RBSE Secondary 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 whole numbers q and r such that a = bq + r, where 0 ≤ r < b.

Terminating decimal condition

A rational number p/q in lowest terms has a terminating decimal expansion exactly when q is of the form 2^m times 5^n, with m and n non-negative integers.

HCF–LCM product rule

For any two positive integers a and b, the product of their HCF and LCM equals the product a × b. This identity does not extend to three numbers.

Irrationality of square root of prime

If p is prime, then √p cannot be written as a ratio of integers; assuming it equals a/b in lowest terms forces p to divide both a and b, a contradiction.

Remainder theorem

When a polynomial p(x) is divided by x − a, the remainder is exactly p(a). This gives a fast way to evaluate remainders without long division.

Zeros and coefficients of quadratic

For ax² + bx + c with zeros α and β, the sum α + β equals −b/a and the product αβ equals c/a.

Cubic zeros relations

For ax³ + bx² + cx + d with zeros α, β, γ: the sum is −b/a, the pairwise product sum is c/a, and the triple product is −d/a.

Consistency of linear pair

Two linear equations are consistent when their coefficient ratios a₁/a₂ and b₁/b₂ differ, or when all three ratios including c₁/c₂ are equal.

Parallel lines condition

Lines a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0 are parallel and give no solution when a₁/a₂ = b₁/b₂ but c₁/c₂ is different.

Elimination method

Multiply each equation by a suitable number so one variable's coefficients become opposites, then add the equations to eliminate that variable and solve for the other.

Discriminant test

In ax² + bx + c = 0, the discriminant D = b² − 4ac determines the roots: two distinct real roots if D > 0, equal roots if D = 0, no real roots if D < 0.

Quadratic formula

The roots of ax² + bx + c = 0 with a ≠ 0 are x = (−b + √(b² − 4ac)) / 2a and x = (−b − √(b² − 4ac)) / 2a.

Completing the square

Rewriting ax² + bx + c as a(x + b/2a)² + (c − b²/4a) converts the quadratic to vertex form and allows solving without factorisation.

nth term of an AP

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

Sum of first n terms

The sum of the first n terms of an AP is Sₙ = (n/2)[2a + (n − 1)d], or equivalently Sₙ = (n/2)(a + l) when the last term l is known.

Arithmetic mean condition

Three numbers a, b, c are in arithmetic progression exactly when 2b = a + c, that is, the middle number is the average of the other two.

Basic Proportionality Theorem

A line drawn parallel to one side of a triangle intersects the other two sides in the same ratio; the converse also holds.

Area ratio of similar triangles

If two triangles are similar with corresponding sides in ratio k, then their areas are in ratio k², the square of the side ratio.

Pythagoras theorem

In a right-angled triangle, the square of the hypotenuse equals the sum of the squares of the other two sides; the converse identifies right angles.

Distance formula

The distance between points (x₁, y₁) and (x₂, y₂) in the plane is the square root of (x₂ − x₁)² + (y₂ − y₁)².

Section formula

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

Centroid coordinates

For a triangle with vertices (x₁, y₁), (x₂, y₂), (x₃, y₃), the centroid is the average of the coordinates: ((x₁ + x₂ + x₃)/3, (y₁ + y₂ + y₃)/3).

Pythagorean identity

For any angle θ, sin²θ + cos²θ = 1. Dividing throughout by cos²θ gives 1 + tan²θ = sec²θ, and dividing by sin²θ gives 1 + cot²θ = cosec²θ.

Angle of elevation

The angle of elevation is the angle between the horizontal line through the observer's eye and the upward line of sight to an object above that level.

Tangent perpendicular to radius

At every point of contact, a tangent to a circle is perpendicular to the radius drawn to that point; this underlies most tangent constructions and proofs.

Equal tangents from external point

The two tangent segments drawn to a circle from an external point are equal in length, and the line joining the point to the centre bisects the angle between them.

Sector area

A sector of a circle of radius r subtending angle θ at the centre has area (θ/360) × πr², a fraction θ/360 of the whole circle.

Arc length

The length of an arc subtending angle θ at the centre of a circle of radius r is (θ/360) × 2πr.

Cone slant height

For a right circular cone of base radius r and height h, the slant height is l = √(r² + h²); curved surface area is πrl and volume is (1/3)πr²h.

Frustum volume

A frustum of a cone with height h and base radii r₁ and r₂ has volume (1/3)πh(r₁² + r₁r₂ + r₂²).

Empirical mean-median-mode relation

For moderately skewed frequency distributions, Mode = 3 Median − 2 Mean, an approximate relation used when one of the three measures is missing.

Probability complement rule

For any event E, the probability of E plus the probability of not E equals 1, so P(not E) = 1 − P(E).