Q50 / 125 · BITSAT 2011 · Mathematics

The term independent of \(x\) in the expansion of \(\left(9 \mathrm{x}-\frac{1}{3 \sqrt{\mathrm{x}}}\right)^{18}, \mathrm{x} > 0\), is a times the corresponding binomial coefficient. Then a is

  • A
    3
  • B
    \(1 / 3\)
  • C
    \(-1 / 3\)
  • Correct answer
    D
    None of these

Solution

\(\begin{aligned} \mathrm{T}_{\mathrm{r}+1} & ={ }^{18} \mathrm{C}_{\mathrm{r}}(9 \mathrm{x})^{18-\mathrm{r}}\left(-\frac{1}{3 \sqrt{\mathrm{x}}}\right)^{\mathrm{r}} \\ & =(-\mathrm{r})^{\mathrm{r}}{ }^{18} \mathrm{C}_{\mathrm{r}} 9^{18-\frac{3 \mathrm{r}}{2}} \mathrm{x}^{18-\frac{3 \mathrm{r}}{2}} \end{aligned}\) is independent of \(\mathrm{x}\) provided \(\mathrm{r}=12\) and then \(\mathrm{a}=1\).

Hence, the correct answer is option: 'D'.

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