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JEE Main20236 Apr 2023Evening ShiftMathematicsBinomial TheoremActual

If the coefficients of x 7 in a x 2 + 1 2 b x 11 and x - 7 in a x - 1 3 b x 2 11 are equal, then

Options

  1. A729 a b = 32
  2. B32 a b = 729
  3. C64 a b = 243
  4. D243 a b = 64

Correct answer

A. 729 a b = 32

Step-by-step solution

The coefficient of x 7 in a x 2 + 1 2 b x 11 and a x - 1 3 b x 2 11 are equal, then T r + 1 = 11 C r a x 2 11 - r 1 2 b x r = 11 C r a 11 - r 1 2 b r x 22 - 3 r Now solving 22 - 3 r = 7 ⇒ r = 5 Coefficient of x - 7 in a x - 1 3 b x 2 11 T r + 1 = 11 C r a x 11 - r - 1 3 b x 2 r = 11 C r a 11 - r - 1 3 b r x 11 - 3 r Now solving 11 - 3 r = - 7 ⇒ r = 6 Now equating coefficient of x 7 and x - 7 we get, ∴   11 C 5 a 6 1 2 b 5 = 11 C 6   a 5 - 1 3 b 6 ⇒ 3 6 a b = 32 ⇒ 729 a b = 32

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