Monday, June 29, 2020

Introduction to Relation

Relation:

A relation R from A to B is a subset of the Cartesian product A x B and is derived by describing a relationship between the first element (say x) and the other element (say y) of the ordered pairs in A & B.

Representation of Relation:

A relation is represented either by Roster method or by Set-builder method. Consider an example of two sets A = {9, 16, 25} and B = {5, 4, 3, -3, -4, -5}. The relation is that the elements of A are the square of the elements of B.

  1. In set-builder form, R = {(x, y): x is the square of y, x ∈ A and y ∈ B}.
  2. In roster form, R = {(9, 3), (9, -3), (16, 4), (16, -4), (25, 5), (25, -5)}.

relations

Terminologies (Terms used):

  1. Image:
    for any ordered pairs, in any Cartesian product (say A × B), the second element is called the image of the first element.
  2. Domain:The set of all first elements of the ordered pairs in a relation R from a set A to a set B.
  3. Range:The set of all second elements in a relation R from a set A to a set B is  called Range.
  4. Co-domain:
    The whole set B is called Co-domain. Range ⊆ Co-domain.

Total Number of Relations:

For two non-empty set, A and B. If the number of elements in A is h i.e., |A| = h & that of B is k i.e., |B| = k, then the number of ordered pair in the Cartesian product will be |A × B| = hk. The total number of relations is 2hk.

Types of Relations:

There are many types of relations.

relationship

Empty Relation:

If no element of set X is related or mapped to any element of Y, then the relation R in A is an empty relation, i.e, R = Φ.

Universal Relation:

A relation R, called universal relation if each element of A is related to every element of B.

Suppose A is a set of all natural numbers and B is a set of all whole numbers. The relation between A and B is universal as every element of A is in set B.

                    Empty relation and Universal relation are sometimes called trivial relation.

Inverse Relation:

Let R be a relation from set A to set B i.e., R ∈ A × B. The relation R-1 is said to be an Inverse relation if R-1 from set B to A is denoted by R-1 = {(b, a): (a, b) ∈ R}.

Identity Relation:

In Identity relation, every element of set A is related to itself only. I = {(a, a), ∈ A}.

For example, If we throw two dice, we get 36 possible outcomes, (1, 1), (1, 2), … , (6, 6). If we define a relation as R: {(1, 1), (2, 2), (3, 3), (4, 4), (5, 5), (6, 6)}, it is an identity relation.

Reflexive Relation:

If every element of set A maps to itself, the relation is Reflexive Relation. For every a ∈ A, (a, a) ∈ R.

Symmetric Relation:

A relation R on a set A is said to be symmetric if (a, b) ∈ R then (b, a) ∈ R, for all a & b ∈ A.

Transitive Relation:

A relation in a set A is transitive if, (a, b) ∈ R, (b, c) ∈ R, then (a, c) ∈ R, for all a, b, c ∈ A

Equivalence Relation:

A relation is said to be equivalence if and only if it is Reflexive, Symmetric, and Transitive.

Ordered Relation:

Partial Ordered Relation:

A relation R⊂X ✖️ X is called partial ordering if it is Reflexive,Anti-Symmetric, and Transitive, that is
   
             (x , x) ∈ R for all x
∈ X
            
(x , y) ∈ R ,(y , x) ∈ R ⇒ x = y
            
(x , y) ∈ R, (y , z) ∈ R
(x , z) ∈ R

Total Ordered Relation:
When all the elements of a partial order relation are comparable, the relation is called a total order Relation.
i.e;  a,b
∈ A and  a≤ b and b≤ a
[
≤ is called relation ]

Linear Ordered:

That is, every element is related with every element one way or the other.
A total order is also called a linear order.

PreOrdered/Quasi Ordered:

A binary relation R on a set A is a quasi order if and only if it is
(1) irreflexive, { (x , x) ∈ R for all x ∉ X }
(2) transitive.  (x , y) ∈ R, (y , z) ∈ R (x , z) ∈ R.


(minimal/maximal element):

Let < A, ≤ > be a poset, where represents an arbitrary partial order. Then an element   b A is a minimal element of A if there is no element  a A that satisfies a b. Similarly an element b A is a maximal element of A if there is no element a A that satisfies b a.
 
(least/greatest element):

Let < A, > be a poset. Then an element b A is the least element of A if for every element a A, b a.

