If the term without x in the expansion of x 2 3 + α x 3 22 is 7315   , then α is equal to _____ .

Question

If the term without x in the expansion of x 2 3 + &#945; x 3 22 is 7315 &#160; , then &#945; is equal to _____ .

Aicharya · Accepted Answer

General term in the binomial expansion of ${(x^{\frac{2}{3}} + \frac{α}{x^{3}})}^{22}$ is

$T_{r + 1} = {}^{22}C_{r} {(x^{\frac{2}{3}})}^{22 - r} {(\frac{α}{x^{3}})}^{r}$

$\Rightarrow T_{r + 1} = {}^{22}C_{r} {(x)}^{\frac{44 - 11 r}{3}} α^{r}$

For term independent of $x$ ,

$\frac{44 - 11 r}{3} = 0$

$\Rightarrow r = 4$

So,

$T_{5} = {}^{22}C_{4} α^{4} = 7315$

$\Rightarrow (\frac{22!}{4! \times 18!}) α^{4} = 7315$

$\Rightarrow 7315 α^{4} = 7315$

$\Rightarrow α^{4} = 1$

$\Rightarrow |α| = 1$

If the term without x in the expansion of x 2 3 + α x 3 22 is 7315   , then α is equal to…