৮.(vi).সমাধান: `81((1 - x)/(1 + x))^3 = (1 + x)/(1 - x)`

    বা, `81 = (1 + x)/(1 - x) xx ((1 + x)/(1 - x))^3`

    বা, `(9)^2 = {((1 + x)/(1 - x))^2}^2`

    বা, `((1 + x)/(1 - x))^2 = 9` [ বর্গমূল করে ]

    বা, `(1 + x)/(1 - x) = +- 3`   [ বর্গমূল করে ]

  হয়, `(1 + x)/(1 - x) = 3`                  

    বা, `1 + x = 3 - 3x`                  

    বা, `4x = 2`                                

    `:. x = 1/2`
    
   অথবা, `(1 + x)/(1 - x) = - 3`
   
    বা, `1 + x = - 3 + 3x`

    বা, `2x = 4`                              
     
   `:. x = 2`

 `:.` নির্ণেয় সমাধান, `x = 2`
                  অথবা `x = 1/2`

সমাধান: ৯.
(i).দেওয়া আছে,`a/b = c/d`

 ধরি,`a/b = c/d = k`

 `:. a = bk` এবং  `c = dk`

 বামপক্ষ `=(a^2 + ab + b^2)/(a^2 - ab + b^2)`

 `=((bk)^2 + bk xx b + b^2)/((bk)^2 - bk xx b + b^2)`

 `=(b^2k^2 + b^2k + b^2)/(b^2k^2 - b^2k + b^2)`

 `=(b^2(k^2 + k +1))/(b^2(k^2 - k + 1))`

 `=(k^2 + k + 1)/(k^2 - k + 1)`

 ডানপক্ষ `=(c^2 + cd + d^2)/(c^2 - cd + d^2)`

 `=((dk)^2 + dk xx d + d^2)/((dk)^2 - dk xx d + d^2)`

 `=(d^2k^2 + d^2k + d^2)/(d^2k^2 - d^2k + d^2)`

 `=(d^2(k^2 + k + 1))/(d^2(k^2 - k + 1))`

 `=(k^2 + k + 1)/(k^2 - k + 1)`

`:. (a^2 + ab + b^2)/(a^2 - ab + b^2) = (c^2 + cd + d^2)/(c^2 - cd + d^2)`( দেখানো হলো )
(ii).দেওয়া আছে,`a/b = c/d`

 ধরি,`a/b = c/d = k`

 `:. c = dk`

    `a = bk`

বামপক্ষ `=(ac + bd)/(ac - bd)`

 `=(bk xx dk + bd)/(bk xx dk - bd)`

 `=(bdk^2 + bd)/(bdk^2 - bd)`

 `=(bd(k^2 + 1))/(bd(k^2 - 1))`

 `=(k^2 + 1)/(k^2 - 1)`

ডানপক্ষ `=(c^2 + d^2)/(c^2 - d^2)`

 `=((dk)^2 + d^2)/((dk)^2 - d^2)`

 `=(d^2k^2 + d^2)/(d^2k^2 - d^2)`

 `=(d^2(k^2 + 1))/(d^2(k^2 - 1))`

 `=(k^2 + 1)/(k^2 - 1)`

`:. (ac + bd)/(ac - bd) = (c^2 + d^2)/(c^2 - d^2)` ( দেখানো হলো )

১০.সমাধান:
(i).দেওয়া আছে,

`a/b = b/c = c/d`

 ধরি,

`a/b = b/c = c/d = k`

`:. c = dk`

`b = ck = dk.k = dk^2`

`a = bk = dk^2.k = dk^3`

বামপক্ষ 

`= (a^3 + b^3)/(b^3 + c^3)`

`= ((dk^3)^3 + (dk^2)^3)/((dk^2)^3 + (dk)^3)`

`= (d^3k^9 + d^3k^6)/(d^3k^6 + d^3k^3)`

`= (d^3k^6(k^3 + 1))/(d^3k^3(k^3 + 1))`

`= k^3` 

ডানপক্ষ

`= (b^3 + c^3)/(c^3 + d^3)`

`= ((dk^2)^3 + (dk)^3)/((dk)^3 + d^3)`

`= (d^3k^6 + d^3k^3)/(d^3k^3 + d^3)`

`= (d^3k^3(k^3 + 1))/(d^3(k^3 + 1))`

`= k^3`

`:. (a^3 + b^3)/(b^3 + c^3) = (b^3 + c^3)/(c^3 + d^3)` ( দেখানো হলো )
(ii).দেওয়া আছে,

