10th EM-Mathematics-Important Concepts and Formulae-Telangana and AndhraPradesh

 REAL NUMBERS

EUCID'S DIVISION ALGORITHM

Given two positive integers a and b, there exist unique pair of whole numbers q and r satisfying a=bq+r, 0≤ r < b

FUNDAMENTAL THEOREM OF ARITHMATIC

Every composite number can be expressed (factorised) as a product of primes and this factorization is unique, apart from the order in which the prime factors occur.

1. x be a rational number whose decimal form is terminating decimal. Then x can be expressed in the form of p/q , where p and q are coprimes, and the prime factorization of q is in the form 2ⁿx 5^m (where n, m are non-negative integer)

2. x = p/q is a rational number, such that the prime factorization of q is in the form 2ⁿx 5^m. (where n and m are non-negative integers) Then, x has terminating decimal.

3. x = p/q is a rational number, such that the prime factorization of q is not in the form 2ⁿx 5^m. (where n and m are non-negative integers) Then, x has a non-terminating and repeating (recurring) decimal.

DEFINITION OF LOGARITHM

a, N are two positive real numbers and a≠1, we define log _a N = x⟺a^x = N

LAWS OF LOGARITHM

Product Rule:

log _a xy = log _a x + log _a y 

Quotient Rule:

log _a (x/y) = log _a x - log _a y

Power Rule:

log _a xⁿ = n log _a x

     SETS

Definition of a Set

Roster form of a Set and Set builder form of a Set

Belongs to and Does not belongs to

Empty set / Null set / Void set

Universal Set and Subset

Equal Sets

Finite Set and Infinite Set

Cardinality of a Finite Set

Union of Sets

Intersection of Sets

Disjoint Sets( A⋂B = ∅ )

Difference of Sets

Venn Diagrams

Formulae:  n(A) + n(B) = n(AUB) + n(A⋂B)

                    (AUB) - (A⋂B) = (A-B) U (B-A)

                                        POLYNOMIAL

Definition of Polynomials

Types of Polynomials:

According to number of terms: Monomial, Binomial, Trinomial and Multinomial

According to degree: Zeroth Polynomial, Constant Polynomial, Linear Polynomial,

                                   Quadratic Polynomial, Cubic Polynomial and n^th degree Polynomial

Value of a Polynomial

Zeroes of a Polynomial:

Linear polynomial has exactly one zero.

Quadratic polynomial can have either two distinct zeroes or two equal zeroes (i.e., one zero) or no zero.  A quadratic polynomial has atmost two zeroes.

Cubic polynomial has atmost three zeroes.

Graph of a Polynomial:

The graph of Linear polynomial is a straight line.

The graph of Quadratic polynomial is Parabola.

The graph of Cubic polynomial is a Curve.

Zeroes are the x-coordinates of the points where the graph of  Polynomial intersects the X-axis.

Relationship between zeroes and coefficients of a Polynomial:

Quadratic Polynomial: ax²+bx+c and its zeroes are ⍺ and 𝛽. 

                1. Sum of zeroes =  - ( coefficient of x ) / ( coefficient of x² ) 

                   

                              ∴      ⍺ + 𝛽 =  -b/a

                2. Product of zeroes =   ( constant term ) / ( coefficient of x² ) 

                   

                              ∴      ⍺ 𝛽 =  c/a

Cubic Polynomial: ax³+bx²+cx+d and its zeroes are ⍺, 𝛽 and 𝛾.

