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Let $\alpha>0,\ \beta>0$ be such that $\alpha^3+\beta^2=4$. If the maximum value of the term independent of $x$ in the binomial expansion of $\left(\alpha x^{1/9}+\beta x^{-1/6}\right)^{10}$ is $10k$, then $k$ is equal to :

(1) $176$

(2) $336$

(3) $352$

(4) $84$
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Ans. (2) 336

Sol. $T_{r+1}=\ ^{10}\!C_r(\alpha x^{1/9})^{10–r}(\beta x^{-1/6})^r$

$T_{r+1}=\ ^{10}\!C_r\alpha^{10–r}\beta^r(x)^{{10-r\over9}-{r\over6}}$

Term independent of $x$

${10-r\over9}-{r\over6}=0\implies r=4$

$T_5=\ ^{10}\!C_4\alpha^6\beta^4$

Now Let $\alpha^3,\ \beta^2$ are 2 numbers.

$A\geq G$

$\implies{\alpha^3+\beta^2\over2}\geq(\alpha^3\beta^2)^{1/2}$

$\implies\alpha^3\beta^2\leq4$

$\implies\alpha^6\beta^4\leq16$

$\implies{T_5\over\ ^{10}C_4}\leq16$

$\implies T_5\leq16.\ ^{10}C_4$

$\implies T_{5\text{ max}}=16×\ ^{10}C_4=10 K$

$\implies K=336$

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