`a/b = b/c = c/d`

 ধরি,

`a/b = b/c = c/d = k`

`:. c = dk`

`b = ck = dk.k = dk^2`

`a = bk = dk^2.k = dk^3`

বামপক্ষ 

`= (a^2 + b^2 + c^2)(b^2 + c^2 + d^2)`

`= {(dk^3)^2 + (dk^2)^2 + (dk)^2}{(dk^2)^2 + (dk)^2 + d^2}`

`= {d^2k^6 + d^2k^4 + d^2k^2}{d^2k^4 + d^2k^2 + d^2}`

`= d^2k^2(k^4 + k^2 + 1) xx d^2(k^4 + k^2 + 1)`

`= d^4k^2(k^4 + k^2 + 1)^2`
 
 ডানপক্ষ

`= (ab + bc + cd)^2`

`= (dk^3 xx dk^2 + dk^2 xx dk + dk xx d)^2`

`= (d^2k^5 + d^2k^3 + d^2k)^2`

`= {d^2k(k^4 + k^2 + 1)}^2`

`= d^4k^2(k^4 + k^2 + 1)^2`

`:. (a^2 + b^2 + c^2)(b^2 + c^2 + d^2) = (ab + bc + cd)^2` ( দেখানো হলো )

১০. সমাধান:
দেওয়া আছে,

 `x = (4ab)/(a + b)`

বা, `x = (2a xx 2b)/(a + b)`

 `:. x/(2a) = (2b)/(a + b)` 

এবং `x/( 2b) = (2a)/(a + b)`

যখন, `x/(2a) = (2b)/(a + b)`

তখন, `(x + 2a)/(x - 2a) = (2b + a + b)/(2b - a - b)` [ যোজন-বিয়োজন করে ]

 `:. (x + 2a)/(x - 2a) = (a + 3b)/(b - a)` .........(i)

আবার, যখন `x/( 2b) = (2a)/(a + b)`

তখন, `(x + 2b)/(x - 2b) = (2a + a + b)/(2a - a - b)` [ যোজন-বিয়োজন করে ]

`:. (x + 2b)/(x - 2b) = (3a + b)/(a - b)`..........(ii)

(i) নং এবং (ii) নং সমীকরণ যোগ করে পাই,

 `(x + 2a)/(x - 2a) + (x + 2b)/(x - 2b) = (a + 3b)/(b - a) + (3a + b)/(a - b)`

 `=  (a + 3b)/(b - a) + (3a + b)/( - (b - a))`

 `= (a + 3b)/(b - a) - (3a + b)/(b - a)`

 `= (a + 3b - (3a + b))/(b - a)`

 `= (2b - 2a)/(b - a)`

 `= (2(b - a))/((b - a))`

 `= 2`

`:. (x + 2a)/(x - 2a) + (x + 2b)/(x + 2b) = 2` ( দেখানো হলো )

১২. সমাধান:
দেওয়া আছে,

    `x = (root(3)(m + 1) + root(3)(m - 1))/(root(3)(m + 1) - root(3)(m - 1))`

বা, `(x + 1)/(x - 1) =(root(3)(m + 1) + root(3)(m - 1) + root(3)(m + 1) - root(3)(m - 1))/(root(3)(m + 1) + root(3)(m - 1) - root(3)(m + 1) + root(3)(m - 1))` [ যোজন-বিয়োজন করে ]

বা, `(x + 1)/(x - 1) = (2root(3)(m + 1))/(2root(3)(m - 1))`

বা, `(x + 1)/(x - 1) = (root(3)(m + 1))/(root(3)(m - 1))` 
 
বা, `((x + 1)/(x - 1))^3 = ((root(3)(m + 1))/(root(3)(m - 1)))^3` [ উভয়পক্ষকে ঘন করে ]

বা, `(x^3 + 3x^2 + 3x + 1)/(x^3 - 3x^2 + 3x - 1) = (m + 1)/(m - 1)`

বা, `(x^3 + 3x^2 + 3x + 1 +x^3 - 3x^2 + 3x - 1)/(x^3 + 3x^2 + 3x + 1 - x^3 + 3x^2 - 3x + 1) = (m + 1 + m - 1)/(m + 1 - m - 1)` 
 [ পুনরায় যোজন-বিয়োজন করে ]

বা, `(2x^3 + 6x)/(2 + 2x^2) = (2m)/2`

বা, `(2(x^3 + 3x))/(2(1 + 3x^2)) = m`

বা, `x^3 + 3x = m + 3mx^2`  [ আড় গুণন করে ]

`:. x^3 - 3mx^2 + 3x - m = 0` ( প্রমাণিত )