                1. Sum of zeroes =  - ( coefficient of x² ) / ( coefficient of x³ ) 

                   

                              ∴      ⍺ + 𝛽 + 𝛾 =  -b/a

                2. Sum of Product of two zeroes =  ( coefficient of x ) / ( coefficient of x³ ) 

                   

                              ∴      ⍺ 𝛽 + 𝛽 𝛾 + 𝛾 ⍺ =  c/a

                

                3. Product of zeroes = - ( constant term ) / ( coefficient of x³ ) 

                   

                              ∴      ⍺ 𝛽 𝛾 =  -d/a

PAIR OF LINEAR EQUATIONS IN TWO VARIABLES

ax+by+c=0 is called general form of a Linear Equations. (a and b are not equal to zero at a time)

a₁x+b₁y+c₁=0 and a₂x+b₂y+c₂=0 are Pair of Linear Equations in Two Variables

Graph of a Linear Equation is represents a Straight Line

Graph of a Pair of Linear Equations is represents Two Straight Lines

Nature of the Equations:

a₁x+b₁y+c₁=0 and a₂x+b₂y+c₂=0 are Pair of Linear Equations in Two Variables

 a₁

a₂

 =

QUADRATIC EQUATIONS

PROGRESSIONS

COORDINATE GEOMETRY

Polynomials-Value of a Polynomial-Zeroes of a Polynomial-Model Problems

 

1.     Find the value of Polynomial P(x)=x³+3x²-7x+10 at x=2

2.     Find the value of Polynomial P(x)=4x³+x²+2x+5 at x= -3

3.     Find the value of Polynomial P(x)=x⁴+2x³+x²+4x+1 at x= -1

4.     Find the value of Polynomial P(x)=2x⁴-5x³-3x²-3x-4 at x= -2

5.     Find the value of Polynomial P(x)=x³-x²-x-2 at x= ⅓ 

6.     Find the value of Polynomial P(x)=4x⁴+5x³+x²+2x+1 at x= ½

7.      

 

 

 

Polynomials-Definitions, its Degree and Types of polynomials

 Before going to know about Polynomials, we learn some Basics

Constant: All the numbers are called as Constants.

Constant term: A term having only number.

Term: A term is separated from other terms by "+ or -" symbols.

Variable: It is a symbol to represent an unknown value and it can take any real number as its value. Generally we use English alphbet to denote variables. (a, b, c, ,,,,,,,,,,x, y, z)

Variable term: The product of constant and variables.

Degree of variable term: The exponent of the variable is the degree of that term. ( If the term having more than one variable, the degree is the "sum of exponents of all variables")

Mathematical Expression:  An expression is combination of numbers and mathametical operations(+, -, x, ÷, .........). This is also called Numerical Expression.

Algebraic Expression: If a numerical expression contains atleast one variable, it is called Algebraic expression.

Types of Algebraic Expressions:

1. MONOMIAL:

If an expression having only ONE term, it is called Monomial.

2. BINOMIAL:

If an expression having only TWO terms, it is called Binomial.

3. TRINOMIAL:

If an expression having only THREE terms, it is called Trinomial.

4. MULTINOMIAL:

If an expression contains more than three terms, it is called Multinomial.

POLYNOMIAL:

A Polynomial is an Algebraic epression which having Non-Negative integers as exponents of the variables.

NOTE:

1. An Algebraic epression is not a Polynomial, If a variable having negative integer as exponents.

2. An Algebraic epression is not a Polynomial, If a variable having fractions as exponents.

3. An Algebraic epression is not a Polynomial, If any variable involved in the denominator.

DEGREE OF A POLYNOMIAL:

Highest degree from all the degrees of terms of a Polynomial is called the Degree of Polynomial.

Types of Polynomials:

1. ZEROTH POLYNOMIAL:

ZERO is called ZEROTH POLYNOMIAL.

Its degree is not exist(Not Defined)

2. CONSTANT POLYNOMIAL:

Any number other than zero is called Constant polynomial

Its degree is "Zero"

3. LINEAR POLYNOMIAL:

A Polynomial having degree "ONE", is called Linear Polynomial.

Its degree is "ONE"

4. QUADRATIC POLYNOMIAL:

A Polynomial which has degree "TWO", is called Quadratic Polynomial.

Its degree is "TWO"

5. CUBIC POLYNOMIAL:

A Polynomial which has degree "THREE", is called Cubic Polynomial.

Its degree is "Three"

6. nth degree POLYNOMIAL:

A Polynomial which has degree "n", is called nth degree Polynomial.

Its degree is "n"