Correct Answer - Option 4 : 1/2
Concept:
Let * be a binary operation on a non-empty set S. If there exists an element e in S such that a * e = e * a = a ∀ a ∈ S. Then the element e is said to be an identity element of S with respect to *.
Let * be a binary operation on a non-empty set S and let e be the identity element. Let a ∈ S then we say that a is invertible if there exists an element b ∈ S such that a * b = b * a = e where b is called the inverse of a and a-1 = b.
Calculation:
Given: O is a binary operation on Q+ which is set of all positive rational numbers such that a O \(\rm b = \frac{{ab}}{2}\)
Let e be the identity element of O with respect to O.
As we know that if e is an identity element of a non-empty set S with respect to a binary operation * then a * e = e * a = a ∀ a ∈ S.
Let a ∈ Q+ and because e is the identity element of Q with respect to given operation O
i.e a O e = a = e O a ∀ a ∈ Q+
According to the definition of O we have
a O e = ae/2 = a
⇒ e = 2 ∈ Q+
So, 2 is the identity element of Q+
Let a = 8 and b be the inverse of a
As we know that, if a is invertible then a * b = b * a = e
Here, a O b = e = 2
According to the definition of O
⇒ a O b = ab/2 = 2
⇒ ab = 4
⇒ b = 1/2 ----------(∵ a = 8)
Hence, inverse of 8 is 1/2
