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Vaibhav Mathur Grade: 12
        

The value of x satisfying (x+1)/(x-1) + (x-2)/(x+2)+(x-3)/(x+3)+(x+4)/(x-4)=4 are




(a)(-5+3451/2)/10


(b) (5+3401/2)/8


(c)(-5+3251/2)/16


(d)none

8 years ago

Answers : (1)

AskIITians Expert Hari Shankar IITD
17 Points
										

Hi


The given equation is


(x+1)/(x-1) + (x-2)/(x+2)+(x-3)/(x+3)+(x+4)/(x-4)=4


Write (x+1) as (x-1)+2. So the first term becomes


1+ 2/(x-1).


Similarly, the second term becomes 1 + 2/(x-2), and so on.


Do this to all four terms on the LHS. Now the equation becomes


1+ 2/(x-1) + 1+ 2/(x-2) + 1+ 2/(x-3) + 1+ 2/(x-4) = 4


or, 4 + 2/(x-1) + 2/(x-1) + 2/(x-1) + 2/(x-1) = 4


or, 2 ( 1/(x-1) + 1/(x-2) + 1/(x-3) + 1/(x-4)) = 0


or, 1/(x-1) + 1/(x-4) + 1/(x-3) + 1/(x-2) = 0


or,


[(x-4)+(x-1)]/[x2-5x+4] + [(x-3)+(x-2)]/[x2-5x+6] = 0


or,


(2x-5)/[x2-5x+4] = -(2x-5)/[x2-5x+6]


Therefore Either 2x-5 = 0 or 1/[x2-5x+4] = -1/[x2-5x+6]


x = 5/2 or [x2-5x+6]=-[x2-5x+4]


x=5/2 or 2x2-10x+10=0


x=5/2 or x2-5x+5=0


x=5/2 or x = (5+root(5))/2


These are the only solutions. I think there is some typing error in the options but I am sure these 3 are the only solution. I have verified it using MATLAB software also.


 

8 years ago
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