General Term

Question

In the binomial expansion of (2x^2 - 1/x)^9, the term that is independent of x, if any exists, has which numerical value as its coefficient?

RankGuru · Accepted Answer

The general term of (a + b)^n is T_{r+1} = C(n, r) a^{n-r} b^r, the cornerstone identity for locating any specific term in a JEE Advanced expansion. Here a = 2x^2, b = -1/x and n = 9, so T_{r+1} = C(9, r) (2x^2)^{9-r} (-1/x)^r = C(9, r) 2^{9-r} (-1)^r x^{2(9-r)-r} = C(9, r) 2^{9-r} (-1)^r x^{18-3r}. The term independent of x requires the exponent 18 - 3r = 0, giving r = 6, a valid integer between 0 and 9, so such a term exists. Substituting r = 6 gives C(9, 6) 2^{9-6} (-1)^6 = C(9, 6) · 2^3 · 1 = 84 × 8 × 1 = 672. Since (-1)^6 = 1, the sign is positive, so the coefficient of the constant term is +672. Option -672 attaches a spurious minus sign even though r = 6 is even and (-1)^6 = +1. Option 84 ignores the factor 2^3 from (2x^2)^3. Option 0 wrongly claims no constant term exists, but r = 6 is a valid integer in [0, 9]. Plausibility check: r = 6 lies in [0, 9], confirming a genuine independent term, and the magnitude 672 = C(9,6) · 2^3 = 84 × 8 is consistent with an even power of (-1) keeping the result positive.

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