ক. দেওয়া আছে, 

  `a = sqrt(13) + 2sqrt(3)`
    
 `:. 1/a = 1/(sqrt(13) + 2sqrt(3)`

 `= (sqrt(13) - 2sqrt(3))/(sqrt(13) + 2sqrt(3)) (sqrt(13) - 2sqrt(3))`

                    [লব ও হরকে `sqrt(13) - 2sqrt(3)`দ্বারা গুণ করে]

 `= (sqrt(13) - 2sqrt(3))/(sqrt(13)^2 - (2sqrt(3)^2)`

 `= (sqrt(13) - 2sqrt(3))/(13 - 12)`

 `= sqrt(13) - 2sqrt(3)`

  :. `1/a = sqrt(13) - 2sqrt(3)`   (Ans)


 খ. ‘ক’ হতে `a = sqrt(13) + 2sqrt(3)` এবং `1/a = sqrt(13) - 2sqrt(3)`

   এখন `(13a)/(a^2 - sqrt(13a + 1))`

    `= (13a)/(a(a - sqrt(13) + 1/a))`

    `= (13)/(a + 1/a - sqrt(13))`

    `= (13)/(sqrt(13) + 2sqrt(3) + sqrt(13) - 2sqrt(3) - sqrt(13)`

   ` = (13)/(sqrt(13) = sqrt(13)`

  :. `(13a)/(a^2 - sqrt(13a) + 1)`

    `= sqrt(13)`  (প্রমাণিত)

 গ. আমরা জানি, 

   `(a + 1/a)^2 = a^2 + 2.a. 1/a + (1/a)^2`

   `sqrt(13) + 2sqrt(3) + sqrt(13) - 2sqrt(3))^2 = a^2 + 1/a^2 + 2`

                         [`:. a = sqrt(13) + 2sqrt(3) 1/a = sqrt(13) - 2sqrt(3)`]

     বা, `(2sqrt(13))^2 = a^2 + 1/a^2 + 2`

    বা, `52 = a^2 + 1/a^2 + 2`

    বা, `a^2 + 1/a^2 = 50`

    বা, `(a^2 + 1/a^2)^2 = (50)^2`   [উভয় পক্ষকে বর্গ করে]

    বা, `(a^2)^2 + 2.a^2. 1/a^2 + (1/a^2)^2 = 2500`

    বা `a^4 + 1/a^4 + 2 = 2500`

    বা,  `a^4 + 1/a^4 = 2500 - 2`

    বা, `a^4 + 1/a^4 = 2498`

     `:. a^4 = 2498 - 1/a^4`  (দেখানো হলো)

ক. দেওয়া আছে, 

 `4x^2 - 2x = - 1`

  বা, `4x^2 + 1 = 2x`

  বা, `(4x^2 + 1)/(2x) = 1`

  বা, `(4x^2)/(2x) + 1/(2x) = 1`

 `:. 2x + 1/(2x) = 1`


 খ.`x^2 + 1/(16x^2) = 1/4. 4(x^2) + 1/(16x^2)`

  `= 1/4 (4x^2 + 1/(4x^2))`

 ` = 1/4 {(2x)^2 + (1/(2x))^2}`

  `= 1/4 {(2x + 1/(2x))^2 - 2.2x. 1/(2x)}`

  `= 1/4 (1^2 - 2)` [’ক’ হতে]

  `= (-1)/4`

 

 গ.`16(x^4 + 1/(256x^4))`

  `= 16{(x^2)^2 + (1/(16x^2))^2}`

  `= 16 {(x^2 + 1/(16x^2))^2 - 2.x^2. 1/(16x^2)}`

  `= 16{(- 1/4)^2 - 1/8}` [’খ’ হতে]

