Mathematics HSSC-I Definitions

Flashcards for vocabulary related to Mathematics HSSC-I Textbook of Algebra and Trigonometry for Class XI.

Rational number

A number which can be written in the form of p/q, where p, q ∈ ℤ, q ≠ 0.

Irrational number

A real number which cannot be written in the form of p/q, where p, q ∈ ℤ, q ≠ 0.

Real number

The field of all rational and irrational numbers.

Terminating decimal

A decimal which has only a finite number of digits in its decimal part.

Recurring decimal

A decimal in which one or more digits repeats indefinitely.

Non-terminating, non-recurring decimal

Decimal which neither terminates nor is it recurring; it cannot be converted into a common fraction and represents an irrational number.

Binary operation

A rule, usually denoted by *, that assigns to any pair of elements of A to another element of A.

Complex number

The number of the form z = x + iy, where x, y ∈ ℝ, i = √-1. Here x is the real part and y is the imaginary part of z.

Real plane or coordinate plane

The geometrical plane on which coordinate system has been specified.

Argand diagram

The figure representing one or more complex numbers on the complex plane.

Modulus of complex number

The distance from the origin of the point representing the complex number.

Set

A well-defined collection of distinct objects.

Descriptive method (of describing a set)

A method by which a set is described in words.

Tabular method (of describing a set)

A set described by listing its elements within brackets.

Set-builder method

Using a letter or symbol for an arbitrary element of a set and stating the property common to all members.

Order of a set

Number of elements in a set.

Equal set

Two sets A and B are said to be equal if each element of set A is an element of set B and both entries are the same.

Equivalent set

Two sets are said to be equivalent if a one-to-one correspondence can be established between them.

Singleton set

A set having one element.

Null set

A set having no element.

Finite set

A set having a finite number of elements.

Infinite set

A set having an infinite number of elements.

Subset

If each element of set A is also an element of set B, then A is called a subset of B.

Proper subset

If A is a subset of B and B contains at least one element which is not in A, then A is called a proper subset of B.

Improper subset

If A is a subset of B, and A = B, then A is an improper subset of B.

Power set

The set of all subsets of set A.

Universal set

The set that contains all the elements and objects involved in the problem under consideration.

Complement of a set

The complement of a set A relative to the universal set U is the set of all elements of U which do not belong to A.

Deduction

To draw general conclusion from well known facts is called deduction.

Induction

To draw general conclusion from limited number of observation or experience is called induction.

Aristotelian logic

Deductive logic in which every statement is regarded as true or false

Non Aristotelian Logic

Deductive logic in which every statement is regarded scope of third or fourth is called non-Aristotelian logic.

Truth Table

A table to drives truth values of a given compound statement in terms of its component parts.

Tautology

A statement which is true for all possible values of variable involved in it

Contradiction

A statement which is always false is called Contradiction or absurdity.

Contingency

A statement which can be true or false depending upon the truth values of variable.

Function

Let A and B be two non-empty sets. If (1) F is a relation from A to B i.e. F is a subset of A × B . (2) Domain of F = A (3) No two ordered pairs of F have same 1st elements .Then F is called a function from A toB and is written as F : Α → Β denoted by y = f(x).

Bijective function

A function f which is both one to one and onto.

Injective function

A function f which is both one to one and into.

Groupoid

A non-empty set which is closed under given Binary Operation ‘*’.

Binary Operation

Any mapping of G × G into G, where G is non empty set.

Semi Group

A non-empty set which is closed under given Binary operation and The Binary operation is associative.

Monoid

A non-empty set which is closed under given Binary operation, the Binary operation is associative, and The set have identity element w.r.t. Binary operation

Group

A non-empty set G id called a group w.r.t Binary operation ‘*’ If it is closed under given Binary operation, the Binary operation is associative, the set have identity element w.r.t. Binary operation, and Every element of G w.r.t Binary operation i.e. a * a' = a' * a = e

Abelian Group

A group G under Binary operation ‘*’ is called Abelian group if Binary operation is commutative i.e. a * b = b * a . if a * b ≠ b * a then this is a Non Abelian group under Binary operation.

Linear Function

The function {( x, y | y = mx + c )} is called a linear function. Geometrical representation of linear function is a straight line.

Quadratic Function

The function {( x, y | y = ax^2 + bx + c )} is called a quadratic function, because it is defined by second degree equation in x and y.

Unary Operation

A mathematical producer that changes one number into another. Or it is an operation which is applied on a single number to give another single number.

Matrix

An arrangement of different elements in the rows and columns, within square brackets.

Order of Matrix

Order of Matrix tells us about no of rows and columns order of a matrix = no. of rows × no. of column.

Row Matrix

A matrix having single row.

Column Matrix

A matrix having single column.

Square Matrix

A matrix in which no of rows and columns are equal.

Rectangular Matrix

A matrix in which no of rows and columns are not equal.