(well order): A total order R on a set A is a well order if every non-empty subset of A has the least element.

Sunday, June 28, 2020

THEOREMS ON PROBABILITY OF EVENTS( PART -2)

[If you are beginner in probability then you should read Probability Theorem (Part-1) before Read the below content]

THEOREM- 4:

                   For any 3 events A , B , C  P( A ⋃ B | C ) = P( A | C ) + P( B | C ) P( A ⋂ B | C )
 

PROOF:

      We have P( A ሀ B ) = P( A ) + P( B ) - P( A ⋂ B )
 ⇒P[( A ⋂ C) ⋃ ( B ⋂ C )] = P( A ⋂ C ) + P( B ⋂ C )     - P( A ⋂ B ⋂ C )
         Dividing both sides by P( C ), we get
 ⇒ P[( A ⋂ C) ⋃ ( B ⋂ C )] / P( C ) = P( A ⋂ C ) / P(         C ) + P( B ⋂ C ) / P( C )  - P( A ⋂ B ⋂ C ) / P( C )                                      , P( C ) > 0
                       
 ⇒  P[( A  ⋃ B ) ⋂ C ] / P( C ) = P( A | C ) + P( B | C ) -  P( A ⋂ B | C )
 ⇒   P[( A  ⋃ B ) | C ] = P( A | C ) + P( B | C ) - P( A ⋂ B | C ) 
                                                                        [Proved]

THEOREM - 5:

           For any 3 events A , B , C  P(A ⋂ B' | C ) + P( A ⋂ B | C ) = P( A | C ) , B' = complementary of event B.

PROOF :

          Given,
         P(A ⋂ B' | C ) + P( A ⋂ B | C )
     = P(A ⋂ B' ⋂ C ) / P( C ) + P( A ⋂ B ⋂ C ) / P( C )
     = [ P(A ⋂ B' ⋂ C ) + P( A ⋂ B ⋂ C ) ] / P( C ) 
     =  P(A ⋂ C ) / P( C ) 
     = P( A | C )                                  [Proved]
 

 THEOREM - 6:

         For any 3 events A , B , C defined on the sample space S such that B ⊂ C and P( A ) > 0 , P( C | A )  ≥ P( B | A ).

 PROOF:

      P( C | A ) = P( C ⋂ A ) / P( A )      [ By definition of conditional probability]

     = [ P( B ⋂ C ⋂ A ) ⋃ P( B' ⋂ C ⋂ A ) ] / P( A )

     =  [ P( B ⋂ C ⋂ A ) ] / P( A )  +  [ P( B' ⋂ C ⋂ A ) ] / P( A )
     =    P [ ( B ⋂ C | A ) + ( B' ⋂ C | A ) ]
    Since B ⊂ C ⇒ B ⋂ C = B
           ⇒ P( C | A ) = P( B | A ) + P( B' ⋂ C | A )
           ⇒ P( C | A ) ≥ P( B |  A ).                                         [Proved]

 Pair-wise Independent Events:

         Definition:  A set of events  A1 , A2  , ... , An  are said to be pair-wise independent  if  
          P(  Ai ⋂  Aj ) = P(  Ai ) . P(  Aj )  ∀ i ≠ j

 Conditions for Mutual Independence of ' n ' Events :

    Let S denote the sample space for a number of events . The events in S are said to be mutually independent if the probability of the simultaneous occurrence of (any) finite number of them is equal to the product of their separate probabilities.
               If  A1 , A2  , ... , An  are ' n ' events , then their mutual independence , we should have  
       (i)  P(  Ai ⋂  Aj ) = P(  Ai ) . P(  Aj ) ,    [ i ≠ j ; i , j = 1,2,...,n]
      (ii) P(  Ai  ⋂  Aj  ⋂  Ak ) = P( Ai ) . P( Aj ) . P( Ak ) ,  [ i ≠ j ≠ k ; i ,      j , k = 1,2,...,n]
                                    .                                     .
                                    .                                     .
                                    .                                     .
 P( A1 ⋂  A2 ⋂ ... ⋂ An ) = P( A1 ) . P( A2 ) . ... . P( An )

 

Remarks : 

