ক. `(x - y)/(xy) + (y - z)/(yz) + (z - x)/(zx)`

    `= (z(x - y) + x(y - z) + y(z - x))/(xyz)`

    `= (zx - yz + xy - zx + yz - xy)/(xyz)`

    `= 0/(xyz) = 0`

   উত্তর: 0

   খ.`(x - y)/((x + y) (y + z)) + (y - z)/((y + z) (z + x)) + (z - x)/((z + x) (x + y))`

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

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

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

      উত্তর: `(x^2 + y^2 + z^2 - xy - yz - zx)/((x + y)(y + z)(z + x))`

    
   গ. ` y/((x - y) (y - z)) + x/((z - x) (x - y)) + z/((y - z) (z - x))`

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

        `= (yz - xy + xy - zx + zx - yz)/((x - y)(y - z)(z - x))`

        `= 0/((x - y)(y - z)(z - x)) = 0`

          উত্তর: 0

  ঘ.`1/(x + 3y) + 1/(x - 3y) - (2x)/(x^2 - 9y^2)`

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

     `= (2x)/(x^2 - 9y^2) - (2x)/(x^2 - 9y^2)`

       = 0

       উত্তর: 0

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

       `= (1/(x - y) + 1/(x + y)) - (2/(2x + y) , + 2/(2x - y))`

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

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

      `= (2x)/(x^2 - y^2) - (8x)/(4x^2 - y^2)`

      `= (2x(4x^2 - y^2) - 8x(x^2 - y^2))/((x^2 - y^2) (4x^2 - y^2))`

      `= (8x^3 - 2xy^2 - 8x^3 + 8xy^2)/((x^2 - y^2) (4x^2 - y^2))`

      `= (6xy)^2/((x^2 - y^2)(4x^2 - y^2))`

      `= (6xy^2)/((x^2 - y^2)(4x^2 - y^2))`


   চ. `1/(x - 2) - (x - 2)/(x^2 + 2x + 4) + (6x)/(x^3 + 8)`

       `= (1(x^2 + 2x + 4) - (x - 2)^2)/((x - 2) (x^2 + 2x + 4) + 6x/(x^3 + 8)`

       `= (x^2 + 2x + 4 - (x^2 - 2.x.2 + 2^2))/((x - 2)(x^2 + x. 2 + 2^2)) + (6x)/(x^3 + 8)`

      `= (x^2 + 2x + 4 - x^2 + 4x - 4)/(x^3 - 2^3) + (6x)/(x^3 + 8)`

      `= (6x)/(x^3 - 8) + (6x)/(x^3 + 8)`

      `= (6x(x^3 + 8) + 6x(x^3 - 8))/((x^3 - 8)(x^3 + 8))`

      `= (6x^4 + 48x + 6x^4 - 48x)/((x^3)^2 - 8^2) = (12x^4)/(x^6 - 64)`

    উত্তর : `(12x^4)/(x^6 - 64)`


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

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

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

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

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

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

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

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

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

    উত্তর `: (8x^4)/(x^8 - 1)`


   জ. `(x - y)/((y - z) (z - x)) + (y - z)/((z - x) (x - y)) + (z - x)/(x - y)(y - z))`

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

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

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

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

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

    উত্তর: `(2(x^2 + y^2 + z^2 - xy - yz - zx))/((x - y) (y - z) (z - x))`

ঝ.` 1/(a - b - c) + 1/(a - b + c) + a/(a^2 + b^2 - c^2 - 2ab)`

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

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

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

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

     `= (3a - 2b)/(a^2 + b^2 - c^2 - 2ab)`

       উত্তর: ` (3a - 2b)/(a^2 + b^2 - c^2 - 2ab)`


  ঞ. `1/(a^2 + b^2 - c^2 + 2ab) + 1/(b^2 + c^2 - a^2 + 2bc) + 1/(c^2 + a^2 - b^2 + 2ca)`