Diagonal Matrix

A square matrix having each of its elements excepts principle diagonal equal to zero and at least one elements in its principle diagonal matrix.

Scalar Matrix

A square matrix having same elements in principle diagonal except 1.

Unit Matrix or Identity Matrix

Let A = [aij] be a square matrix of order n. If aij = 0 for all i ≠ j and aij = 1 for all i = j, then the matrix A is called a unit matrix or identity matrix of order n. It is denoted by In.

Null Matrix or Zero Matrix

A square or rectangular matrix whose each element is zero.

Equal Matrix

Two matrix are said to be equal if they are of same order with the same correspondence elements.

Upper Triangular Matrix

If all elements below the principle diagonal of square matrix are zero then it is called upper triangular matrix.

Lower Triangular Matrix

If all elements above the principle diagonal of square matrix are zero then it is called lower triangular matrix.

Singular Matrix

A square matrix Α is called singular if |Α| = 0

Non-Singular matrix

A square matrix Α is called non-singular if |Α| ≠ 0

Adjoint of a 2x2 matrix

The adjoint of a matrix A = [[a, b], [c, d]] is denoted by adj A and is defined as adj A = [[d, -b], [-c, a]]

Symmetric Matrix

Let ‘A’ be the square matrix if tA = A then ‘A’ is called symmetric matrix.

Skew Symmetric Matrix

Let ‘A’ be the square matrix if tA = -A then ‘A’ is called skew symmetric matrix.

Hermitian Matrix

Let ‘A’ be the square matrix if tA = A then ‘A’ is called Hermitian matrix .

Skew Hermitian Matrix

Let ‘A’ be the square matrix if tA = -A then ‘A’ is called skew Hermitian matrix .

Rank

Non zero row in a matrix is called rank of the matrix.

Quadratic Equation

An equation of second degree polynomial in a certain variable

Exponential Equation

Equations in which variable occur in exponents.

Reciprocal Equation

An equation which remains unchanged when x is replaced by 1/x.

Radical Equation

Equation involving radical expression of the variable.

Remainder Theorem

If a polynomial f(x) of degree n ≥1 is divided by (x - a) till no x term exits in the remainder then f(a) is remainder .

Polynomial function

A polynomial in x is an expression of the form an x^n + a{n-1} x^{n-1} + … + a1 x + a0, where n is a non-negative integer and the coefficients a_i are real numbers.

Factor Theorem

The polynomial (x - a) is a factor of the polynomial f(x) if and only if f(a) = 0.

Partial Fraction

Partial fraction is an expression of a single rational function as a sum of two or more single rational fraction.

Identity

It is an equation which holds good for all values of the variable.

Rational Fraction

The Quotient of two polynomials P(x) / Q(x) where Q(x) ≠ 0 , with no common factor.

Proper Rational Fraction

A rational Fraction P(x) / Q(x) is called. if the degree of polynomial P(x) is less degree of polynomial Q(x).

Improper Rational Fraction

A Improper rational Fraction P(x) / Q(x) is called. if the degree of polynomial P(x) is greater than or equal to the degree of polynomial Q(x).

Conditional Equation

It is an equation which is true for particular values of variable.

Sequence

Sequence is a function whose domain is subset of the set of natural numbers.

Real Sequence

If all members of a sequence are real numbers, then it is called a real sequence.

Finite Sequence

If the domain of a sequence is a finite set, then the sequence is called finite sequence.

Infinite Sequence

If the domain of a sequence is an infinite set, then the sequence is called infinite sequence.

Series

The sum of an indicated number of terms in a sequence is called series.

Arithmetic Sequence

A sequence {an} is an Arithmetic Sequence or Arithmetic progression if an - a{n-1} is the same number for all n∈ Ν and n >1. .

Arithmetic Mean

A number Α is said to be the Α Μ. . between the two numbers a and b. If a , Α, b are in Α Ρ. .

Geometric Progression

A sequence {an} is geometric sequence or geometric progression if an / a{n-1} is the same non zero number of all n∈ Ν & n >1.

Geometric Mean

A number is said to be geometric means between two numbers a and b if a, G, b are in G P..

Harmonic Progression

A sequence of numbers is called harmonic progression or harmonic sequence if the reciprocal of its terms are in arithmetic progression.

Harmonic Means

A number H is said to be the harmonic means ( H M. ) between two numbers a and b , if a, H, b are in H.P.

Permutation

An ordering arrangement of n objects.

Circular Permutation

The permutation of things which can be represents by the points on a circle.

Probability

Probability is the numerical evaluation of a chance that a particular event would occur.

Sample Space

The set S consisting of all possible outcome of a given experiment.

Combination

When a selection of objects is the made without paying regard to the order of selection.

Event

An event is a subset of sample space.

Equally Likely

Two events Α and Β are said to be Equally Likely if one event is as likely to occur as other.