       1. It may be observed that pair-wise or mutual independence of events A1 , A2  , ... , An  is defined only when P(  A1 ) ≠ 0 , for i = 1,2,...,n .
       2. If the events A and B are such that P( A )  ≠ 0 , P( B ) ≠ 0 and A independent of B , then B is independent of A.
      Proof :  We are given that , P( A | B) = P( A )       
                   [∵ A is independent of B ]
                   ⇒ P( A ⋂ B ) / P( B ) = P( A )
                   ⇒  P( A ⋂ B ) = P( A ) . P( B )
         ⇒ P( B ⋂ A ) / P( A ) = P( B )    [∵ P( A ) ≠ 0 ,                    A ⋂ B =  B ⋂ A ]
         ⇒ P( B | A ) = P( B )  


Saturday, June 27, 2020

A QUESTION FROM IMO 1995 | TITU'S LEMMA | CAUCHY-SCHWARZ INEQUALITY

INTRODUCTION:
In this session we will discuss about a question which had appeared in IMO 1995. 
So before staring the session i will discuss about ' Titu's Lemma ' and  ' Cauchy-Schwarz Inequality '.

Cauchy-Schwarz Inequality :
 The Cauchy-Schwarz inequality also known as Cauchy-Bunyakovsky-Schwarz inequality states that for all sequence of real number aand bi,we have
Condition:
Equality holds if and only if  ai = kbi, for non-zero constant K € R.

TITU'S LEMMA:
Titu's lemma is a direct consequence of cauchy-Schwarz inequality .
It is also known as Engle's Form, or Sedrakyan's inequality.
      Titu's Lemma states that for positive real number a1,a2,a3.....aand b1 ,b2 ,b3....bis
This form is specially helpful when the inequality involvs fractions where the numeretor is a perfect square.
Then it becomes
           
QNs.
Let  a,b,c are positive real numbers such that a.b.c = 1.
Prove that
ANS.
There are two ways to solve the question 
1.By Titu's Lemma
2. By AM-GM inequality

Method-1:
We have already discussed the inequality at the top.
Substitute a= 1/x  , b= 1/y , c=1/z

by the use of Cauchy-Schwarz inequality 
And the Titu's lemma
we know that a.b.c =1 so ,the  result will be 3/2.

Method-2:

Consider (a+b) , (b+c) ,(c+a).Then by the AM - GM inequality, we have the first set of inequalities.
Multiplying altogether 
But  abc=1 ,then
       (a+b)(b+c)(c+a) >=   8

By the power mean inequality ( AP > GP >HP), and considering the arithmetic and harmonic means of and its permutations

This is how we prove.

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Friday, June 26, 2020

BOLZANO-WEIERSTRASS THEOREM FOR SETS | REAL ANALYSIS

INTRODUCTION:

Have you remembered  Bounded set ,Order completeness of Real numbers.
If not just Click to Read Real Analysis part-1 .

Bounded Set:

A set is said to be bounded if it is bounded above and below.
" I will  not explain this just read about it in the above link "

Bolzano-Weierstrass Theorem:

Statement: Every infinite bounded set of real numbers has a limit point.

Proof: Let ' S ' be any infinite bounded set of real numbers.
                 
                  m ≤ x
≤ M ,      ∀  x ∈S  ---(1)
                     
    m = infimum of S       M=supremum of S


Let us assume T is a set of Real numbers.
 
          T = { x : x is greater than only for finite number of elements } ------(2)

 
x ∈ T { x ≥ only a finite number of elements of S } --(3)
{ x ≤ infinite number of elements of S         }---(4)


  [ Don't confuse with 'S' and 'T' .   S is a infinitely bounded set while T is a finitely bounded set and T is the subset of S also element 'x' is common in both sets]

From equation (1) and (4)
               
                      m
∈ T  ⇔  T ≠ Φ

From equation (1)
               

                       M ≥ x  ,    
∀  x ∈S
 
Let us assume another element 'y' from set T.
                  y
∈ T  ⇔  y ≥ only a finite number of elements of S  [From equation (3)]

    M ≥ y ,    ∀  y ∈ T 
⇔ T is bounded above.

 Now we have prove that T is bounded above

So it will follow Order Completeness Property and we know that " There exist a supremum of  a set of which set follows the completeness property ".

       Let  P be the Supremum of T .
 
Claim: ( P  is a limit point  of set S )

          Let є > 0 be arbitrary number.

