If in the expansion of \(\Big(\text{x}-\frac{1}{3\text{x}^{3}}\Big)^{9},\) the term independent of x is:

If in the expansion of \(\Big(\text{x}-\frac{1}{3\text{x}^{3}}\Big)^{9},\) the term independent of x is:

  1. None of these.

  2. \(\text{T}_{5}\)

  3. \(\text{T}_{3}\)

  4. \(\text{T}_{4}\)

Solution

Solution:

Suppose Tr+1 is the term in the given expression that is independent of x.

Thus, we have

\(\text{T}_{\text{r}+1}={^\text{9}}\text{C}_{\text{r}}\ \text{x}^{9-\text{r}}\Big(\frac{-1}{3\text{x}^{2}}\Big)^{\text{r}}\)

\(=(-1)^{\text{r}}\ {^\text{9}}\text{C}_{\text{r}}\frac{1}{3^{\text{r}}}\ \text{x}^{9-\text{r}-2\text{r}}\)

For this term to be independent of x, we must have

\(9-3\text{r}=0\)

\(\Rightarrow\text{r}=3\)

Hence, the required term is the 4th term i.e. \(\text{T}_{4}\)

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