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

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

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

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

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

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

  `= 2/(x - 2)`

  উত্তর:  `= 2/(x - 2)`


 জ. ` 1/(x^2 - 1) + 1/(x^4 - 1) + 4/(x^8 - 1)`

      `= 1/(x^2 - 1) + 1/(x^2 - 1)(x^2 + 1)) + 4/(x^8 - 1)`

      `= (x^2 + 1 + 1)/((x^2 - 1)(x^2 + 1)) + 4/(x^8 - 1)`

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

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

     `= (x^6 + x^2 + 2x^4 + 2 + 4)/((x^4)^2 - 1^2)`

     `= (x^6 + 2x^4 + x^2 + 6)/(x^8 - 1)`

    উত্তর:  `= (x^6 + 2x^4 + x^2 + 6)/(x^8 - 1)`

ক. `a/(x - 3) - a^2/(x^2 - 9)`

     `= a/(x - 3) - a^2/(x^2 - 3^2)`

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

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

    ` = (ax + 3a - a^2)/((x - 3)(x + 3))`

     `= (ax + 3a - a^2)/(x^2 - 9)`

 উত্তর: `= (ax + 3a - a^2)/(x^2 - 9)`


  খ. `1/(y(x - y)) - 1/(x(x + y))`

     `= (x(x + y) - y(x - y))/(xy(x - y)(x + y))`

     `= (x^2 + xy - xy + y^2)/(xy(x - y)(x + y))`

     `= (x^2 + y^2)/(xy(x - y)(x + y))`

     `= (x^2 + y^2)/(xy(x^2 - y^2)`

      উত্তর: `= (x^2 + y^2)/(xy(x^2 - y^2)`

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

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

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

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

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

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

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

   উত্তর :` 2/(x^4 + x^2 + 1)`


 ঘ.  `(a^2 + 16b^2)/(a^2 - 16b^2) - (a - 4b)/(a + 4b)`

 `= (a^2 + 16b^2)/(a^2 - 16b^2) - (a - 4b)/(a + 4b)`

 `= (a^2 + 16b^2)/((a + 4b) (a - 4b)) - (a - 4b)/(a + 4b)`

 `= (a^2 + 16b^2 - (a - 4b)^2)/((a + 4b) (a - 4b))`

 `= (a^2 + 16b^2 - (a^2 - 2.a.4b + 16b^2))/((a + 4b) (a - 4b))`

 `= (a^2 + 16b^2 - (a^2 - 8ab + 16b^2))/((a + 4b) (a - 4b))`

 `= (a^2 + 16b^2 - a^2 + 8ab - 16b^2)/(a^2 - (4b)^2`

 `= (8ab)/(a^2 - 16b^2)`

    উত্তর: `(8ab)/(a^2 - 16b^2)`

   
  ঙ. `1/(x - y) - (x^2 - xy + y^2)/(x^3 + y^3)`

  `= ((x^3 + y^3) - (x - y) (x^2 - xy + y^2))/((x - y) (x^3 + y^3))`

  `= (x^3 + y^3 - (x^3 - x^2y + xy^2 - x^2y + xy^2 - y^3))/((x - y)(x^3 + y^3))`

  `= (x^3 + y^3 - x^3 + x^2y - xy^2 + x^2y - xy^2 + y^3)/((x - y)(x^3 + y^3))`

  `= (2x^2y - 2xy^2 + 2y^3)/((x - y)(x^3 + y^3))`

  `= (2y(x^2 - xy + y^2))/((x - y)(x + y)(x^2 - xy + y^2))`

  `= (2y)/(x^2 - y^2)`

   উত্তর :`(2y)/(x^2 - y^2)`

প্রদত্ত সমীকরণ

    x + y = 4 .............(i)

    x - y = 2 ...............(ii)

   সমীকরণ (i) হতে পাই,

   y = 4 - x ...........(iii)

   সমীকরণ (iii) হতে প্রাপ্ত y এর মান (ii) এ বসিয়ে পাই,

    x - (4 - x) = 2

  বা, x - 4 + x = 2

  বা, 2x - 4 = 2

  বা, 2x = 2 + 4

  বা,  2x = 6

   বা, x =` 6/2`

   :. x = 3

   x এর মান সমীকরণ (iii) এ বসিয়ে পাই, 

   y = 4 - 3

   :. y = 1

   :. নির্ণেয় সমাধান (x, y) = (3, 1)

প্রদত্ত সমীকরণ

   2x + y = 5...............(i)

   x - y = 1.............(ii)

   সমীকরণ (i) হতে পাই,

    y = 5 - 2x..............(iii)

   সমীকরণ (iii) হতে প্রাপ্ত y এর মান সমীকরণ (ii) এ বসিয়ে পাই,

    x - (5 - 2x) = 1

    x - 5 + 2x = 1

    x + 2x = 1 + 5

    3x = 6

    x = `6/3`

    :. x = 2

    x এর মান সমীকরণ (iii) এ বসিয়ে পাই,

        y = 5 - 2.2

    বা, y = 5 - 4

    বা, y = 1

   :. নির্ণেয় সমাধান (x, y) = (2, 1)

প্রদত্ত সমীকরণ

  3x + 2y = 10............(i)

   x - y = 0................(ii)

   সমীকরণ (ii) হতে পাই,

    x = y

   বা, y = x..............(iii)

   সমীকরণ (iii) হতে প্রাপ্ত y এর মান (i) এ বসিয়ে পাই,

    3x + 2x = 10

   বা, 5x = 10

    বা, x = `(10)/5`

     :. x = 2

    সমীকরণ (iii) এ x এর মান বসিয়ে পাই,

      y = 2

    :. নির্ণেয় সমাধান (x, y) = (2, 2)