All Key Concepts and Formulas, MATHS, NCERT, CLASS X(10)

Class – 10

Chapter – 1 Real numbers

Types of Number

1 Natural Numbers N = {1,2,3,4,5..........}
It is the counting numbers

2 Whole number W= {0,1,2,3,4,5........}
It is the counting numbers + zero

3 Integers Z={...-7,-6,-5,-4,-3,-2,-1,0,1,2,3,4,5,6...}

4 Positive integers Z+= {1,2,3,4,5........}

5 Negative integers Z-= {...-7, -6, -5, -4, -3, -2, -1}

6 Rational Number: Any number is called a rational if it can be represented in the form p/q where p
and q are integers (q>0).

7 Irrational Numbers: Any number is called a irrational if it cannot be represented in the form p/q
where p and q are integers (q>0).

8 Real Numbers: All rational and all irrational number makes the collection of real number. It is
denoted by the letter R

Euclid’s Division Lemma: It is defined as “for a given positive integer a and b, there exist unique
integers q and r satisfying [a = bq + r,0 ≤ r <b]

HCF of two positive integers can be find using the Euclid’s Division Lemma algorithm for any two
integers a. b. we can write following expression

a = bq + r, 0 ≤ r <b

If r = 0, then

HCF (a. b) =b

If r ≠ 0, then

HCF (a. b) = HCF (b. r)

Again, expressing the integer b. r in Euclid’s Division Lemma, we get

B = pr + r1

HCF (b, r) = HCF (r, r1)

Important Notes

HCF (a, b) =1 – Then a and b are co primes.

,Product of primes Theorem of Arithmetic – Composite Number = Product of Primes

HCF and LCM by prime factorization method:
HCF = Product of the smallest power of each common factor in the numbers
LCM = Product of the greatest power of each prime factor involved in the number

Important Formula HCF (a, b) × LCM (a, b) =a × b

Important concept for rational Number – Terminating decimal expression can be written in the form
p/2n5m



Chapter – 2 Polynomials



Polynomial: A polynomial is a type of expression in which the exponents of all variables should be a
whole number.

Degree of a Polynomial: The highest sum of the exponents is known as the degree of a polynomial.

Types of Polynomials

Based on the numbers of terms, there are mainly three types of polynomials that are listed below:

Monomials – Type of polynomial with single term

Binomials - Type of polynomial with two terms

Trinomials - Type of polynomial with three terms

Based on the degree of the polynomial, polynomials can be classified into 4 major types:

· Zero or Constant polynomial - Polynomials with 0 degree are called zero polynomials

· Linear polynomial - Polynomials with 1 degree are called linear polynomials

· Quadratic polynomial - Polynomials with 2 degree are called quadratic polynomials

· Cubic polynomial - Polynomials with 3 degree are called cubic polynomials

Important Notes on Polynomials

· Terms in a polynomial can be only separated by the '+' or '-' sign.

· For any expression to become a polynomial, the power of the variable should be a whole
number.

· The addition and subtraction of a polynomial are possible between like terms only.

· All the numbers in the universe are called constant polynomials

, Division algorithm for polynomial: Let’s p(x) and q(x) are any two polynomials with q(x) 0, then we
can find polynomial s(x) and r(x) such that

P(x) = s(x) q(x) + r(x)
Where r(x) can be zero or degree of r(x) < degree of g(x)
Dividend = Quotient × Divisor + Remainder

Chapter – 3 Linear equations in two variables

The linear equations in two variables are of the highest exponent order of 1 and have one, none or
infinitely many solutions. The standard form of linear equation is ax+ by+ c= 0 where x and y are the
two variables.


Methods of Solving Linear Equations in Two Variables

Graphical Method

Method of Substitution

Cross Multiplication Method

Method of Elimination

Determinant Method

Graphical Method

The steps to solve linear equations in two variables graphically we use the below steps:

Step 1: To solve a system of two equations in two variables graphically we graph each equation.

Step 2: To graph an equation manually, first convert it to the form y = mx+b by solving the equation
for y.

Step 3: Identify the point where both lines meet.

Step 4: The point of intersection is the solution of the given system.