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JEE Main202528 Jan 2025Evening ShiftMathematicsBinomial TheoremActual

Let the coefficients of three consecutive terms T_r, T_ r+1 and T_ r+2 in the binomial expansion of (a+b)¹² be in a G.P. and let p be the number of all possible values of r . Let q be the sum of all rational terms in the binomial expansion of ( [4] 3 + [3] 4 )¹² . Then p + q is equal to :

Options

  1. A283
  2. B287
  3. C295
  4. D299

Correct answer

A. 283

Step-by-step solution

Coefficient of aligned & T_r, T_ r+1 , T_ r+2 G P & ( ¹² C_r )^2= 12 r-1 12 r+1 aligned ( ¹² C_r )^2= 12 r-1 12 r+1 but no three consecutive binomial coefficient are in GP P=0 Now for (3^ 1 / 4 +4^ 1 / 3 )¹², T_ r+1 = ¹² C_r(4)^ K / 3 (3)^ 12-K 4 for rational terms K =0,12 sum of rational terms aligned & = ¹² C ₀ 4^0 3^3+ ¹² C ₁₂ 4^4 3^0 & =27+256=283=q & p+q=283 aligned

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