Given equation is,
?28 - 6?3 = ?3a + b
By expanding equation,
? 1 + 27 - 6?3 = ?3a + b
? ?(1)2 + (3?3)2 - 2 x 3?3 = ? ?3a + b
? ?(1 - 3?3)2 = ?3a + b
? 1 - 3?3 = ?3a + b
On comparing the left and right side equation, we get,
a = -3, b = 1
? a + b = -3 + 1 = - 2
Given, x + a/x = 1
? x2 + a = x .................(i)
? x2 - x = -a ................(ii)
Now, ( x2 + x + a ) / ( x3 - x2)
( x2 + a + x ) / ( x3 - x2)
= ( x + x )/ x(x2 - x) [from Eq. (i)]
= 2x/x(-a) [from Eq. (ii) ]
= 2/-a
= -2/a
Given expression can be written as
= 1/(1 x 4) + 1/(4 x 7) + 1/(7 x 10) + 1/(10 x 13) + 1/(13 x 16)
= 1/3{ 3/(1 x 4) + 3/(4 x 7) + 3/(7 x 10) + 3/(10 x 13) + 3/(13 x 16) }
= 1/3{ (4 - 1)/(1 x 4) + (7 - 4)/(4 x 7) + (10 - 7 )/(7 x 10) + (13 - 10)/(10 x 13) + (16 - 13/(13 x 16) }
= 1/3[(1 - 1/4) + (1/4 - 1/7) + (1/7 - 1/10) + (1/10 - 1/13) + (1/13 - 1/16)]
= 1/3[1 - 1/4 + 1/4 - 1/7 + 1/7 - 1/10 + 1/10 - 1/13 + 1/13 - 1/16]
= 1/3[1 - 1/16]
= 1/3[ (16 - 1)/16]
= 1/3[ (15)/16]
= 1 x 15/ 3 x 16
= 5/16
Given that,
x = ( ?2 + 1 ) / ( ?2 - 1 )
Expanding given equation by multiplying and dividing with ?2 + 1
we will get,
? x = ( ( ?2 + 1 ) / ( ?2 - 1 ) ) x ( ( ?2 + 1 ) / ( ?2 + 1 ) )
? x = ( ?2 + 1 ) x ( ?2 + 1 ) / ( ?2 - 1 ) x ( ?2 + 1 )
? x = ( ?2 + 1 ) 2 / ( ?2 - 1 ) x ( ?2 + 1 ) [ Use the formula (A + B)(A - B) = A2 - B2 ]
? x = ( ?2 + 1 ) 2 / ( ?22 - 12 ) [ Use the formula (A + B)2 = A2 + 2AB + B2 ]
? x = ( 2 + 1 + 2?2 ) / (2 - 1)
? x = 3 + 2?2 .....................(i)
and given in question x - y = 4 ?2
? y = x - 4?2
Put the value of x [ from Equation (i) ]
? y = 3 + 2?2 - 4?2
? y = 3 - 2?2
Now, x2 + y2 = (3 + 2?2)2 + (3 - 2?2)
Use the Algebra formula and solve the equation.
[ Use the formula (A + B)2 = A2 + 2AB + B2 ]
[ Use the formula (A - B)2 = A2 - 2AB + B2 ]
? x2 + y2 = 9 + 8 + 12?2 + 9 + 8 - 12?2
? x2 + y2 = 34
Let a = 8.73 and b = 4.27
Then, given expression =( a3 + b3 ) / ( a2 - ab + b2 )
As we Know the Algebra formula,
a3 + b3 = (a + b) (a2 - ab + b2)
Apply the above algebra formula in given above expression,
= (a + b) (a2 - ab + b2) / ( a2 - ab + b2 )
= (a + b)
= 8.73 + 4.27 = 13
We know that, a3 + b3 + c3 - 3abc = (a + b + c ) ( a2 + b2 + c2 - ab - bc - ca )
? (a + b + c) = ( a3 + b3 + c3 - 3abc ) / ( a2 + b2 + c2 - ab - bc - ca ) .......................(i)
Given that in question,
(2.247)3 + (1.730)3 + (1.023)3 - 3 x 2.247 x 1.730 x 1.023 / (2.247)2 + (1.730)2 + (1.023)2 - (2.247 x 1.730) - (1.730 x 1.023) - (2.247 x 1.023)
Apply the formula given in Equation (i) and we will get,
= ( a + b + c )
= ( 2.247 + 1.730 + 1.023) [ from eq. (i) ]
= 5.000 = 5
Given that,
a = (?2 - 1)1/3
By cubing the both side of the given equation.
