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JEE Main 2019 · 11 Jan · Shift I

JEE Main 2019 Mathematics question on Binomial Theorem

The sum of the real values of xx for which the middle term in the binomial expansion of (x33+3x)8\left(\frac{x^3}{3} + \frac{3}{x}\right)^8 equals 5670 is
  1. A.
    4
  2. B.
    0
  3. C.
    6
  4. D.
    8
Show answer and solution

Correct option: B

Solution

The expansion has 9 terms, so the middle term is the 5th term (T₅).

T₅ = C(8,4) · (x³/3)⁴ · (3/x)⁴ = 70 · (x¹²/81) · (81/x⁴) = 70x⁸.

Setting 70x⁸ = 5670 gives x⁸ = 81 = 3⁴, so x⁴ = ±9. Since x⁴ must be non-negative, x⁴ = 9, giving x = ±√3.

The sum of real values = √3 + (−√3) = 0.

[JEE Main 2019 · 11 Jan · Shift I]

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