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  • Find the term independent of x in the expansion of(1+x+2x^3)(3/2x^2 – 1/3x)^9

Find the term independent of x in the expansion of(1+x+2x3)(3x2/2 – 1/3x)9

 

Answers (1)

Given: (1+x+2x3)( 3x2/2 – 1/3x)9

Now, let us consider (3x2/2 – 1/3x)9

Tr+1 = 9Cr (3/2 x2)9-r (-1/3x)r

                 = 9Cr (3/2)9-r (-1/3)r x18-3r

Thus, in the expansion of (1+x+2x3)(3/2x2 – 1/3x)9 the general term is:

9Cr (3/2)9-r (-1/3)r x18-3r + 9Cr (3/2)9-r (-1/3)r x19-3r + 2.9Cr (3/2)9-r (-1/3)r x21-3r

Now, for the independent term x,

Let us put 18 – 3r = 0,

19 – 3r = 0

& 21 – 3r = 0,

We get,

R = 6 & r = 7

Now, since the second term is not independent of x, it will be:

9C6 (3/2)9-6 (-1/3)6 + 2.9Cr (3/2)9-7 (-1/3)7

= 9x8x7x6!/6!x3x2 . 1/23.33 – 2. 9x8x7!/7!x2x1. 32/22.1/37

= 84/8.1/33 – 36/4.2/35

= 17/54

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