Tamil Nadu SSLC Mathematics flashcards

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

Cartesian product

The Cartesian product A×B is the set of all ordered pairs (a,b) with a in A and b in B; its size is n(A)·n(B).

Equivalence relation

A relation that is reflexive, symmetric, and transitive. It partitions its underlying set into disjoint equivalence classes, and every element is related to all members of its own class.

Bijective function

A function that is both one-one and onto. Every element of the codomain has exactly one preimage, so the function has an inverse function.

Composite function

(f∘g)(x)=f(g(x)); it is defined when the range of g lies within the domain of f. Composition is associative but generally not commutative.

Euclid division lemma

For positive integers a and b, there exist unique integers q and r such that a=bq+r, where 0≤r<b; repeated use gives the HCF.

Fundamental theorem of arithmetic

Every integer greater than 1 can be expressed as a product of primes uniquely, apart from the order of the prime factors.

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 GP

For a geometric progression with first term a and common ratio r≠1, the sum of n terms is S_n=a(r^n−1)/(r−1).

Discriminant of quadratic

For ax²+bx+c=0, Δ=b²−4ac. If Δ>0 the roots are real and distinct; if Δ=0 they are real and equal; if Δ<0 there are 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; the quadratic with these roots is x²−(α+β)x+αβ=0.

Determinant of 2×2 matrix

For A=[[a,b],[c,d]], det A=ad−bc. A is invertible exactly when det A≠0; otherwise it is singular.

Cramer's rule

It solves a linear system using determinants: x=Δ_x/Δ and y=Δ_y/Δ when Δ, the coefficient determinant, is nonzero.

Basic proportionality theorem

A line parallel to one side of a triangle divides the other two sides in the same ratio; its converse is also true.

Pythagoras theorem

In a right triangle, the square of the hypotenuse equals the sum of the squares of the other two sides; the converse also holds.

Tangent–radius theorem

A tangent to a circle is perpendicular to the radius drawn to the point of contact, and tangents from an external point are equal in length.

Centroid of triangle

The point where the three medians of a triangle meet. It divides each median in the ratio 2:1 from the vertex and is the triangle's centre of gravity.

Distance formula

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

Section formula

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

Slope of a line

For two points (x₁,y₁) and (x₂,y₂), the slope is m=(y₂−y₁)/(x₂−x₁), provided x₂≠x₁; it measures steepness and direction.

Perpendicular lines condition

Two nonvertical lines are perpendicular exactly when the product of their slopes is −1; a horizontal line is perpendicular to a vertical line.

Pythagorean identity

For any angle θ, sin²θ+cos²θ=1. Dividing by cos²θ or sin²θ gives 1+tan²θ=sec²θ and 1+cot²θ=cosec²θ, provided the functions involved are defined.

Complementary angle identity

For complementary angles, sin(90°−θ)=cosθ, cos(90°−θ)=sinθ, and tan(90°−θ)=cotθ; the cosecant, secant, and cotangent cofunction identities follow similarly.

Angle of elevation

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

Tangent ratio

In a right triangle, tanθ=opposite side/adjacent side=sinθ/cosθ; it is undefined at 90° because cos90°=0 in the standard right-triangle definition.

Volume of a sphere

The volume enclosed by a sphere of radius r is V=(4/3)πr³; its surface area is 4πr².

Frustum volume

For a frustum of height h and base radii R and r, V=(1/3)πh(R²+Rr+r²); its slant height is √[h²+(R−r)²].

Cone slant height

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

Similar solids ratio

If two similar solids have corresponding lengths in ratio k, their surface areas are in ratio k² and their volumes are in ratio k³.

Probability of complement

For any event E, P(E′)=1−P(E), where E′ is the event that E does not occur; hence 0≤P(E)≤1.

Standard deviation

The positive square root of variance: σ=√[Σ(xᵢ−x̄)²/n] for a population of n observations; it is in the same units as the data.

Addition rule of probability

For events A and B, P(A∪B)=P(A)+P(B)−P(A∩B); if A and B are mutually exclusive, P(A∩B)=0 and the last term vanishes.

Independent events

Events A and B are independent when P(A∩B)=P(A)P(B); knowing that one occurred does not change the probability of the other.