বীজগণিতীয় ভগ্নাংশ, গণিত

বীজগণিতীয় ভগ্নাংশ

Question:সরল কর:

 ক. `(x^2 + 2x - 15)/(x^2 + x - 12) -: (x^2 - 25)/(x^2 - x - 20) xx (x - 2)/(x^2 - 5x + 6)`

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

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

 গ. `(x^2 + 2x - 3)/(x^2 + x - 2) -: (x^2 + x - 6)/(x^2 - 4)`

 ঘ. `(a^4 - b^4)/(a^2 + b^2 - 2ab) xx ((a + b)^2 - 4ab)/(a^3 - b^3) `

                                                     `-:  (a + b)/(a^2 + ab + b^2)` 
Answerক. `(x^2 + 2x - 15)/(x^2 + x - 12) -: (x^2 - 25)/(x^2 - x - 20) xx (x - 2)/(x^2 - 5x + 6)`

   `= (x^2 + 5x - 3x - 15)/(x^2 + 4x - 3x - 12) -: (x^2 - 5^2)/(x^2 - 5x + 4x - 20) xx (x - 2)/(x^2 - 3x - 2x + 6)`

   `= (x(x + 5) - 3(x + 5))/(x(x + 4) - 3(x + 4) - 3(x + 4)) -: ((x + 5)(x - 5))/(x(x - 5) + 4(x - 5)) `

                                                                                            `xx (x - 2)/(x(x - 3) - 2(x - 3))`

 `= ((x + 5)(x - 3))/((x + 4) (x - 3)) -: ((x + 5)(x - 5))/((x - 5)(x + 4)) xx (x - 2)/((x - 3) (x - 2))`

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

  `= 1/(x - 3)`

     উত্তর: `1/(x - 3)`

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

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

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

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

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

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

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

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

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

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

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

     উত্তর: `(3x^2 + y^2)/(2xy)`

 গ. `(x^2 + 2x - 3)/(x^2 + x - 2) -: (x^2 + x - 6)/(x^2 - 4)`

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

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

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

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

     `= (x + 3)/(x + 2) xx (x + 2)/(x + 3)`

       = 1

         উত্তর: 1

 ঘ. `(a^4 - b^4)/(a^2 + b^2 - 2ab) xx ((a + b)^2 - 4ab)/(a^3 - b^3) -:  (a + b)/(a^2 + ab + b^2)`

    `= ((a^2 + b^2)(a^2 - b^2))/(a^2 - 2ab + b^2) xx (a^2 + 2ab + b^2 - 4ab)/((a - b)(a^2 + ab + b^2) -: (a + b)/(a^2 + ab + b^2)`

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

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

   `= a^2 + b^2`

     উত্তর: ` a^2 + b^2`

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Question:`(a^4 - b^4)/(a^2 - 2ab + b^2), (a - b)/(a^3 - b^3), (a + b)/(a^3 + b^3)` তিনটি বীজগাণিতিক রাশি।

  ক. ১ম রাশিকে লঘিষ্ট আকারে প্রকাশ কর।

  খ. দেখাও যে, রাশি তিনটির গুণফল `(a^2 + b^2)/(a^2 - ab + b^2)^2`

  গ. ১ম রাশিকে `(a^3 + a^2b + ab^2 + b^3)/((a + b)^2 - 4ab)` দ্বারা ভাগ করে 

     ভাগফলের সাথে `a^2/(a + b)` যোগ কর। 
Answerক. ১ম রাশি `= (a^4 - b^4)/(a^2 - 2ab + b^2)`

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

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

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

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

   নির্ণেয় লঘিষ্ট রুপ: `((a + b) (a^2 + b^2))/(a - b)` (Ans)

   খ. নির্ণেয় গুণফল 

           `= (a^4 - b^4)/(a^2 - 2ab + b^2) xx (a - b)/(a^3 + b^3) xx (a + b)/(a^3 + b^3)`

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

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

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

   গ. `{(a^4 - b^4)/(a^2 - 2ab + b^2) -: (a^3 + a^2b + ab^2 + b^3)/((a + b)^2 - 4ab)} + a^2/(a + b)`

