Thursday, December 29, 2022
Friday, December 2, 2022
10th class- english-sliptest - attitude is altitude-model test
Model sliptest on "Attitude is Altitude"
Prepared by P. Dharamdeep SA(English)
ZPHS Kathalapur, Jagtial District
Wednesday, October 26, 2022
Friday, September 16, 2022
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 5m (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 5m. (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 5m. (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⟺ax = 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 nth 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:
COORDINATE GEOMETRY
Thursday, September 15, 2022
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.
Sunday, September 11, 2022
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"