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# Find which of the operations given above has identity.

Q. 10 Find which of the operations given above has identity.

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An element  $p \in Q$ will be identity element for operation *

if $a*p = a = p*a$   for all $a \in Q$

$(v) a * b = \frac{ab}{4}$

$a * p = \frac{ap}{4}$

$p * a = \frac{pa}{4}$

$a*p = a = p*a$    when  $p=4$  .

Hence, $(v) a * b = \frac{ab}{4}$  has identity as 4.

However, there is no such element  $p \in Q$  which satisfies above condition for all rest five operations.

Hence, only (v) operations have identity.

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