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द्विपद प्रमेय द्वारा विस्तार कीजिये - (i) ` ( 2x - 3y ) ^ 4 ` (ii) ` ( 1 + x + x^ 2) ^ 3 ` (iii) ` (1 - x + x ^ 2 ) ^ 4 ` (iv) ` (x + (1 )/ (y) ) ^(11) ` (v) ` (x ^ 2 - (2 )/ (x) ) ^ 7 ` (vi ) ` (1 + x - x ^ 2 ) ^ 4 ` |
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Answer» (i) द्विपद प्रमेय का प्रयोग करने पर, ` (2x- 3y ) ^ 4 = {2x + ( -3y ) }^ 4 ` ` = ""^4 C _ 0 (2x) ^ 4 ( - 3y ) ^( 0 ) + ""^4C _ 1 ( 2x ) ^ 3 ( - 3y ) + ""^ 4 C _ 2 ( 2x ) ^ 2 ( - 3y ) ^ 2 + ""^ 4 C _ 3 ( 2x ) ^ 1 ( -3y ) ^ 3 + ""^ 4C _ 4 (2x ) ^0 ( -3y ) ^ 4 ` ` = 16 x ^ 4 + 4 ( 8x ^ 3 ) ( - 3 y ) + 6 ( 4x ^ 2 ) ( 9y ^ 2) + 4 ( 2x) ( - 27 y^3 ) + 81 y^ 4 ` ` = 16 x^ 4 - 96 x ^ 3 y + 216 x ^ 2 y ^ 2 - 216 xy ^ 3 + 81 y ^ 4 ` माना ` y = x + x ^ 2 ` तब ` (1+ x + x ^ 2 ) ^ 3 = ( 1 + y ) ^ 3 = ""^3C _ 0 + ""^3C _ 1 y + ""^3 C _ 2 y ^ 2 + ""^ 3C _ 3 y ^ 3 ` ` = 1 + 3y + 3y^ 2 + y ^ 3 ` ` = 1 + 3 ( x + x ^ 2 ) + 3 ( x + x^ 2 ) ^ 2 + ( x + x ^ 2 ) ^ 3 ` ` = 1 + 3 ( x + x ^ 2 ) + 3 ( x ^ 2 + 2x ^ 3 + x ^ 4 ) + {""^ 3 C _ 0 x ^3 ( x ^ 2 ) ^0 + ""^ 3 C _ 1 x ^ ( 3 - 1 ) ( x ^2 ) ^ ( 1 ) + ""^ 3 C _ 2 x ^( 3 - 2 ) ( x ^ 2 ) ^ 2 + ""^ 3 C _3 x ^0 ( x ^ 2 ) ^ 3 } ` ` = 1 + 3 ( x + x^ 2) + 3 ( x^ 2 + 2x ^ 3 + x ^ 4 ) + ( x ^ 3 + 3x^ 4 + 3x ^ 5 + x ^ 6 ) ` ` = x ^ 6 + 3x ^ 5 + 6x ^ 4 + 7 x ^ 3 + 6 x ^ 2 + 3x + 1 ` (iii) माना ` y = - x + x ^ 2 ` , तब ` ( 1 - x + x^ 2 ) ^ 4= ( 1 + y) ^ 4 ` ` = ""^4 C _ 0 + ""^ 4 C _ 1 y ^ 2 + ""^ 4 C _ 3 y ^ 3 + ""^4C _ 4 y ^ 4 ` ` = 1 + 4 y + 6y ^ 2 + 4 y ^ 3 + y ^ 4 ` ` = 1 + 4 ( - x + x ^ 2 ) + 6 ( - x + x ^ 2 ) ^ 2 + 4 ( - x + x ^ 2) ^3 + ( - x + x^ 2 )^ 4 ` `= 1 + 4x + 4x ^ 2 + 6 x ^ 2 ( 1 - 2x + x ^ 2 ) - 4x ^ 3 ( 1 - 3x + 3x^ 2 - x ^ 3 ) + x ^ 4 ( 1 - 4x + 6x ^ 2 - 4x ^ 3 + x ^ 4 ) ` ` = 1 - 4x + 4x ^ 2 + 6x ^ 2 - 12 x ^3 + 6 x ^ 4 - 4 x ^ 3 + 12 x ^ 4 - 12 x ^ 5 + 4 x ^ 6 + x ^ 4 - 4x ^ 5 + 6 x ^ 6 - 4x ^ 7 + x ^ 8 ` ` = 1 - 4x + 10 x ^ 2 - 16 x ^ 3 + 19 x ^ 4 - 16 x ^ 5 + 10 x ^ 6 - 4 x ^ 7 + x ^ 8 ` (iv) द्विपद प्रमेय का प्रयोग करने पर, ` ( x + (1 ) /(y))^(11) = ""^(11) C _ 0 x ^( 11) ((1 ) /( y ))^(0) + ""^(11)C_1 x ^(10) ((1)/(y)) + ""^(11)C_2 x ^(9) ((1 )/( y ))^ 2 + ""^11 C _ 3 x ^ 8 (( 1 )/( y))^ 3 + ""^(11)C _ 4 x ^ 7 ((1 )/(y) ) ^4 + ""^11C _ 5 x ^ 6 ((1 )/( y)) ^ 5 + ""^11C _ 6 x ^ 5 (( 1 ) /( y))^ 6 + ""^ (11)C _ 6 x ^4 (( 1 ) /(y))^ 7 + ""^(11) C _ 8 x ^3 ((1 ) /( y)) ^(8) + ""^(11) C _ 9 x ^ 2 (( 1 )/(y))^9 + ""^(11)C _ (10) x ((1 ) /( y) ) ^(10) + ""^(11)C _(11) ((1)/(y))^(11)` ` = x ^(11) + 11 ( x^(10))/( y ) + 55 (x^9)/( y ^ 2 ) + 165 (x ^8) /( y ^ 3) + 330 ( x^7 ) /( y ^ 4 ) + 462 (x^6) /(y^5) + 462 (x ^ 5 ) /( y ^ 6) + 330 ( x ^ 4 ) /( y ^7) + 165 ( x ^ 3)/(y^8) + 55 (x ^ 2)/ ( y^9) + 11 ( x )/( y^(10)) + (1 ) /( y ^( 11 )) ` (v ) ` ( x ^ 2 - ( 2 ) /( x )) ^ 7 = ""^7 C _ 0 ( x ^ 2 ) ^ 7 (- ( 2 ) /( x ) ) ^0 + ""^7 C _ 1 ( x ^ 2 ) ^ 6 ( - (2 ) / (x)) + ""^7 C _ 2 ( x ^ 2 ) ^ 5 ( - ( 2 ) /( x ) ) ^ 2 + ""^ 7 C _ 3 ( x ^ 2 ) ^ 4 ( - (2 ) / ( x ) ) ^ 3 + ""^7 C _ 4 ( x ^ 2 ) ^3 ( - ( 2 ) /( x) ) ^ 4 + ""^7C _ 5 ( x ^ 2) ^ 2 ( - (2 ) /( x ) ) ^ 5 + ""^7C _ 6 ( x ^ 2 ) ^(1) ( - ( 2 ) /( x ) ) ^(6 ) + ""^7 C _ 7 ( - ( 2 ) /( x ) ) ^(7 ) ` ` = x ^( 14 ) - 7 x ^( 12 ) (2 ) /(x ) + 21 x ^(10) ( 4 ) /( x ^ 2 ) - 35 x ^8 ( 8 ) /( x ^ 3 ) + 35 x ^ 6 (16 ) / ( x ^ 4 ) - 21 x ^ 4 ( 32 ) /(x ^ 5 ) + 7x ^ 2 (64 ) / ( x ^ 6 ) - (128 ) /( x ^( 7 ) ) ` ` = x ^( 14 ) - 14 x ^( 11) + 84 x ^( 8) - 280 x ^ 5 + 560 x ^ 2 - 672 x ^( -1 ) + 448 x ^( - 4) - 128 x ^( - 7 ) ` (vi) माना ` x - x^ 2 = y ` अब ` ( 1 + x - x ^ 2 ) ^ 4 = ( 1 + y ) ^ 4 = ""^4 C _ 0 + ""^ 4 C _ 1 y + ""^ 4 C _ 2 y ^ 2 + ""^ 4 C _ 3 y ^ 3 + ""^ 4 C _ 4 y ^ 4 ` ` = 1 + 4 y + 6y ^ 2 + 6 y ^ 3 + y ^ 4 ` ` = 1 + 4 ( x - x ^ 2 ) + 6 ( x - x ^ 2 ) ^ 2 + 4 ( x - x ^ 2 ) ^ 3 + ( x - x ^ 2 ) ^ 4 ` ` = 1 + 4 ( x - x ^ 2 ) + 6 ( x^ 2 - 2x ^ 3 + x ^ 4 ) + 4 ( x ^ 3 - 3x ^ 4 + 3x ^ 5 - x ^ 6 ) + x ^ 4 ( 1 - x ) ^ 4 ` ` = 1 + 4 ( x - x ^ 2 ) + 6 ( x^ 2 - 2x^ 3 + x ^ 4 ) + 4 ( x ^ 3 - 3x^ 4 + 3x ^ 5 - x ^ 6 ) + x ^ 4 ( 1 - 4 x + 6x ^ 2 - 4x ^ 3 + x ^ 4 ) ` ` = 1 + 4x + 2x ^ 2 - 8 x ^3 - 5x ^ 4 + 8x ^ 5 + 2x ^ 6 - 4x ^7 + x ^8 ` |
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