InterviewSolution
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InterviewSolution
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In the expansion of `(x^3-1/(x^2))^n ,n in N`, if the sum of the coefficients of `x^5a n dx^(10)`, then `n`isa. 25 b. 20 c. 15 d. none of theseA. 5005B. 7200C. -5005D. -7200 |
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Answer» Correct Answer - C `(x^(3)-(1)/(x^(2)))^(n)` General term, `T_(r+1)=.^(n)C_(r)(x^(3))^(n-r).(-(1)/(x^(2)))^(r)` `=.^(n)C_(r).x^((3n-3r)).(-1)^(r).x^(-2x)` `=.^(n)C_(r).(-1)^(r).x^((3n-5r))" "...(i)` For the coefficient `x^(5)` Put 3n - 5r = 5 5r = 3n - 5 `therefore r = (3n)/(5)-1` `therefore "Coefficient of" x^(5)=.^(n)C_(((3n)/(5)-1))(-1)^(((3n)/(5)-1))` For the cefficient of `x^(10)` Put 3n - 5r = 10 5r = 3n - 10 `therefore r = (3n)/(5)-2` `therefore "Coefficient of" x^(10)=.^(n)C_(((3n)/(5)-2))(-1)^(((3n)/(5)-2))` The sum of the coefficient of `x^(5) and x^(10)=0` `rArr .^(n)C_(((3n)/(5)-1))(-1)^(((3n)/(5)-1))+.^(n)C_(((3n)/(5)-2))(-1)^(((3n)/(5)-2))=0` `rArr (-1)^((3n)/(5))[.^(n)C_(((3n)/(5)-1)).(-1)^(-1)+.^(n)C_(((3n)/(5)-2)).(-1)^((-2))]=0` `rArr -.^(n)C_(((3n)/(5)-1))+.^(n)C_(((3n)/(5)-2))=0" "...(ii)` For the independent term, put `3n - 5 r = 0 " "["from eq. (i)"]` `rArr 5r = 3n = 3 xx 15` `5r = 3 xx 3 xx 5` r = 9 Putting the value of r in eq. (i), we get `T_(9+1)=.^(15)C_(9).(-1)^(9).x^((3 xx 15 - 5 xx 9))` `rArr T_(10)=-.^(15)C_(9).x^(0)=-.^(15)C_(9)` `rArr T_(10)=-.^(15)C_(6)" "[because .^(n)C_(r)=.^(n)C_(n-r)]` `=(-15!)/(6!9!)" "[because .^(n)C_(r)=(n!)/(r!(n-r)!)]` `=-5005` |
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