The sum of the real values of x for which the middle term in the binomial expansion of $\left(\frac{x^{3}}{3}+\frac{3}{x}\right)^{8}$ equals 5670 is:

JEE Main · 2019 · 11 Jan 2019 · Shift 1 · Binomial Theorem

The sum of the real values of x for which the middle term in the binomial expansion of

(\frac{x ^{3}}{3} + \frac{3}{x})^{8}

equals 5670 is:

Answer: (A)

The middle term of the eight-power expansion is the fifth,

T_{5} = (4 8) (\frac{x ^{3}}{3})^{4} (\frac{3}{x})^{4} = (4 8) x^{8}

, because the powers of

3

cancel. With

(4 8) = 70

, setting

70 x^{8} = 5670

gives

x^{8} = 81

. Over the reals this factors as

(x^{2} - 3) (x^{2} + 3) (x^{4} + 9) = 0

, whose only real roots are

x = \pm 3

, and their sum is

0

JEE Main · 2019 · 11 Jan 2019 · Shift 1 · Single correct option