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  • The term independent of x in expansion of <img alt="\left ( \frac{x+1}{x^{2/3}-x^{1/3}+1} -\frac{x-1}{x-x^{1/2}}\right )^{10}" src="http://entrancecorner.oncodecogs.com/gif.latex?%5Cdpi%7B100%7D%20

The term independent of x in expansion of \left ( \frac{x+1}{x^{2/3}-x^{1/3}+1} -\frac{x-1}{x-x^{1/2}}\right )^{10} is :

Option: 1

310


Option: 2

4


Option: 3

120


Option: 4

210


Answers (1)

best_answer

As we have learned

General Term in the expansion of (x+a)^n -

T_{r+1}= ^{n}c_{r}\cdot x^{n-r}\cdot a^{r}
 

 

Now,S = \left ( \frac{\left ( x^{1/3}+1 \right )(x^{2/3}- x^{1/3}+1)}{(x^{2/3}- x^{1/3}+1)} - \frac{(x^{1/2}- 1)(x^{1/2}+ 1)}{x^{1/2}(x^{1/2}- 1)}\right )^{10}

= \left ( (x^{1/3}+ 1) - (1+ x^{-1/2 })\right )^{10 }

= \left ( x^{1/3}- x^{-1/2 }\right )^{10 }

T_{r+1} = ^{10}C_r (x^{1/3})^{10 - r } (-x ^{-1/2 })^r

=(-1)^r. ^{10} C_r . x^{\frac{10-r}{3} - \frac{r}{2}}

For term independent of x,

\frac{10-r }{3}-\frac{r}{2}= 0 \Rightarrow 5r = 20 \\\Rightarrow r = 4

So , \: \: T_{r+1} = ^{10}C_4 = 210

 

 

 

 

 

 

 

 

 

Posted by

Gaurav

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