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

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

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

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

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

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

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

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Question:`A = x^2 - 5x + 6, B = x^2 - 7x + 12`

 `C = x^2 - 9x + 20` তিনটি বীজগাণিতিক রাশি।

  ক. `x/y` থেকে `(x + y)/y` এর বিয়োগ কর।

  খ. `1/B + 1/C` কে লঘিষ্ট আকারে প্রকাশ কর।

  গ. `1/A, 1/B, 1/C` কে সাধারণ হর বিশিষ্ট ভগ্নাংশে প্রকাশ কর। 
Answerক. নির্ণেয় বিয়োগফল `= x/y - (x + y)/y`

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

                       `= (-y)/y`

                         = - 1

              উত্তর: - 1

  খ. দেওয়া আছে, 

          `B = x^2 - 7x + 12` এবং `C = x^2 - 9x + 20`

        `:. 1/B + 1/C = 1/(x^2 - 7x + 12) + 1/(x^2 - 9x + 20)`

        `= 1/(x^2 - 4x - 3x + 12) +  1/(x^2 - 5x - 4x + 20)`

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

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

        `= (x - 5 + x - 3)/((x - 3)(x - 4)(x - 5))`

        `= (2x - 8)/((x - 3)(x - 4)(x - 5))`  (Ans)

  গ. `1/A = 1/(x^2 - 5x + 6)`

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

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

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

    `1/B = 1/(x^2 - 7x + 12)`

          `= 1/(x^2 - 4x - 3x + 12)`

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

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

    এবং `1/C = 1/(x^2 - 9x + 20)`

              `= 1/(x^2 - 4x - 5x + 20)`

              `= 1/(x(x - 4) - 5(x - 4))`

              `= 1/((x - 4) (x - 5))`

         প্রদত্ত ভগ্নাংশগুলোর হর `x^2 - 5x + 6, x^2 - 7x + 12` ও `x^2 - 9x + 20` 

        এর ল,সা,গু হলো (x - 2) (x - 3) (x - 4) (x - 5)

        এখন, হরগুলোর ল,সা,গু/প্রথম ভগ্নাংশের হর

         `= ((x - 2) (x - 3) (x - 4) (x - 5))/((x - 2) (x - 3))`

         `= (x - 4) (x - 5)`

      `:. 1/A = (1 xx (x - 4) (x - 5))/((x - 2)(x - 3)(x - 4) (x - 5))`

         `= ((x - 4)(x - 5))/((x - 2) (x - 3) (x - 4) (x - 5))`

           হরগুলোর ল,সা,গু/দ্বিতীয় ভগ্নাংশের হর

        `= ((x - 2) (x - 3) (x - 4) (x - 5))/((x - 3) (x - 4))`

        `= (x - 2) (x - 5)`

     `:. 1/B = (1 xx (x - 2) (x - 5))/((x - 3) (x - 4) (x - 2) (x - 5))`

        `= ((x - 2) (x - 5))/((x - 2) (x - 3) (x - 4) (x - 5))`

      এবং হরগুলোর ল,সা,গু/তৃতীয় ভগ্নাংশের হর

          `= ((x - 2)(x - 3)(x - 4)(x - 5))/((x - 4) (x - 5))`

          `= (x - 2) (x - 3)`

      `:. 1/C = (1 xx (x - 2) (x - 3))/((x - 4)(x - 5) xx (x - 2) (x - 3))`

               `= ((x - 2) (x - 3))/((x - 2) (x - 3)(x - 4)(x - 5))`

         :. নির্ণেয় সাধারণ হরবিশিষ্ট ভগ্নাংশগুলো হলো:

          `((x - 4) (x - 5))/((x - 2) (x - 3) (x - 4) (x - 5))`

          `((x - 2) (x - 5))/((x - 2) (x - 3) (x - 4) (x - 5))`

          `((x - 2) (x - 3))/((x - 2) (x - 3)(x - 4)(x - 5))` (Ans)

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Question:`A = x - 2, B = x^2 + 2x + 4, C = x^2 - 8` তিনটি বীজগাণিতিক রাশি।

