RBSE Praveshika 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. It underlies the HCF algorithm.

HCF–LCM product rule

For any two positive integers a and b, HCF(a, b) × LCM(a, b) = a × b. Knowing one value lets you find the other directly.

Terminating decimal condition

A rational number in lowest terms has a terminating decimal exactly when its denominator's only prime factors are 2 and 5, each to any power.

Irrational number test

A real number is irrational if its decimal expansion is non-terminating and non-repeating. The square root of any prime is irrational.

Remainder Theorem

When a polynomial p(x) is divided by (x − a), the remainder equals p(a). Substituting a into the polynomial gives the remainder without long division.

Factor Theorem

(x − a) is a factor of polynomial p(x) if and only if p(a) = 0. This converts factor questions into simple substitution checks.

Sum and product of roots

For ax² + bx + c = 0, the two roots satisfy sum = −b/a and product = c/a. These relations let you build or verify quadratics.

Discriminant and nature of roots

D = b² − 4ac. If D > 0 the quadratic has two distinct real roots, if D = 0 equal real roots, and if D < 0 no real roots.

Condition for unique solution

A pair of linear equations has exactly one solution when a₁/a₂ ≠ b₁/b₂. Graphically the two lines intersect at a single point.

Parallel lines condition

Two linear equations have no solution when a₁/a₂ = b₁/b₂ ≠ c₁/c₂. The lines are distinct and parallel, so they never meet.

Coincident lines condition

The pair has infinitely many solutions when a₁/a₂ = b₁/b₂ = c₁/c₂. Both equations describe the same line, so every point on it satisfies both.

Cross-multiplication formula

For a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0, x/(b₁c₂ − b₂c₁) = y/(c₁a₂ − c₂a₁) = 1/(a₁b₂ − a₂b₁).

Distance formula

The distance between points (x₁, y₁) and (x₂, y₂) is √((x₂ − x₁)² + (y₂ − y₁)²). It follows from Pythagoras' theorem on the coordinate grid.

Section formula

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

Area of a triangle formula

For vertices (x₁,y₁), (x₂,y₂), (x₃,y₃), area = ½|x₁(y₂ − y₃) + x₂(y₃ − y₁) + x₃(y₁ − y₂)|. A zero result means the points are collinear.

Perpendicular slope relation

Two non-vertical lines are perpendicular exactly when the product of their slopes is −1, so one slope is the negative reciprocal of the other.

Basic Proportionality Theorem

A line drawn parallel to one side of a triangle intersects the other two sides in the same ratio. It is also called Thales' theorem.

Similar triangle area ratio

In two similar triangles, the ratio of their areas equals the square of the ratio of any pair of corresponding sides.

Tangent perpendicular to radius

A tangent to a circle is perpendicular to the radius drawn to the point of contact. This gives right triangles in tangent problems.

Equal tangents from a point

The two tangent segments drawn from an external point to a circle are equal in length, and they subtend equal angles at the centre.

Heron's formula

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

Cone volume and slant height

Volume of a cone is (1/3)πr²h, and its slant height is l = √(r² + h²). Curved surface area equals πrl.

Sphere surface and volume

A sphere of radius r has surface area 4πr² and volume (4/3)πr³. A hemisphere has half the volume and curved area 2πr².

Sector area and arc length

For a sector of angle θ in a circle of radius r: area = (θ/360)πr² and arc length = (θ/360)2πr. Both scale with θ.

Pythagorean identity

For every angle θ, sin²θ + cos²θ = 1. Dividing through by cos²θ gives 1 + tan²θ = sec²θ; dividing by sin²θ gives 1 + cot²θ = cosec²θ.

Complementary angle relations

For any angle θ: sin(90° − θ) = cos θ, cos(90° − θ) = sin θ, tan(90° − θ) = cot θ. These convert between co-functions.

Angle of elevation

The angle measured upward from the horizontal observer's line to the line of sight of an object above. Used with tan = height/distance.

Grouped data median formula

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, h the class width.

Empirical mean-median-mode relation

For moderately skewed distributions, Mode = 3 Median − 2 Mean. It allows one measure to be estimated when the other two are known.

Classical probability rule

If all outcomes are equally likely, P(E) = number of favourable outcomes ÷ total number of outcomes. Probability always lies between 0 and 1.

Rationalising factor

A surd multiplier that removes the root from a denominator: √a for √a, a − √b for a + √b, and √a − √b for √a + √b.

Compound interest formula

Amount = P(1 + R/100)ⁿ for principal P, annual rate R and n years, with interest compounded yearly. Compound interest is amount minus principal.