 Since   P +
є > P
     
         
P + є
finite number of elements of S   -----(5)
   [ Because P is the Supremum of T and T is the subset of S]
 
Since    
P - є < P
      
        
P - є ≤  infinite number of elements of S ----(6)


From equation (5) and (6) that  nbh
      
                ] 
P - є , P + є [   of  P contains infinite number of elements of S.

Hence , P is a limit point.


Example:


  1.  Z  has no limit point .{it has not bounded}
    2.   S = { -1 n(n/(n+1)  : n ∈ N}
Ans. 
           S = { -1/2 ,2/3 , -3/4 .....}
          infinite | Bounded   -1 < x < 1

hence lmit point will  be 0.


If any doubt , Comment below

THEOREMS ON PROBABILITY OF EVENTS (PART-1)

  Before you readout this blog I suggest you to read it's previous session i.e. " Introduction to Probability" , CLICK HERE to read. 

 THEOREM-1:

             Probability of the impossible event is zero i.e. p(Φ) = 0.
 

    PROOF:

              Impossible event contains no sample  point and hence the certain event S and   the impossible   event Φ are mutually     exclusive.  
     Hence      S U Φ = S  , [where S=    Sample space ]
             ∴      P( S U Φ ) = P( S )
             ⇒    P( S ) + P(Φ ) = P( S )
             ⇒    P( Φ ) = 0                                                                                           [Proved]

REMARK:

    It may be noted P( A )=0, does not imply  that  A is necessarily an empty set. In practice, probability ' 0 ' is assigned to the events which are so rare for example , let us consider the   random tossing of a coin. The event that the coin will stand erect on its edge,  is assigned the probability 0.

 

THEOREM -2:

           Probability of the complementary event Ā  of A is given by  P( Ā ) =1-P( A ).

    PROOF:

                 We know that A and Ā are dis-joint                          events.
             Moreover,          A U Ā = S 
                                    ⇒ P( A U Ā ) = P( S )
                                    ⇒ P( A ) + P( Ā ) = 1
                                    ⇒ P( Ā ) = 1 - P( A )                                                                               [Proved]

                                                

THEOREM - 3:

                      For any event A, 0 ≤ P( A ) ≤ 1.

PROOF:

         Clearly for any event A , 
               we have  Φ ⊆ A⊆ S
                         ⇒ | Φ | ⊆ | A | ⊆ | S |
                         ⇒ | Φ |/ | S | ≤ | A |/ | S | ≤ | S |/ | S |
                         ⇒  0  ≤ P( A ) ≤ 1        [Proved]

ADDITION RULE:

    For any two events A and B                

            P( A ⋃ B ) = P( A ) + P( B ) - P( A ⋂ B ).

  ★    If A and B are two mutually exclusive  events then,
                      P( A ⋃ B ) = P( A ) + P( B )                                            [ ∵ P( A ⋂ B ) = 0 ]
 ★    For any three events A  , B , C ,
                        
                  P( A ⋃ B ⋃ C )=P( A ) + P( B ) + P( C ) -      P( A ⋂ B ) - P(B ⋂ C ) - P(C ⋂ A ) + P(A ⋂ B ⋂ C)
 
    PROOF:
           Let us take A ⋃ B = D
      P( A ⋃ B ⋃ C ) = P( D ⋃ C ).                                   
      = P( D ) + P( C ) - P ( D ∩ C )
     = P( A ⋃ B ) + P( C ) -  P(( A ⋃ B ) ∩ C ))                 
     = P( A ⋃ B ) + P( C ) -  [ P(( A ⋂ C )  ⋃ ( B ⋂ C )) ]
     = P( A ) + P( B ) + P( C ) -P( A ⋂ B ) - [ P( A ⋂ C )  + P( B ⋂ C ) - P((A ⋂ B ) ⋂ ( B ⋂ C)) ].  
     =  P( A ) + P( B ) + P( C ) -P( A ⋂ B ) - P(B ⋂ C ) -  P(C ⋂ A ) + P(A ⋂ B ⋂ C)
                                                                                                                                            [Proved]
 ★  If A , B , C are the pair-wise mutually exclusive events , then 
                            
             P( A ⋃ B ⋃ C )=P( A ) + P( B ) + P( C ) .

