If term without x in expansion of left( x ^{frac{2}{3}}+frac{α}{ x ^3}right)^{22} is 7315 , then |?

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7.Binomial Theorem

hard

If the term without $x$ in the expansion of $\left( x ^{\frac{2}{3}}+\frac{\alpha}{ x ^3}\right)^{22}$ is $7315$ , then $|\alpha|$ is equal to $...........$.

A

$2$

B

$1$

C

$4$

D

$6$

(JEE MAIN-2023)

Solution

$T _{ r +1}={ }^{22} C _{ r } \cdot\left( x ^{\frac{2}{3}}\right)^{22- r } \cdot(\alpha)^{ r }, x ^{-3 r }$

$={ }^{22} C _{ r } \cdot x ^{\frac{44}{3}-\frac{2 r }{3}-3 r }(\alpha)^{ r }$

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

$r =4$

${ }^{22} C _4 \cdot \alpha^4=7315$

$\frac{22 \times 21 \times 20 \times 19}{24} \cdot \alpha^4=7315$

$\alpha=1$

Standard 11

Mathematics

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