The term independent of ' x ' in the expansion of ( x + 1 x 2 / 3 - x 1 / 3 + 1 - x - 1 x - x 1 / 2 ) 10 , where x 0 ,1 is equal to

Question

The term independent of $' x'$ in the expansion of ${(\frac{x + 1}{x^{2 / 3} - x^{1 / 3} + 1} - \frac{x - 1}{x - x^{1 / 2}})}^{10}$ , where $x \neq 0$ ,1 is equal to ________.

Smart Achievers · Accepted Answer

(210)

$The term independent of x in the expansion of$

${(\frac{x + 1}{x^{2 / 3} - x^{1 / 3} + 1} - \frac{x - 1}{x - x^{1 / 2}})}^{10}$

$= {[(x^{1 / 3} + 1) - \frac{(x^{1 / 2} + 1)}{x^{1 / 2}}]}^{10} = {(x^{1 / 3} - x^{- 1 / 2})}^{10}$

$T_{r + 1} = {}^{10}C_{r} {(x^{1 / 3})}^{r} \cdot {(- x^{- 1 / 2})}^{10 - r}$

$= {}^{10}C_{r} {(- 1)}^{10 - r} \cdot x^{r / 3 + r / 2 - 5}$

$For this term to be independent,$ $\frac{r}{3} + \frac{r}{2} = 5$

$\Rightarrow r = 6$

$∴$ $Term independent is T_{7} = {}^{10}C_{6} = 210$

Solution

Q.
The term independent of $' x'$ in the expansion of ${(\frac{x + 1}{x^{2 / 3} - x^{1 / 3} + 1} - \frac{x - 1}{x - x^{1 / 2}})}^{10}$ , where $x \neq 0$ ,1 is equal to ________.

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Solution

Q. The term independent of 'x' in the expansion of (x+1x2/3-x1/3+1-x-1x-x1/2)10, where x≠0,1 is equal to ________.

Ans. (210) The term independent of x in the expansion of (x+1x2/3-x1/3+1-x-1x-x1/2)10 =[(x1/3+1)-(x1/2+1)x1/2]10=(x1/3-x-1/2)10 Tr+1=Cr10(x1/3)r·(-x-1/2)10-r =Cr10(-1)10-r·x r/3+r/2-5 For this term to be independent, r3+r2=5 ⇒ r=6 ∴ Term independent is T7=C610=210

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Q.
The term independent of $' x'$ in the expansion of ${(\frac{x + 1}{x^{2 / 3} - x^{1 / 3} + 1} - \frac{x - 1}{x - x^{1 / 2}})}^{10}$ , where $x \neq 0$ ,1 is equal to ________.