Real Numbers Exercise -1.1 SSC Examination

REAL NUMBERS Class 10 Notes pdf

Real Numbers Class 10 Notes pdf

REAL NUMBERS :

• Positive or negative, Large or small, whole numbers or decimal numbers are all Real Numbers Class 10 pdf.

They are called “Real Numbers” Real Numbers Class 10 pdf because they are not Imaginary Numbers

Real Numbers Class 10 Notes pdf

EXERCISE – 1.1

Real Numbers Class 10 Solutions pdf

1Q.  Use Euclid’s algaritham to find the HCF of

i). 900 and 270

Sol.    900 = 270 X 3 + 90

          270 = 90 X 3 + 0

 HCF = 90

ii)   196 and 38220

    38220 = 196 X195 + 0

196 is the

HCF of 196 and 38220

iii)    1651 and 2032

2032 = 1651 X 1 + 381

         1651 = 381 X 4 + 127

           381 = 127 X 3 + 0

          HCF  = 127

Real Numbers Class 10 Solutions pdf

2Q.  Use division algorithm to show that any positive odd integer  is of the form 6q + 1 or 6q + 3 or 6q + 5, where q is some integer.

Sol : Let ‘a’ be an odd

positive integer.

         Let us now apply division algorithm with a

and b = 6

          0 < r < 6, the possible remainders are 0, 1, 2, 3, 4 and 5

          i.e., ‘a’

can be 6q or 6q + 1 or 6q + 2 or 6q + 3 or 6q + 4 or 6q +5, where q is the

quotient.

          But ‘a’ is taken as an odd number.

           :. A can’t be 6q or 6q + 2 or 6q + 4.

           :. Any odd integer is of the form

           6q + 1, 6q + 3 or 6q + 5.

3Q.     Use division algorithm to show that the square of any positive integer is of the form 3p, 3p + 1.

Sol :    Let ‘a’  be the square of an integer.     

Applying  Euclid’s  division lemma with a and b = 3

            Since 0 <

r < 3, the  possible remainders are 0,

1 and 2.

             :. Any square

number is of the form 3q, 3q + 1 or 3q + 2, where q is the quotient.

                                                               

OR

           Let ‘a’ be

a positive even integer then a is of the form 2p.

            and  a² = (2p)² = 4p²

= 2 (2p²)                    

= 2k ( where k = 2p² )

            again an even integer.

            If ‘a’ is

an odd number, then ‘a’ is of the form 2p + 1.

            a²   = (2p+1)² = 4p² + 4p + 1

                  = 2( 2p² + 2p ) + 1

                  = 2k + 1 ( where k = 2p² + 2p )

            again an odd integer.

Now applying  Euclid’s  division lemma with a and  b = 3

                a = 3p

                   = 3p + 1

                   = 3p + 2

Where p is quotient and 0, 1, 2 are the possible remainders.

‘a’ can be considered as square of an integer.

:.  Any square number is of the form 3p or 3p + 1.

Real Numbers Class 10 Solutions pdf

Real Numbers Class 10 Solutions pdf

4Q.  Use division algorithm to show that the cube of any positive integer is of the form 9m, 9m +1 or 9m +8.

Sol : 

Let ‘a’ be

positive integer Then from Euclidean lemma a = bq +r ;

Now consider b = 9

then 0 < r <9, it means remainder will be 0, or 1, 2, 3, 4, 5, 6, 7, or 8.

• so a = bq + r   
• a = 9q + r (for b = 9 )
• Now cube of a = a³ = ( 9q + r )³
• =  (9q)³
  + 3.(9q)² r + 3. 9q.r + r³
• =  9³
  ( q³ ) + 3.9² (q² r ) + 3.9 ( q. r ) + r³
• =  9 (9²
  q³ + 3.9. q² r + 3.q.r ) 
  + r³
• a³ = 9m + r³    ( where  ‘m’ = 9². q³ + 3.9.q²
  r + 3.q.r )
• If r = 0
• r³ = 0, then
• a³ = 9m + 0 = 9m

• and for r = 1
• r³ = 1³ , then
• a³ 
  = 9m + 1 =  9m + 1
• and for r = 2
• r³ = 2³ , then
• a³ 
  = 9m + 8 = 9m + 8
• For r = 3
• r³ 
  = 3³
• a³ = 9m + 27
• 9 (m) where m = ( 9m + 3 )
• For r = 4
• r³ = 4³ ,     r = 4
• a³ = 9m + 64
• = ( 9m + 63 ) + 1  ,  9m
  +1
• For 
  r  = 5
• r³ = 125 ,     a³ = 9m + 125
•  =  ( 9m + 117 ) + 8 = 9m + 8
• For  r = 6
  
• r³ 
  = 216,    a³ = 9m
  +  216 = 9m + 9 (24) = 9m
• For r = 7
• r³ 
  =  243 ,    a³ = 9m + 9 (27) = 9m
• For  r = 8
  
• r³ 
  = 512 ,    a³ = 9m +
  9(56) + 8 ,     9m+8

So from the above it is clear that  a³  is either in the form of 9m or 9m +1 or 9m + 8.

        Hence  Approved

Real Numbers , Real Numbers Class 10 Important Questions With Solutions

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Real Numbers Class 10 Important Questions With Solutions

Real Numbers Class 10 Notes NCERT Solutions

What is a real numbers Class 10?

Genuine numbers are just the mix of judicious and silly numbers, in the number framework. As a rule, every one of the math tasks can be performed on these numbers and they can be addressed in the number line, too. Simultaneously, What is a real numbers Class 10? the nonexistent numbers are the un-genuine numbers, which can't be Real Numbers Class 10 Important Questions With Solutions communicated in the number line and is usually used to address a mind boggling number. A portion of the instances of genuine numbers are 23, - 12, 6.99, 5/2, π, etc. In this article, What is a real numbers Class 10? we will talk about the meaning of genuine numbers, properties of genuine numbers and the instances of the What is a real numbers Class 10? with complete clarifications.

What is rational number for class 10th?

Real Numbers Class 10 Notes NCERT Solutions Rational numbers are numbers that can be written in the form p/q, where p and q are integers and q≠0. Examples -1/2, 4/5, 1,0,−3 and so on. REAL NUMBERS Class 10 Notes pdf

Who is the father of real numbers?

Mathematician Richard Dedekind asked these questions 159 years ago at ETH Zurich, and became the first person to define Real Numbers Class 10 Notes NCERT Solutions.

What is set of real number?

The set of real numbers includes every number, negative and decimal included, that exists on the number line. The set of real numbers is represented by the symbol R . The set of integers includes all whole numbers (positive and negative), Real Numbers Class 10 Notes pdf Real Numbers Class 10 Important Questions With Solutions

including 0 . Real Numbers Class 10 pdf The set of integers is represented by the symbol Z 

What are the types of real number?

Real Numbers Class 10 Important Questions With Solutions

Real Number System

• Natural numbers.
• Whole numbers.
• Integers.
• Fractions.
• Rational numbers.
• Irrational numbers.