1. Question:যোগ নির্ণয় কর: ৩.ক. `(a - b)/a + (a + b)/b` খ. `a/(bc) + b/(ca) + c/(ab)` গ. `(x - y)/x + (y - z)/y + (z - x)/z` ঘ. `(x + y)/(x - y) + (x - y)/(x + y)` 

    Answer
    ক. `(a - b)/a + (a + b)/b`
    
             `= (b(a - b) + a(a + b))/(ab)`
    
            `= (ab - b^2 + a^2 + ab)/(ab)`
    
            `= (a^2 + 2ab - b^2)/(ab)`
    
        উত্তর: `(a^2 + 2ab - b^2)/(ab)`
    
    
    
       খ. `a/(bc) + b/(ca) + c/(ab)`
    
              `= (a^2 + b^2 + c^2)/(abc)`
    
        উত্তর: `= (a^2 + b^2 + c^2)/(abc)`
    
       গ. `(x - y)/x + (y - z)/y + (z - x)/z`
    
               `= (yz(x - y) + zx(y - z) + xy(z - x))/(xyz)`
    
               `= (xyz - y^2z + xyz - z^2x + xyz - x^2y)/(xyz)`
    
               `= (3xyz - x^2y - y^2z - z^2x)/(xyz)`
    
       উত্তর:  `= (3xyz - x^2y - y^2z - z^2x)/(xyz)`
    
       ঘ.  `(x + y)/(x - y) + (x - y)/(x + y)`
    
             `= ((x + y)^2 + (x - y)^2)/((x - y)(x + y))`
    
            `= (x^2 + 2xy + y^2 + x^2 - 2xy + y^2)/((x + y)(x - y))`
    
             `= (2x^2 + 2y^2)/(x^2 - y^2)`
    
             `= (2(x^2 + y^2))/(x^2 - y^2)`
    
       উত্তর: `= (2(x^2 + y^2))/(x^2 - y^2)`

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  2. Question:যোগ নির্ণয় কর: ৩.ঙ.`1/(x^2 - 3x + 2) + 1/(x^2 - 4x + 3) + 1/(x^2 - 5x + 4)` 

    Answer
    `1/(x^2 - 3x + 2) + 1/(x^2 - 4x + 3) + 1/(x^2 - 5x + 4)`
    
     `= 1/(x^2 - 2x - x + 2) + 1/(x^2 - 3x - x + 3) + 1/(x^2 - 4x - x + 4)`
    
     `= 1/(x(x - 2) - 1(x - 2)) + 1/(x(x - 3) - 1(x - 3)) `
    
                                     `+ 1/(x(x - 4) - 1(x - 4))`
    
     `= 1/((x - 2)(x - 1)) + 1/((x - 3)(x - 1)) + 1/((x - 4)(x - 1))`
    
     `= ((x - 3)(x - 4) + (x - 2)(x - 4) + (x - 2)(x - 3))/((x - 1)(x - 2)(x - 3)(x - 4))`
    
     `= (x^2 - 7x + 12 + x^2 - 6x + 8 + x^2 - 5x + 6)/((x - 1)(x - 2)(x - 3)(x - 4))`
    
      উত্তর: `= (3x^2 - 18x + 26)/((x - 1)(x - 2)(x - 3)(x - 4))`

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  3. Question:যোগ নির্ণয় কর: ৩.চ. `1/(a^2 - b^2) + 1/(a^2 + ab + b^2) + 1/(a^2 - ab + b^2)` 

    Answer
    `1/(a^2 - b^2) + 1/(a^2 + ab + b^2) + 1/(a^2 - ab + b^2)`
    
      `= 1/(a^2 - b^2) + (a^2 - ab + b^2 + a^2 + ab + b^2)/((a^2 + ab + b^2)(a^2 - ab + b^2)`
    
      `= 1/(a^2 - b^2) + (2a^2 + 2b^2)/((a^2 + b^2 + ab)(a^2 + b^2 - ab))`
    
      ` = 1/(a^2 - b^2) + (2a^2 + 2b^2)/((a^2 + b^2)^2 - (ab)^2)`
    
      ` = 1/(a^2 - b^2) + (2a^2 + 2b^2)/(a^4 + 2a^2b^2 + b^4 - a^2b^2)`
    
       `= 1/(a^2 - b^2) + (2a^2 + 2b^2)/(a^4 + a^2b^2 + b^4)`
    
       `= (a^4 + a^2b^2 + b^4 + 2(a^2 + b^2)(a^2 - b^2))/((a^2 - b^2)(a^4 + a^2b^2 + b^4))`
    
        `= (a^4 + a^2b^2 + b^4 + 2(a^4 - b^4))/((a^2 - b^2)(a^4 + a^2b^2 + b^4))`
    
        `= (a^4 + a^2b^2 + b^4 + 2a^4 - 2b^4)/((a^2 - b^2) {(a^2)^2 + a^2b^2 + (b^2)^2})`
    
       `= (3a^4 + a^2b^2 - b^4)/((a^2)^3 - (b^2)^3)`
    
       `= (3a^4 + a^2b^2 - b^4)/(a^6 - b^6)`
    
       উত্তর: `(3a^4 + a^2b^2 - b^4)/(a^6 - b^6)`

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  4. Question:যোগ নির্ণয় কর: ৩.ছ. `1/(x - 2) + 1/(x + 2) + 4/(x^2 - 4)` জ.` 1/(x^2 - 1) + 1/(x^4 - 1) + 4/(x^8 - 1)` 

