Let $\alpha>0$, be the smallest number such that the expansion of $\left(x^{\frac{2}{3}}+\frac{2}{x^{3}}\right)^{30}$ has a term $\beta x^{-\alpha}, \beta \in \mathbb{N}$. Then $\alpha$ is equal to

JEE Main · 2023 · 31 Jan 2023 · Shift 1 · Binomial Theorem

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

Answer: 2

The general term is

(r 30) (x^{2/3})^{30 - r} (2 x^{- 3})^{r} = (r 30) 2^{r} x^{(60 - 11 r) /3}

. A negative power of

x

needs

60 - 11 r < 0

, i.e.

r > \frac{60}{11}

, so the first such term occurs at

r = 6

, where the exponent is

\frac{60 - 66}{3} = - 2

. The coefficient

(6 30) 2^{6}

is a natural number, and no smaller positive

α

is possible, so

α = 2

JEE Main · 2023 · 31 Jan 2023 · Shift 1 · Integer answer