 Example:
          A dice is thrown is thrown twice , find the 
          i) P( getting total is 8)
         ii) P( getting total is 10)
        iii) P( total is 8 and 10 )
        iv) P ( total is 8 or 10 )
         v) P( total is an even number)
        vi) P( total is not 8)
       vii) P(  tota is an even number but not 8 )

Answer-
  Since the dice is thrown twice , the sample space ( S ) = | 36 |
       i) Let A is the event of getting total is 8
         A={ (2,6) , (3,5) , (4,4) , (5,3) , (6,2) } , | A | = 5
                 so, P( A )= | A | / | S | = 5/36.

      ii) Let B is the event of getting total is 10
                   B={ (4,6) , (5,5) , (6,4) } , | B |=3
                 so, P( B )= | B | / | S | = 3/36= 1/12

     iii) P( getting total is 8 and 10)
           = P( A ⋂ B ) = P(Φ ) = 0

      iv)P( getting total is 8 pr 10)
          = P( A ⋃ B ) = P( A ) + P( B ) - P( A ⋂ B ) 
          = 5/36 + 3/36 - 0 = 8/36=2/9

       v) Let C be the event of total is an even      no. 
              P( C ) = 18/ 36 = 1/2

      vi) P( getting total is not 8)
            = 1 - P( A ) = 1- (5/36) = 31/ 36

   vii) P( getting total is an even no. but not 8)
             = P(getting total is an even no.) - 
                                                P(getting total is 8)
             = (18/36) - (5/36) = 13/36.
 
    
 
  

Thursday, June 25, 2020

Introduction to Geometric Progression

GEOMETRIC PROGRESSION(G.P.):

⮚A progression is called a G.P. if the ratio of it's each term to it's previous term is always constant. If       'a'  be the first term and 'r' be the common ratio then a, ar, ar2, ... ,arn-1 is a sequence of G.P . tn =               arn-1. 

GENERAL TERM OF A G.P.:

  •   nth term of G.P. i.e.  tn = arn-1. , where the common ratio (r) =  t2 /  t1 =  t3 /  t2 = ...
  • If G.P.  consists of ' n ' terms , then  pth term from the end = (n-p+1)th term from the beginning = arn-p. Also , the   pth   term from the end of the G.P. with the last term ' l ' and common ratio 'r'  is =  l(1/r)n-1 . 

SUM OF FIRST ' n ' TERMS OF A G.P.:

 Sum of first ' n ' terms of a G.P. is given by
             Sn = ∑   tn 
                  =   t1 +   t2  +   t3 + ... +   tn    
                  =   a + ar + ar2+ ... +arn-1
  •  Sn =  a(1-rn )/(1-r)  and  Sn = (a-lr)/(1-r) , [when | r | < 1. ] 
  •  Sn =  a(rn -1)/(r-1)  and  Sn = (lr-a)/(r-1) , [when | r | > 1. ]
  •  Sn = na  [ when r=1 ]

SUM OF INFINITE TERMS OF G.P.:

⮚ When n→∞ and | r | < 1 ( -1 < r < 1),      S  = a/(1-r).

⮚ When n→∞ and | r | ≥ 1 ,  S  does not exist.

GEOMETRIC MEAN(G.M.):

⮚ If a , b , c are in G.P.  , then  b/a = c/b  ⇒ b2= ac ⇒ b= √(ac)  is the G.M. of a and c.

      Similarly  G.M. of a , b , c  is   (abc)1/3 .

      G.M.  of   a1 , a2  , a3 , ... , an  is     (a1. a . a3  ...  .a)1/n         

  ⮚ Let n G.M.s are inserted between a & b . Let    G1 , G2  , G3 , ... , Gn   are n G.M.s ,

      then a ,  G1 , G2 , G3 , ... , Gn , b are in G.P. .

       Then common ratio ( r ) =  (b/a )1/n+1          

         So,   G1  =   ar   =   a . (b/a )1/n+1 

                  G2  =   ar2  =   a . (b/a )2/n+1 

                  ___________________________

                 Gn  =   arn =   a . (b/a )n/n+1 

PROPERTIES OF G.P. :

✪  If all the terms of a G.P. be multiplied or divided by the same non-zero constant , then it                 remains a G.P.  , with the same common ratio ' r '.  

✪  The reciprocal of the terms of a given G.P.  form a G.P. with common ratio as reciprocal of the         common ratio(r) of the original G.P. i.e. ' 1/r '.

