Prove that there is no term involving $x^{6}$ in the expansion of ${(2 x^{2} - \frac{3}{x})}^{11}$ , where $r \neq 0$ .

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Solution

Suppose $x^{6}$ occurs in $(r + 1)^{t h}$ term in the expansion of $(2 x^{2} - \frac{3}{x})^{11}$ .

Now, $T_{r + 1} = 11_{C_{r}} (2 x^{2})^{11 - r} (- \frac{3}{x})^{r}$

$= 11_{C_{r}} (- 1)^{r} 2^{11 - r} 3^{r} x^{22 - 3 r}$

For this term to contain $x^{6}$ ,

$22 - 3 r = 6$

$r = \frac{16}{3}$ , which is a fraction.

But, r is a natural number. Hence, there is no term containing $x^{6}$ .

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