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Question

If the coefficient of x3 and x4 in the expansion of (1+ax+bx2)(12x)18 in powers of x are both zero, then (a,b) is equal to

A
(16,2513)
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B
(14,2513)
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C
(14,2723)
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D
(16,2723)
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Solution

The correct option is D (16,2723)
Given expression is (1+ax+bx2)(12x)18

=(1+ax+bx2)(118C1(2x)+18C2(2x)218C3(2x)3+....18C18(2x)18)

Now, coefficient of x3

=18C1(2b)+a×18C2(4)18C3(8)=0

34a=b+34×163...(i)

Similarly, a coefficient of x4

18C4(4)18C3(2a)+18C2(b)=0...(ii)

From (i) and (ii), a=16 and b=2723

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