✪  If each term of a G.P.  with common ratio ' r ' be raised to the same power k , the resulting              sequence also forms a G.P. with common ratio  rk  .

✪  In a finite G.P. , the product of terms equidistant from the beginning and the end is always the       same and is equal to the product of the first and last term i.e. if   a1 , a2  , a3 , ... , an   be in G.P.        . Then                a1. an  = a2 . an-1  =  a. an-2  = ... = ar . an-r+1          

✪  If  a1 , a2  , a3 , ... , an , ... is a G.P. of non-zero , non-negative terms , then  loga1 , loga2  ,log a3          , ... ,  logan , ...  is in A.P. and vice-versa.

✪  Three non-zero numbers a , b , c are in G.P. iff  b2= ac .

✪  If    ax1 , ax2, ax3, ... , axn   are in G.P. , then  x1 , x2  , x3 , ... , xn are in A.P. .

✪  If the first term of a G.P. of n terms is ' a ' and last term is ' l ' , then the product of all terms         of  the G.P.  is  (al )n/2 .

  ⮚ So, in the below tables , these are terms taken , which should be used in some kinds of questions in        G.P. as same as in the  A.P. .

                  TABLE-1 : When the product is given.

            No. of terms                        Terms taken 

 3 
  a/r , a , ar 
 4         a/ r3 , a/r , a/r , a/ r3 
 5 a/ r2 , a/r , a , a/r , a/ r2  

                   TABLE -2 : When the product is not given. 
       
         No. of terms                        Terms taken 
 3 
   a , ar , ar2 
 4    
             a , ar , ar2 ,  ar3 
 5
        a , ar , ar2, ar3ar4  


       So guys , this is all about the introduction of G.P. .      And on later, we will learn about                         Harmonic  Progression and it's properties . 
       Thanks for reading this post .😇😇😇 
                                     And Stay tuned guys . ✌✌✌



Newton-Raphson Method ---Solution of Polynomial And Transcendental equation| Engineering Mathematics

Introduction:
≫In the previous session we read about Bisection method which is used to solve the equation for finding root.
if you are not read that blog i suggest you to read that blog first
Click Here to read.


Newton-Raphson Method:

Concept:

We will choose a +ve point much nearer to cut of the curve to X-axix . Let that point is xand it intersect the curve at (xn,∱(xn)) .
              If we draw a tagent at the point (xn,∱(xn)) then the slope will be ∱'(xn) and cut the X-axis at xn+1.
     Similarly,the point xn+1 cuts the curve at (xn+1,∱(xn+1)) and if we draw a tangent at there then the slope will be ∱'(xn+1) and the tangent cut the X-axix at xn+2 .
         And The point xn+2 cuts the curve at (xn+2,∱(xn+2)). 
We can see that at every step we move towards the root point of ∱(x)=0.
This method is very easy and less-effort than Bisection-Method.


Working Rule:

1. Given ∱(x)=0 ------(1)
   Find initial root
x0 such that ∱(x)~0
[ i.e;
x0 is near to the root
∱(x)=0]

2. Find
∱(x0) and ∱'(x0)

3. First approximate root by Newton-Raphson method:
            x1 =x0 - (x0)'(x0)      

Find
∱(x1) and ∱'(x1)

4. Similarly, Second approximate root is

x2 =x1 - (x1)'(x1)              

5.Also Third approximate root is

         
x3 =x2 - (x2)'(x2)                 

General Formula:


xn =xn-1 - (xn-1)'(xn-1)



Example

QnS. ∱(x) = x - 2 + lnx has a root near x =1.5 .Use Newton-Raphson Method to obtain a better estimate 2 digit after decimal point.

Ans.

In this question we don't need to figure out the x0, because it has given that the root is lies near x=1.5 .

Here x0 =1.5 and ∱(1.5) = -0.5 + ln(1.5) = -0.0945

∱'(x)= 1+(1/x)

∱'(1.5) = 5/3

Hence using the general formula

x1=1.5567

The Newton-Raphson formula can be used again.

so ∱(x1) = ∱(1.5567) = -0.0007

∱'(x1) = ∱'(1.5567) = 1.6424

{ Calculation Part is skipped for Readers}

x2= 1.5571

⇒We will note that the 2digits after decimal point of x1 is same as x2 .So

our answer will be x2.

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