Complex Numbers

Step 1: Modulus of .
Since , the modulus of is . Step 2: Evaluating the given options.
We need to check which of the given options has a modulus of 1: - has modulus 1, because , - is a real number, and since , . Thus, , which has modulus 1 for specific values of . Step 3: Conclusion.
The correct answer is (3) .

Step 1: Understand the given condition.
We are given that , which means that lies on the unit circle in the complex plane. This implies that can be written in the form , where is a real number.

Step 2: Analyze the expression.
The expression represents a complex number. By substituting , we can simplify this expression to show that it results in a purely imaginary number. The real part cancels out due to the properties of complex numbers on the unit circle.

Step 3: Conclusion.
Thus, the value of is purely imaginary, which corresponds to option (C).

Step 1: Use Euler's formula.
We are given that , which is the standard form of Euler’s formula. This means that .

Step 2: Simplify the expression.
To find , we use the following property of complex exponentials: Thus, we have:
Step 3: Conclusion.
Thus, the value of is , which corresponds to option (A).