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KCET2017MathematicsBinomial Theorem

Binary operation * on ( R- -1 ) defined by ( a^ * b= a b+1 ) is

Options

  1. A( ^ * ) is associative and commutative
  2. B( ^ * ) is associative but not commutative
  3. C( ^ * ) is neither associative nor commutative
  4. D( ^ * ) is commutative but not associative

Correct answer

C. ( ^ * ) is neither associative nor commutative

Step-by-step solution

Given that [ array l a * b= a b+1 So, b * a= b a+1 array ] Since, ( a * b b^ * a ) It is not commutative. Now, ( (a^ * b )^ * c= ( a b+1 )^ * c ) ( = ( a b+1 ) c+1 = a (b+1)(c+1) ) ( a *(b * c)=a^ * b c+1 ) ( = a ( b c+1 +1 ) = a(c+1) (b+c+1) ) Since, ( (a * b)^ * C a * (b^ * c ) ) It is not associative. Therefore, ( ^ * ) is neither commutative nor associative. Therefore, ( ^ * ) is neither commutative nor associative.

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