    `= 1/((a^2 + + 2ab + b^2) - c^2)  + 1/((b^2 + 2bc + c^2) - a^2) `

           `+ 1/((c^2 + 2ca + a^2) - b^2)`

    `= 1/((a + b)^2 - c^2) + 1/((b + c)^2 - a^2) + 1/((c + a)^2 - b^2)`

    `= 1/((a + b + c) (a + b - c)) + 1/((b + c + a) (b + c - a)) `

            `+ 1/((c + a + b) (c + a - b))`

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

  এখন লব, =

      `bc + ab - b^2 + c^2 + ca - bc - ca - a^2 + ab + ca + a^2 - ab + bc + ab `
                       
           `- b^2 - c^2 - ca + bc + ab + ca - a^2 + b^2 + bc - ab - bc - c^2 + ca`

      `= 2ca + 2bc + 2ab - b^2 - c^2 - a^2`

     `= 2ab + 2bc + 2ca - a^2 - b^2 - c^2`

   :. প্রদত্ত রাশি = `(2ab + 2bc + 2ca - a^2 - b^2 - c^2)/((a + b + c) (a + b - c) (b + c - a) (c + a - b))`

    উত্তর: `(2ab + 2bc + 2ca - a^2 - b^2 - c^2)/((a + b + c) (a + b - c) (b + c - a) (c + a - b))`

ক. নির্ণেয় গুণফল

    `= (9x^2y^2)/(7y^2z^2) xx (5b^c^2)/(3z^2x^2) xx (7c^2a^2)/(x^2y^2)`

     `= (9x^2y^2 xx 5b^2c^2 xx 7c^2a^2)/(7y^2z^2 xx 3z^2x^2 xx x^2y^2)`

    ` = (15a^2b^2c^4)/(x^2y^2z^4)`

       উত্তর: `(15a^2b^2c^4)/(x^2y^2z^4)`


  খ. নির্ণেয় গুণফল

   `= (16a^2b^2)/(21z^2) xx (28z^4)/(9x^3y^4) xx (3y^7z)/(10x)`

   `= (16a^2b^2 xx 28z^4 xx 3y^2z)/(21z^2 xx 9x^3y^4 xx 10x)`

   `= (16 xx 28 xx 3 xx a^2b^2y^7z^5)/(21 xx 9 xx 10 xx x^4y^4z^2)`

   `= (32a^2b^2y^3z^3)/(45x^4)` 

    উত্তর:` (32a^2b^2y^3z^3)/(45x^4)` 


 গ. নির্ণেয় গুণফল
   
    `= (yz)/x^2 xx (zx)/y^2 xx (xy)/z^2`

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

      = 1

      উত্তর: 1

 ঘ.   নির্ণেয় গুণফল
   
     `= (x - 1)/(x + 1) xx (x - 1)^2/(x^2 + x) xx x^2/(x^2 - 4x + 5)`

     `= (x - 1)/(x + 1) xx ((x - 1) (x - 1))/(x (x + 1)) xx x^2/(x^2 - 4x + 5)`

     `= (x (x - 1))^3/((x + 1)^2 (x^2 - 4x + 5))`

     উত্তর: `(x (x - 1))^3/((x + 1)^2 (x^2 - 4x + 5))`


 ঙ.  নির্ণেয় গুণফল

  `= (x^4 - y^4)/(x^2 - 2xy + y^2) xx (x - y)/(x^3 + y^3) xx (x + y)/(x^3 + y^3)`

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

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

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

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

 চ.নির্ণেয় গুণফল
 
   `= (1 - b^2)/(1 + x) xx (1 - x^2)/(b + b^2) xx (1 + (1 - x)/x)`

   `= ((1 + b) (1 - b))/(1 + x) xx ((1 + x) (1 - x))/(b (1 + b)) xx (x + 1 - x)/x`

   `= ((1 - b) (1 - x))/b xx 1/x`

   `= ((1 - b) (1 - x))/(bx)`

      উত্তর:` ((1 - b) (1 - x))/(bx)`


 ছ. নির্ণেয় গুণফল

   `= (x^2 - 3x + 2)/(x^2 - 4x + 3) xx (x^2 - 5x + 6)/(x^2 - 7x + 12) xx (x^2 - 16)/(x^2 - 9)`

   `= (x^2 - 2x - x + 2)/(x^2 - 3x - x + 3) xx (x^2 - 3x - 2x + 6)/(x^2 - 4x - 3x + 12) xx (x^2 - 4^2)/(x^2 - 3^2)`

