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JEE Main20253 Apr 2025Morning ShiftMathematicsBinomial TheoremActual

If _ r =1 ^9 ( r +3 2^ r ) . ^9 C _ r = ( 3 2 )^9- , , N , then ( + )^2 is equal to

Options

  1. A27
  2. B9
  3. C81
  4. D18

Correct answer

C. 81

Step-by-step solution

_ r =1 ^9 ( r +3 2^ r ) ^9 C _ r = ( 3 2 )^9- , , N Now, aligned & _ r =1 ^9 ( r +3 2^ r ) ^9 C _ r = _ r =1 ^9 ( r 2^ r ) ^9 C _ r + _ r =1 ^9 ( 3 2^ r ) ^9 C _ r & = _ r =1 ^9 ( 9 2^ r ) ^8 C _ r -1 +3 _ r =1 ^9 ^9 C _ r ( 1 2 )^ r [ U ~g ^9 C _ r ^8 C _ r -1 = 9 r ] & = 9 2 _ r =1 ^9 ^8 C _ r -1 ( 1 2 )^ r -1 +3 ( _ r =0 ^9 ( ^9 C _ r ( 1 2 )^ r )-1 ) & = 9 2 (1+ 1 2 )^8+3 ( (1+ 1 2 )^9-1 ) aligned = 9 2 ( 3 2 )^8+3 ( 3 2 )^9-3=6 ( 3 2 )^9-3 Hence, =6, =3 Thus ( + )^2=81

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