  `= 16 (1/(16) - 1/8)`

  `= 16 ((1 - 2)/(16))`

  `= - 1`

ক. দেওয়া আছে, 

 `2a = sqrt(7) + sqrt(5) 2b = sqrt(7) - sqrt(5)`

 `:. a + b = (2a + 2b)/2 = (sqrt(7) + sqrt(5) + sqrt(7) - sqrt(5))/2 = sqrt(7)`

 এবং `a - b = (2a - 2b)/2 = (sqrt(7) + sqrt(5)) - (sqrt(7) - sqrt(5))/2`

 `= (sqrt(7) + sqrt(5) - sqrt(7) + sqrt(5))/2 = (2sqrt(5))/2 = sqrt(5)`



 খ. ‘ক’ হতে পাই, `a + b = sqrt(7) a - b = sqrt(5)`

  বামপক্ষ 

  =`8ab(a^2 + b^2)`

  = `4ab . 2(a^2 + b^2)`

  =`{(a + b)^2 - (a - b)^2} {(a + b)^2 + (a - b)^2}`
                                              [সূত্র প্রয়োগ করে]
  
 =` {(sqrt(7))^2 - (sqrt(5))^2} {(sqrt(7))^2 + (sqrt(5))^2}`

                                             [মান বসিয়ে]

 =` (7 - 5) (7 + 5)`

 =` 2 xx 12`

 =` 24`

 = ডানপক্ষ

  :. `8ab (a^2 + b^2) = 24`  (প্রমাণিত)


 গ. বামপক্ষ =`ab(a^4 - b^4)`

       =`ab{(a^2)^2 - (b^2)^2}`

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

       =` (8ab(a^2 + b^2)/8  (a + b) (a - b)`

       =` (24)/8. sqrt(7). sqrt(5). ` [’ক’ ও ‘খ’ এর মান বসিয়ে]

       =` 3sqrt(35)`

       = ডানপক্ষ

     :. `ab (a^4 - b^4) = 3sqrt(35)`  ( দেখানো হলো)

ক. দেওয়া আছে, 

  `x^2 - sqrt(2) x = 1`

   বা, `x^2 - 1 = sqrt(2x)`

   বা, `(x^2 - 1)/x = sqrt(2)`

   বা, `x^2/x - 1/x = sqrt(2)`

   :. `x - 1/x = sqrt(2)`   (দেখানো হলো)


 খ. বামপক্ষ

  `= 7(x^2 + 1/x^2)`

 ` = 7{(x)^2 + (1/x)^2}`

 ` = 7{(x - 1/x)^2 + 2.x. 1/x}`

 ` = 7{7(sqrt(2)^2 + 2}`  [’ক’ হতে]

  `= 7 (2 + 2)`

  `= 7 xx 4`

  `= 28`



  ডানপক্ষ 

 `= 2(x^4 + 1/x^4) = 2{(x^2)^2 + (1/x^2)^2}`

 `= 2{(x^2 + 1/x^2)^2 - 2.x^2. 1/x^2}`

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

 `= 2[{(x - 1/x)^2 + 2. x. 1/x}^2 - 2]`

 `= 2[{(sqrt(2))^2 + 2}^2 - 2]` [’ক’ হতে]

 `= 2[(4)^2 - 2]`

 `= 2 xx 14`

   = 28

    :.` 7 (x^2 + 1/x^2) = 2(x^4 + 1/x^4)`  (দেখানো হলো)

   
  গ. `(x^4 - 1/x^4)/(x + 1/x) = ((x^2)^2 - (1/x^2)^2)/(x + 1/x)`

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

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

   `= 4 xx sqrt(2)` [’খ’ হতে]

   `= 4sqrt(2)`

প্রদত্ত রাশির ঘন 

 `= (3x + 2y)^3`

 `= (3x)^3 + 3. (3x)^2. 2y + 3.3x.(2y)^2 + ()2y)^3`

 `= 27x^3 + 3.9x^2.2y + 3.3x.4y^2 + 8y^3`

 `= 27x^3 + 54x^2y + 36xy^2 + 8y^3` (Ans)