a3 = ?2 - 1
Then 1/a3 = 1/?2 - 1
Now multiply and divide the above equation by ?2 + 1
? 1/a3 = ( ?2 + 1 ) / ( ?2 + 1 ) x 1/(?2 - 1 )
? 1/a3 = ( ?2 + 1 ) x 1 / ( ?2 + 1 ) x (?2 - 1 ) [ Use the formula (A + B)(A - B) = A2 - B2 ]
? 1/a3 = ( ?2 + 1 ) / ( 2 - 1 )
? 1/a3 = ?2 + 1
According to the question
( a - a-1 )3 + 3 ( a - a-1 )
= (a - 1/a)3 + 3(a - 1/a) Use the formula [ ( A - B)3 = A3 - B3 + 3AB(A - B) ]
= a3 - 1/a3 - 3 a x 1/a (a -1/a) + 3(a - 1/a)
= a3 - 1/a3 - 3 (a -1/a) + 3(a - 1/a)
= a3 - 1/a3
Pu the value of a3 and 1/a3
= ?2 -1 - ( ?2 + 1 )
= ?2 -1 - ?2 - 1
= -2
Given, x = ( ?3 + 1) / ( ?3 - 1 )
y = ( ?3 - 1 ) / ( ?3 + 1 )
Then , x + y = { ( ?3 + 1) / ( ?3 - 1 ) } + { ( ?3 - 1 ) / ( ?3 + 1 ) }
? x + y = { (?3 + 1)2 + (?3 - 1)2 } / (?3 - 1) (?3 + 1)
? x + y = ( 3 + 1 +2?3 + 3 + 1 - 2?3 ) / ( 3 - 1 )
? x + y = 8/2
? x + y = 4 --------------------(1)
Now xy = ( ?3 - 1 / ?3 + 1 ) x ( ?3 + 1 / ?3 - 1 )
xy = (3 - 1) / (3 - 1) = 2/2 = 1------------------(2)
Given in question,
? x2/y + y2/x
= ( x3 + y3 )/xy ------------------- (3)
Apply the algebra formula, A3 + B3 = (A + B)3 - 3AB(A + B)
= (x + y)3 - 3xy(x + y)/xy
Put the value of x + y from (1) and xy from (2) in Equation (3), we will get
= (4)3 - 3 x 1 x (4)/1
= 64 - 3 x 1 x 4
= 64 - 12
= 52
Given in question,
x + a/x = b
? ( x2 + a )/x = b
? x2 + a = bx .........................(i)
? bx - x2 = a .........................(ii)
Now Given in question,
( x2 + bx + a ) / ( bx2 - x3 )
? ( x2 + a + bx ) / ( bx2 - x3 )
Put the value of x2 + a from equation (i) in above equation,
? ( bx + bx )/(bx2 - x3)
= (2bx)/( bx2 - x3 )
= (2bx)/x( bx - x2 )
= 2b/(bx - x2)
Put the value of bx - x2 from equation (ii) in above equation, we will get
= 2b/a
51.300 + 7.078 + 1.380 + 0.900 = 60.658
For the product of 3.5413 x 2.1
Consider 35413 x 21 = 743673
Total decimal places = 4 + 1
? 3.5413 x 2.1=7.43673
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