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If the coefficients of x^3 and x^4 in th...

If the coefficients of `x^3` and `x^4` in the expansion of `(1""+a x+b x^2)""(1-2x)^(18)` in powers of x are both zero, then (a, b) is equal to (1) `(16 ,(251)/3)` (3) `(14 ,(251)/3)` (2) `(14 ,(272)/3)` (4) `(16 ,(272)/3)`

A

`(16,251/3)`

B

`(14,251/3)`

C

`(14, 272/3)`

D

`(16,272/3)`

Text Solution

Verified by Experts

The correct Answer is:
D
`(1+ax+bx^(2))(1-2x)^(18)`
`= 1(1-2x)^(18)+ax(1-2x)^(18)+bx^(2)(1-2x)^(18)` ltbr gt Coefficient of `x^(3) : (-2)^(3) .^(18)C_(3) + a(-2)^(2).^(18)C_(2)+b(-2).^(18)C_(1)=0`
`(4xx(17xx16))/((3xx2)) - 2a.(17)/(2)+b = 0" "(1)`
Coefficient of `x^(4) : (-2)^(4).^(18)C_(4)6+a(-2)^(3).^(18)C_(3)+b(-2)^(2).^(18)C_(2) = 0`
`(4 xx 20 ) - 2a . (16)/(3) + b = 0`
From equation (1) and (2) , we get
`4((17xx8)/(3)-20)+2a(16/3-17/2) = 0`
`rArr a = 16`
`rArr b = (2xx16xx16)/(3) - 80 = 272/3`
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