InterviewSolution
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InterviewSolution
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Find the term independent of `x`in the expansion of `(1+x+2x^3)[(3x^2//2)-(1//3)]^9`A. `1//3`B. `19//54`C. `17//54`D. `1//4` |
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Answer» Correct Answer - c We have, `(1 + x + 2x^(3))((3)/(2) x^(2)-(1)/(3x))^(9)` `(1 + x + 2x^(3)){sum_(r=0)^(9)""^(9)C_(r) ((3)/(2) x^(2))^(9-r)(-(1)/(3x))^(r)}` `(1 + x + 2x^(3)){sum_(r=0)^(9)""^(9)C_(r) ((3)/(2))^(9-r)(-(1)/(3))^(r)x^(18 - 3r)}` `= {sum_(r=0)^(9)""^(9)C_(r) ((3)/(2))^(9-r)(-(1)/(3))^(r)x^(18 - 3r)}` `+ {sum_(r=0)^(9)""^(9)C_(r) ((3)/(2))^(9-r)(-(1)/(3))^(r)x^(19 - 3r)}` `+2 {sum_(r=0)^(9)""^(9)C_(r) ((3)/(2))^(9-r)(-(1)/(3))^(r)x^(21 - 3r)}` Clearly, first and third expansions contain term idepen- dent of x and obtained by equation 18 - 3r = 0 and 21 -3r = 0 respectivley. So, coefficient of the term independent of x `""^(9)C_(6) ((3)/(2))^(9-6)(1-(1)/(3))^(6) +2{""^(9)C_(7)((3)/(2))^(9-7) (-(1)/(3))^(7)}` ` (7)/(18) - (2)/(27) = (17)/(54)` . |
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