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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shorol karo: ka.`(x - y)/(xy) + (y - z)/(yz) + (z - x)/(zx)` kh.`(x - y)/((x + y) (y + z)) + (y - z)/((y + z) (z + x)) + (z - x)/((z + x) (x + y))` ga. ` y/((x - y) (y - z)) + x/((z - x) (x - y)) + z/((y - z) (z - x))` gh.`1/(x + 3y) + 1/(x - 3y) - (2x)/(x^2 - 9y^2)` ঙ.` 1/(x - y) - 2/(2x + y) + 1/(x + y) - 2/(2x - y)` ch. `1/(x - 2) - (x - 2)/(x^2 + 2x + 4) + (6x)/(x^3 + 8)`