Let 0 , be the smallest number such that the expansion of ( x 2 3 + 2 x 3 ) 30 has a term x - N . Then is equal to ________ . [2023]

Question

Let $α > 0$ , be the smallest number such that the expansion of ${(x^{\frac{2}{3}} + \frac{2}{x^{3}})}^{30}$ has a term $β x^{- α}, β \in N .$ Then $α$ is equal to ________ . [2023]

Smart Achievers · Accepted Answer

(2)

$Here, T_{r + 1} = {}^{30}C_{r} {(x^{2 / 3})}^{30 - r} {(\frac{2}{x^{3}})}^{r}$

$= {}^{30}C_{r} x^{(20 - \frac{2 r}{3} - 3 r)} {(2)}^{r} = {}^{30}C_{r} {(2)}^{r} x^{(\frac{60 - 11 r}{3})}; 0 \leq r \leq 30$

$For β x^{- α}, α > 0, r = 6, and T_{7} = {}^{30}C_{6} {(2)}^{6} x^{- 2} \Rightarrow α = 2$

Solution

Q.
Let $α > 0$ , be the smallest number such that the expansion of ${(x^{\frac{2}{3}} + \frac{2}{x^{3}})}^{30}$ has a term $β x^{- α}, β \in N .$ Then $α$ is equal to ________ . [2023]

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Solution

Q. Let α>0, be the smallest number such that the expansion of (x23+2x3)30 has a term βx-α,β∈N. Then α is equal to ________ . [2023]

Ans. (2) Here, Tr+1=Cr30(x2/3)30-r(2x3)r =Cr30 x(20-2r3-3r)(2)r=Cr30(2)rx(60-11r3); 0≤r≤30 For βx-α, α>0, r=6, and T7=C630(2)6x-2⇒α=2

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Q.
Let $α > 0$ , be the smallest number such that the expansion of ${(x^{\frac{2}{3}} + \frac{2}{x^{3}})}^{30}$ has a term $β x^{- α}, β \in N .$ Then $α$ is equal to ________ . [2023]