 ক. যোগফল নির্ণয় কর: `a/(bc) + b/(ca) + c/(ab) + (a - b)/(ac)`

 খ. সরল কর: `1/A xx (x - 2)/B + (6x)/C`

 গ. প্রমাণ কর যে, `1/A xx (x + 2) -: (x + 2)/C = 1` 
Answerক. নির্ণেয় যোগফল 

       `= a/(bc) + b/(ca) + c/(ab) + (a - b)/(ac)`

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

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

       `= (a^2 + ab + c^2)/(abc)` (Ans)

 খ. দেওয়া আছে,

              A = x - 2

              B =` x^2 + 2x + 4`

              C =` x^3 - 8`

                 =` x^3 - 2^3`

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

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

  প্রদত্ত রাশি,`= 1/A xx (x - 2)/B + (6x)/C`

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

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

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

                =` (7x - 2)/(x^3 - 8)` (Ans)

  গ. বামপক্ষ `= 1/A xx (x + 2)/B -: (x + 2)/C`

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

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

               `= (x^3 - 8)/(x^3 - 2^3)`

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

                 = 1

                 = ডানপক্ষ

             `:. 1/A xx (x + 2)/B -: (x + 2)/C = 1`  (প্রমাণিত)

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Question:`A = (x^2 + 3x - 4)/(x^2 + 7x + 12), B = (x^2 + 2x - 3)/(x^2 + 6x - 7),`

`C = (x^2 + 12x + 35)/(x^2 + 4x - 5)` তিনটি বীজগাণিতিক রাশি।

 ক.  A কে লঘিষ্ট আকারে প্রকাশ করো।

 খ.  A + B কে সরল করো।

 গ. দেখাও যে, `B xx C -: (x^2 - 9)/(x - 1) = 1/(x - 3)` 
Answerক.`A = (x^2 + 3x - 4)/(x^2 + 7x + 12)`

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

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

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

       `= (x - 1)/(x + 3)`  (Ans)

  খ. `A + B = (x^2 + 3x - 4)/(x^2 + 7x + 12) + (x^2 + 2x - 3)/(x^2 + 6x - 7)`

      `= (x - 1)/(x + 3) + (x^2 + 3x - x - 3)/(x^2 + 7x - x - 7)` [ক থেকে]

      `= (x - 1)/(x + 3) + (x(x + 3) - 1(x + 3))/(x(x + 7) - 1(x + 7))`

      `= (x - 1)/(x + 3) + ((x + 3) (x - 1))/((x + 7)(x - 1))`

      `= (x - 1)/(x + 3) + (x + 3)/(x + 7)`

      `= ((x - 1)(x + 7) + (x + 3)^2)/((x + 3) (x + 7))`

      `= (x^2 + 7x - x - 7 + x^2 + 6x + 9)/(x^2 + (3 + 7)x + 3.7)`

      `= (x^2 + 6x - 7 + x^2 + 6x + 9)/(x^2 + 10x + 21)`

      `= (2x^2 + 12x + 2)/(x^2 + 10x + 21)`  (Ans)

 গ. বামপক্ষ` = B xx C -: (x^2 - 9)/(x - 1)`

    `= (x^2 + 2x - 3)/(x^2 + 6x - 7) xx (x^2 + 12x + 35)/(x^2 + 4x - 5) -: (x^2 - 9)/(x - 1)`

    `= (x^2 + 2x - 2)/(x^2 + 7x - x - 7) xx (x^2 + 7x + 5x + 35)/(x^2 + 5x - x - 5) -: (x - 1)/(x^2 - 9)`

    `= (x^2 + 2x - 2)/(x(x + 7) - 1(x - 7)) xx (x(x + 7) + 5(x + 7))/(x(x + 5) - 1(x + 5)) xx (x - 1)/(x^2 - 3^2)`

    `= (x^2 + 2x - 2)/((x + 7) (x - 1)) xx ((x + 7)(x + 5))/((x + 5)(x - 1)) xx (x - 1)/((x + 3)(x - 3))`

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

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

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

    `= 1/(x - 3)`

      = ডানপক্ষ

    `:. B xx C -: (x^2 - 9)/(x - 1) `

            `= 1/(x - 3)` (প্রমাণিত)

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