    Answer
    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)`

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  5. Question:বিয়োগফল নির্ণয় কর: ক. `a/(x - 3) - a^2/(x^2 - 9)` খ. `1/(y(x - y)) - 1/(x(x + y))` গ. `(x + 1)/(1 + x + x^2) - (x - 1)/(1 - x + x^2)` ঘ. `(a^2 + 16b^2)/(a^2 - 16b^2) - (a - 4b)/(a + 4b)` ঙ. `1/(x - y) - (x^2 - xy + y^2)/(x^3 + y^3)` 

    Answer
    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)`

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  6. Question:সরল কর: ক.`(x - y)/(xy) + (y - z)/(yz) + (z - x)/(zx)` খ.`(x - y)/((x + y) (y + z)) + (y - z)/((y + z) (z + x)) + (z - x)/((z + x) (x + y))` গ. ` y/((x - y) (y - z)) + x/((z - x) (x - y)) + z/((y - z) (z - x))` ঘ.`1/(x + 3y) + 1/(x - 3y) - (2x)/(x^2 - 9y^2)` ঙ.` 1/(x - y) - 2/(2x + y) + 1/(x + y) - 2/(2x - y)` চ. `1/(x - 2) - (x - 2)/(x^2 + 2x + 4) + (6x)/(x^3 + 8)` 

    Answer
    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))`

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  7. Question:সরল কর: ৫. ঝ.` 1/(a - b - c) + 1/(a - b + c) + a/(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)` 

    Answer
    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))`

    1. Report
  8. Question:গুণ কর: ক. `(9x^2y^2)/(7y^2z^2), (5b^c^2)/(3z^2x^2)` এবং `(7c^2a^2)/(x^2y^2)` খ. `(16a^2b^2)/(21z^2), (28z^4)/(9x^3y^4)` এবং `(3y^7z)/(10x)` গ. `(yz)/x^2, (zx)/y^2,` এবং `(xy)/z^2` ঘ. `(x - 1)/(x + 1), (x - 1)^2/(x^2 + x)` এবং `x^2/(x^2 - 4x + 5)` ঙ. `(x^4 - y^4)/(x^2 - 2xy + y^2), (x - y)/(x^3 + y^3)` এবং `(x + y)/(x^3 + y^3)` চ.`(1 - b^2)/(1 + x),(1 - x^2)/(b + b^2)` এবং `(1 + (1 - x)/x)` ছ.`(x^2 - 3x + 2)/(x^2 - 4x + 3), (x^2 - 5x + 6)/(x^2 - 7x + 12)` এবং `(x^2 - 16)/(x^2 - 9)` জ.`(x^3 + y^3)/(a^2b + ab^2 + b^3), (a^3 - b^3)/(x^2 - xy + y^2)` এবং `(ab)/(x + y)` ঝ.` (x^3 + y^3 + 3xy (x + y))/(a + b)^3, (a^3 + b^3 + 3ab (a + b))/(x^2 - y^2)` এবং `(x - y)^2/(x + y)^2` 

    Answer
    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)

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  9. Question:ভাগ কর (১ম রাশিকে ২য় রাশি দ্বারা): ক.`(3x^2)/(2a), (4y^2)/(15zx)` খ. `(9a^2b^2)/(4c^2), (16a^3b)/(3c^3)` গ. `(21a^4b^4c^4)/(4x^3y^3z^3), (7a^2b^2c^2)/(12xyz)` ঘ. `x/y, (x + y)/y` ঙ. `(a + b)^2/(a - b)^2, (a^2 - b^2)/(a + b)` চ. `(x^3 - y^3)/(x + y), (x^2 + xy + y^2)/(x^2 - y^2)` ছ. `(a^3 + b^3)/(a - b), (a^2 - ab + b^2)/(a^2 - b^2)` জ. `(x^2 - 7x + 12)/(x^2 - 4), (x^2 - 16)/(x^2 - 3x + 2)` ঝ.` (x^2 - x - 30)/(x^2 - 36), (x^2 + 13x + 40)/(x^2 + x - 56)` 

    Answer
    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)`

    1. Report
  10. Question:সরল কর: ক. `(1/x + 1/y) xx (1/y - 1/x)` খ. `(1/(1 + x) + (2x)/(1 - x^2)) (1/x - 1/x^2)` গ. `(1 - c/(a + b)) (a/(a + b + c) - a/(a + b + c))` ঘ. `(1/(1 + a) + a/(1 - a)) (1/(1 + a^2) - 1/(1 + a + a^2))` ঙ.`(x/(2x - y) + x/(2x + y)) (4 + (3y^2)/(x^2 - y^2))` চ.`((2x + y)/(x + y) - 1) -: (1 - y/(x + y))` ছ. `(a/(a + b) + b/(a - b)) -: (a/(a - b) - b/(a + b))` জ. `((a^2 + b^2)/(2ab) - 1) -: ((a^3 - b^3)/(a - b) - 3ab)` ঝ.`((x + y)^2 - 4xy)/((a + b)^2 - 4ab) -: (x^3 - y^3 - 3xy(x - y))/(a^3 - b^3 - 3ab(a - b))` ঞ. `(a/b + b/a + 1) -: (a^2/b^2 + a/b + 1)` 

    Answer
    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`

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