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Question

# The coefficient of the term independent of x in the expansion of (1+x+2x3)(32x2âˆ’13x)9

A
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C
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Solution

## The correct option is C The general term in the expansion of (32x2−13x)9 is T(r+1)=9Cr(32x2)9−r(−13x)r=9Cr(32)9−r(−13)x18−3r Now, the coefficient of the term independent of x in the expansion of (1+x+2x3)(32x2−13x)9 = Sum of the coefficient of the terms x0,x−1 and x−3 in (32x2−13x)9 .....(ii) For x0 in (i) above, 18–3r=0⇒r=6.x−1 in (i) above, there exists no value of r and hence no such term exits. For x−3 in (i), 18–3r=–3⇒r=7 ∴ For term independent of x, in (ii) the coefficient =1×9C6(−1)6(32)9−6(13)6+2×9C7(−1)7(32)9−7(13)7 =9.8.71.2.3.3323.136+29.81.2(−1)3222.137=718−227=1754

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