If the fourth term in the expansion of $\left(x+x^{\log_{2} x}\right)^{7}$ is $4480$, then the value of $x$ where $x \in \mathbb{N}$ is equal to:

JEE Main · 2021 · 17 Mar 2021 · Shift 1 · Binomial Theorem

If the fourth term in the expansion of

(x + x^{l o g_{2} x})^{7}

4480

, then the value of

x

where

x \in N

is equal to:

Answer: (B)

Put

t = lo g_{2} x

, so that

x = 2^{t}

. The fourth term of the expansion is

T_{4} = (3 7) x^{4} (x^{l o g_{2} x})^{3} = 35 x^{4 + 3 t} = 35 \cdot 2^{t (4 + 3 t)} .

Setting this equal to

4480

gives

2^{3 t^{2} + 4 t} = 128 = 2^{7}

, so

3 t^{2} + 4 t - 7 = 0

, that is

(t - 1) (3 t + 7) = 0

. The roots

t = 1

and

t = - \frac{7}{3}

give

x = 2

and

x = 2^{- 7/3}

, and only the first is a natural number. Hence

x = 2

JEE Main · 2021 · 17 Mar 2021 · Shift 1 · Single correct option