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Properties of binary operations - class-XI

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Multiple Choice

Let $\ast$ be a binary operation on the set $Q$ of rational numbers as follows:
(i) $a\ast b = a - b$ (ii) $a\ast b = a^{2} + b^{2}$
(iii) $a\ast b = a + ab$ (iv) $a\ast b = (a - b)^{2}$
(v) $a\ast b = \dfrac {ab}{4}$ (vi) $a\ast b = ab^{2}$
Find which of the binary operations are commutative and which are associative

  1. $ii, iv, v$ are commutative and $v$ associative
  2. $ii, iv, v$ are not commutative and $v$ associative
  3. $iii, iv, v$are commutative and $v$ associative
  4. $vi, iv, v$are commutative and $v$ associative
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