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

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

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

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

     উত্তর: ` ((x - 2)^2 (x + 4))/((x - 3)^2 (x + 3))`

 জ. নির্ণেয় গুণফল

   `= (x^3 + y^3)/(a^2b + ab^2 + b^3) xx (a^3 - b^3)/(x^2 - xy + y^2) xx (ab)/(x + y)`

   `= ((x + y)(x^2 - xy + y^2))/(b(a^2 + ab + b^2)) xx ((a - b) (a^2 + ab + b^2))/(x^2 - xy + y^2) xx (ab)/(x + y)`

   `= a (a - b)`

     উত্তর:` a (a - b)`
 
 ঝ. নির্ণেয় গুণফল

  ` = (x^3 + y^3 + 3xy (x + y))/(a + b)^3 xx (a^3 + b^3 + 3ab (a + b))/(x^2 - y^2) xx ((x - y))^2/((x + y))^2`

   `= (x^3 + y^3 + 3x^2y + 3xy^2)/(a + b)^3 xx (a^3 + b^3 + 3a^2b + 3ab^2)/(x^2 - y^2) xx (x - y)^2/(x + y)^2`

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

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

     = (x - y)

     উত্তর: (x - y)

ক. নির্ণেয় ভাগফল

   `= (3x^2)/(2a) -: (4y^2)/(15zx)`

   `= (3x^2)/(2a) xx (15zx)/(4y^2)`

   `= (45zx^3)/(8ay^2)`

    উত্তর: `(45zx^3)/(8ay^2)`

    

 খ. নির্ণেয় ভাগফল

     `= (9a^2b^2)/(4c^2) -:  (16a^3b)/(3c^3)`

     `= (9a^2b^2)/(4c^2) xx (3c^3)/(16a^3b)`

     `= (27a^2b^2c^3)/(64a^3bc^2)`

     `= (27bc)/(64a)`

    উত্তর: `(27bc)/(64a)`

 গ. নির্ণেয় ভাগফল 

   `= (21a^4b^4c^4)/(4x^3y^3z^3) -: (7a^2b^2c^2)/(12xyz)`

   `= (21a^4b^4c^4)/(4x^3y^3z^3) xx (12xyz)/(7a^2b^2c^2)`

   `= 3a^2b^2c^2 xx 3/(x^2y^2z^2)`

   `= (9a^2b^2c^2)/(x^2y^2z^2)`

       উত্তর:`(9a^2b^2c^2)/(x^2y^2z^2)`

 ঘ. নির্ণেয় ভাগফল 

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

   `= x/y xx y/(x + y)`

   `= x/(x + y)`

   উত্তর: `x/(x + y)`
 ঙ. নির্ণেয় ভাগফল 

     `= (a + b)^2/(a - b)^2 -: (a^2 - b^2)/(a + b)`

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

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

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

      উত্তর: `(a + b)^2/(a - b)^3`

 চ.  নির্ণেয় ভাগফল 

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

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

   `= (x - y)^2`

     উত্তর:`(x - y)^2`

     

 ছ.   নির্ণেয় ভাগফল 

   `= (a^3 + b^3)/(a - b) -: (a^2 - ab + b^2)/(a^2 - b^2)`

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

   `= (a + b)^2`

     উত্তর: `(a + b)^2`

 জ.  নির্ণেয় ভাগফল 

    `= (x^2 - 7x + 12)/(x^2 - 4) -: (x^2 - 16)/(x^2 - 3x + 2)`

    `= (x^2 - 4x - 3x + 12)/(x^2 - 2^2) -: (x^2 - 4^2)/(x^2 - 2x - x + 2)`

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

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

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

      উত্তর: `((x - 3) (x - 1))/((x + 2)(x + 4))`


 ঝ. নির্ণেয় ভাগফল 
   
     ` = (x^2 - x - 30)/(x^2 - 36) -: (x^2 + 13x + 40)/(x^2 + x - 56)`

      `= (x^2 - 6x + 5x - 30)/(x^2 - 6^2) -: (x^2 + 8x + 5x + 40)/(x^2 + 8x - 7x - 56)`

