Question

Let \( G = \mathbb{R}^* \times \mathbb{R} \), where \( \mathbb{R} \) is the set of all real numbers and \( \mathbb{R}^* = \mathbb{R} - \{0\} \). Define an operation \( \circ \) on \( G \) as \( (a,b) \circ (c,d) = (ac, bc + d) \). Then, the identity element of the group \( (G, \circ) \) is _____.

• A. \( (0,0) \)
• B. \( (0,1) \)
• C. \( (1,0) \)
• D. \( (1,1) \)

Hint:

For operations involving pairs, the identity of the first component usually matches the standard multiplicative identity (1) if the operation is product-based, and the second component often matches the additive identity (0).

Answer 1

The Correct Option is C