First slide

Binomial theorem for positive integral Index

Question

$The term independent of x (x > 0, x \neq 1) in the expansion of$ ${[\frac{(x + 1)}{(x^{\frac{2}{3}} - x^{\frac{1}{3}} + 1)} - \frac{(x - 1)}{(x - \sqrt{x})}]}^{10}$

Moderate

A

105
B

210
C

315
D

420

Solution

$\begin{array}{l} {[\frac{(x + 1)}{(x^{\frac{2}{3}} - x^{\frac{1}{3}} + 1)} - \frac{(x - 1)}{(x - \sqrt{x})}]}^{10} = {[(x^{\frac{1}{3}} + 1) - \frac{\sqrt{x} + 1)}{\sqrt{x}}]}^{10} \\ = {(x^{\frac{1}{3}} + 1 - 1 - \frac{1}{\sqrt{x}})}^{10} = {(x^{\frac{1}{3}} - \frac{1}{\sqrt{x}})}^{10} \\ If T_{r + 1} is the term independent of x then r = \frac{n p}{p + q} = \frac{10 (\frac{1}{3})}{\frac{1}{3} + \frac{1}{2}} = 4 \\ ∴ required t e r m = T_{5} = {}^{10}C_{4} = 210 \end{array}$

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