      `= (x(x - 6) + 5(x - 6))/((x + 6)(x - 6)) -: (x(x + 8) + 5(x + 8))/(x(x + 8) - 7(x + 8))`

      `= ((x - 6)(x + 5))/((x + 6) (x - 6)) -: ((x + 8) (x + 5))/((x + 8) (x - 7))`

      `= (x + 5)/(x + 6) xx (x - 7)/(x + 5)`

      `= (x - 7)/(x + 6)`

          উত্তর: `(x - 7)/(x + 6)`

ক. `(1/x + 1/y) xx (1/y - 1/x)`

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

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

      উত্তর: `(x^2 - y^2)/(x^2y^2)`

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

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

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

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

      = `1/x^2`

      উত্তর: `1/x^2`

 গ. `(1 - c/(a + b)) (a/(a + b + c) - a/(a + b + c))`

     `= (a + b - c)/(a + b) xx (a(a + b - c) - a(a + b + c))/((a + b + c) (a + b - c))`

     `= (a + b - c)/(a + b) xx (a^2 + ab - ac - a^2 - ab - ac)/((a + b + c)(a + b - c))`

     `= (- 2ac)/((a + b) (a + b + c))`

      উত্তর: `(- 2ac)/((a + b) (a + b + c))`

 ঘ. `(1/(1 + a) + a/(1 - a)) (1/(1 + a^2) - 1/(1 + a + a^2))`

    `= (1(1- a) + a(1 + a))/((1 + a)(1 - a)) xx (1(1 + a + a^2) - 1(1 + a^2))/((1 + a^2) (1 + a + a^2))`

   `= (1 - a + a + a^2)/((1 + a) (1 - a)) xx (1 + a + a^2 - 1 - a^2)/((1 + a^2) (1 + a + a^2))`

   `= (1 + a^2)/((1 + a) (1 - a)) xx a/((1 + a^2) (1 + a + a^2))`

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

     উত্তর: `a/((1 - a^2) (1 + a + a^2))`

 ঙ.`(x/(2x - y) + x/(2x + y)) (4 + (3y^2)/(x^2 - y^2))`

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

    `= (2x^2 + xy + 2x^2 - xy)/((2x - y)(2x + y)) xx (4x^2 - 4y^2 + 3y^2)/(x^2 - y^2)`

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

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

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

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

      উত্তর: `(4x^2)/(x^2 - y^2)`

 চ.`((2x + y)/(x + y) - 1) -: (1 - y/(x + y))`

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

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

    `= x/(x + y) xx x/(x + y)`

      = 1

       উত্তর: 1

 ছ. `(a/(a + b) + b/(a - b)) -: (a/(a - b) - b/(a + b))`

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

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

     `= (a^2 + b^2)/((a + b)(a - b)) xx ((a - b)(a + b))/(a^2 + b^2) = 1`

       উত্তর: 1

 জ. `((a^2 + b^2)/(2ab) - 1) + ((a^3 - b^3)/(a - b) - 3ab)`

     `= (a^2 + b^2 - 2ab)/(2ab) -: (a^3 - b^3 - 3ab(a - b))/(a - b)`

     `= (a - b)^2/(2ab) -: (a - b)^3/(a - b)`

     `= (a - b)^2/(2ab) xx (a - b)/(a - b)^3`

     `= 1/(2ab)`

      উত্তর: `1/(2ab)`

 ঝ.`((x + y)^2 - 4xy)/((a + b)^2 - 4ab) -: (x^3 - y^3 - 3xy(x - y))/(a^3 - b^3 - 3ab(a - b))`

    `= (x - y)^2/(a - b)^2 -: (x - y)^3/(a - b)^3`

    `= (x - y)^2/(a - b)^2 xx (a - b)^3/(x - y)^3`

    `= (a - b)/(x - y)`

      উত্তর: ` (a - b)/(x - y)`



 ঞ. `(a/b + b/a + 1) -:  (a^2/b^2 + a/b + 1)`

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

      `= (a^2 + ab + b^2)/(ab) xx b^2/(a^2 + ab + b^2)`

      `= b/a` 

        উত্তর:` b/a`