If the constant term, in binomial expansion of $\left(2 x^{r}+\frac{1}{x^{2}}\right)^{10}$ is 180, then $r$ is equal to ____

JEE Main · 2021 · 22 Jul 2021 · Shift 2 · Binomial Theorem

If the constant term, in binomial expansion of

(2 x^{r} + \frac{1}{x ^{2}})^{10}

is 180, then

r

is equal to ____

Answer: 8

The general term of the expansion is

T_{k + 1} = (k 10) (2 x^{r})^{10 - k} (\frac{1}{x ^{2}})^{k} = (k 10) 2^{10 - k} x^{r (10 - k) - 2 k}

so its numerical coefficient depends on

k

alone. Running

k

from

0

10

, the value

(k 10) 2^{10 - k}

equals

180

only at

k = 8

, where

(8 10) 2^{2} = 45 \times 4 = 180

. That term is constant when its exponent vanishes, so

r (10 - 8) - 16 = 0

, giving

2 r = 16

and

r = 8

JEE Main · 2021 · 22 Jul 2021 · Shift 2 · Integer answer