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JEE Main202524 Jan 2025Morning ShiftMathematicsBinomial TheoremActual

For some n 10 , let the coefficients of the 5 th, 6 th and 7 th terms in the binomial expansion of (1+ x )^ n +4 be in A.P. Then the largest coefficient in the expansion of (1+ x )^ n +4 is:

Options

  1. A20
  2. B10
  3. C35
  4. D70

Correct answer

C. 35

Step-by-step solution

aligned & (1+x)^ n+4 & ^ n+4 C₄, ^ n+4 C₅, ^ n+4 C₆, A.P. & 2 ^ n+4 C₅= ^ n+4 C₄+ ^ n+4 C₆ & 4 ^ n+4 C₅= ( ^ n+4 C₄+ ^ n+4 C₅ )+ ( ^ n+4 C₅+ ^ n+4 C₆ ) & 4 ^ n+4 C₅= ^ n+5 C₅+ ^ n+5 C₆ aligned aligned & 4 ( n +4)! 5! ( n -1)! = ( n +6)! 6! n ! & 4= ( n +6)( n +5) 6 n & n ^2+11 n +30=24 n & n ^2-13 n +30=0 & n =3,10( rejected ) & n 10 aligned Largest binomial coefficient in expansion of aligned & (1+ x )^7 & ( n +4=7) aligned is coeff. of middle term ^7 C ₄= ^7 C ₃=35

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