` (( 4x ) /( 5 ) - ( 5 )/( 2x ) ) ^(9) ` के विस्तार में प्रारम्भ से `6` वां पद ज्ञात कीजिए

Options

A. ` (5040)/(x^2 )`
B. ` (5040)/(x )`
C. `- (5040)/(x )`
D. `- (5040)/(x^2 )`

Answer

C

Explanation

हम जानते है कि ` T _ ( r + 1 ) = ( - 1 ) ^r ""^n C _ r x ^( n - r ) a ^ r `
` therefore ((4x )/(5 ) - (5 ) /( 2x) ) ^(9) ` में ` T _ ( r + 1 ) = ( - 1 ) ^r ""^ 9C _ r ((4x )/ (5 )) ^( 9 - r ) ((5 )/( 2x )) ^ r `
(i) प्रारम्भ से 6 वां पद
` T _ 6 = T _ ( 5 + 1 ) = ( - 1 ) ^ 5 ""^ 9 C _ 5 (( 4x )/(5 ))^( 9 - 5 ) ((5 ) /( 2x ) ) ^ (5 )" " (because r = 5 ) `
` = - (9* 8 * 7 *6 ) /( 1* 2 *3*4) xx ( 4 ^ 4 x^ 4 )/(5 ^ 4 ) xx ( 5 ^ 5 ) /( 2 ^ 5 x ^ 5 ) = - (5040)/(x ) " " ( because ""^9 C _ 5 = ""^9 C_ 4 ) `
(ii) ` (x - a ) ^n ` के विस्तार में अन्त में p वां पद = प्रारम्भ से ` (n - p + 2 ) ` वां पद = ` T _ (n - p + 2 ) `
` therefore (( 4x ) / (5 ) - (5 ) /( 2x ) ) ^9 ` के विस्तार में अन्त से चौथा पद
` = T _ ( 9 - 4 + 2 ) = T _ ( 6 + 1 ) `
` = ( - 1 ) ^( 6) ""^ 9 C _ 6 ((4x ) /( 5 ) ) ^( 9 - 6 ) (( 5 ) /( 2x ) ) ^ 6 " " ( because r = 6 ) `
` = (9*8*7 ) /( 1* 2*3 ) xx ( 4 ^ 3 x ^ 3 ) /( 5 ^ 3 ) xx ( 5 ^ 6 ) /( 2 ^ 6 xx x ^ 6 ) =( 10500)/(x ^3) " " ( because ""^ 9 C _ 6 = ""^9 C _ 3 ) `