line joining the point to the centre.

Angle in a semicircle

An angle inscribed in a semicircle is a right angle; also, the angle subtended by an arc at the centre is twice the angle at the circumference.

Pythagorean identity

For any angle θ, sin²θ + cos²θ = 1; dividing this identity by cos²θ gives 1 + tan²θ = sec²θ.

Complementary angle identities

For acute angles, sin(90° − θ) = cos θ, tan(90° − θ) = cot θ, and sec(90° − θ) = cosec θ.

Angle of elevation

The angle of elevation is the angle between the horizontal line of sight and the line of sight to an object above the horizontal level.

Angle of depression

The angle of depression is the angle between the horizontal line of sight and the line of sight to an object below the horizontal level.

45-45-90 triangle ratio

In a 45°–45°–90° triangle, the two legs are equal and the hypotenuse is √2 times each leg, so the side ratio is 1 : 1 : √2.

Sector area

The area of a sector with radius r and central angle θ degrees is (θ/360) × πr².

Arc length

The length of an arc with radius r and central angle θ degrees is (θ/360) × 2πr.

Cone slant height

For a right circular cone with radius r and height h, the slant height is l = √(r² + h²), and the curved surface area is πrl.

Sphere volume and surface area

The volume of a sphere of radius r is (4/3)πr³, and its total surface area is 4πr².

Frustum volume

The volume of a frustum of a cone with base radii R and r and height h is (1/3)πh(R² + Rr + r²).

Probability of an event

For equally likely outcomes, the probability of an event E is the number of favourable outcomes divided by the total number of outcomes, and 0 ≤ P(E) ≤ 1.

Complement probability rule

The probability that an event E does not occur is P(not E) = 1 − P(E), because an event and its complement together include all possible outcomes.

Median of grouped data

For grouped data, the median is l + [(n/2 − cf)/f] × h, where l is the lower limit of the median class, cf is the cumulative frequency before it, f is its frequency, and h is the class width.

Mode of grouped data

For grouped data, the mode is l + [(f1 − f0)/(2f1 − f0 − f2)] × h, where f1 is the modal frequency, f0 and f2 are adjacent class frequencies, and h is the class width.