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Manipal MET2015MathematicsBinomial Theorem

If there is a term containing x^ 2 r in (x+ 1 x^2 )^ n-3 , then

Options

  1. An-2 r is a positive integral multiple of 3.
  2. Bn-2 r is even
  3. Cn-2 r is odd
  4. DNone of the above

Correct answer

A. n-2 r is a positive integral multiple of 3.

Step-by-step solution

Suppose (s+1) th term contains x^ 2 r . Then, we have aligned T_ s+1 & = ^ n-3 C_s x^ n-3-s ( 1 x^2 )^s & = ^ n-3 C_s x^ n-3-3 s aligned This will contain x^ 2 r , if n-3-3 s=2 r array lr & s= n-3-2 r 3 & s= n-2 r 3 -1 & s+1= n-2 r 3 & n-2 r=3(s+1) array Hence, (n-2 r) is a positive integral multiple of 3.

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