Complex Numbers

We are given that , where is a complex number. This describes a circle in the complex plane centered at (which corresponds to the point in the complex plane) with radius 15.

Step 1: Geometrical interpretation
The inequality represents the set of all complex numbers whose distance from the point is less than or equal to 15. This is a circle centered at with radius 15.

Step 2: Maximum and minimum arguments of
To find the maximum and minimum arguments of , we consider the lines from the center of the circle (i.e., the point ) to the points on the circle that are farthest and closest to the positive real axis. - The maximum argument corresponds to the direction that is furthest counterclockwise from the positive real axis. - The minimum argument corresponds to the direction that is furthest clockwise from the positive real axis.

Step 3: Calculation of the angle
The maximum angular separation between the maximum and minimum arguments occurs along the vertical line through the center of the circle (since the center is on the imaginary axis). The distance from the origin to the center of the circle is (the imaginary part of ), and the radius of the circle is . We can use the following relationship to calculate the angle between the maximum and minimum arguments: Thus, the maximum difference between the arguments is:

Answer:

We are given that and . Let's first write down the equation in terms of complex numbers: Expanding using the binomial theorem: Since and , we get: This simplifies to: So, comparing the real and imaginary parts with , we have the equations: Step 1: Express Now, substitute the expressions for and into : Simplifying each term: Step 2: Simplifying the expression Now, we need to evaluate the full expression: This simplifies to:

Answer:

Let and be two non-zero complex numbers, and let represent the argument of the complex number .

Step 1: Product of complex numbers
For two complex numbers and , we know that: However, the principal value of the argument is restricted to the interval , so when calculating the principal value of , it may not be exactly equal to due to the periodicity of the argument function. Hence, the principal value of may not equal the sum of the principal values of and .

Step 2: Division of complex numbers
Similarly, for the division of two complex numbers and , we know that: However, due to the periodicity of the argument function, the principal value of the argument of the quotient may not exactly equal the difference of the principal values of and .

Conclusion
Thus, both the principal value of and may not exactly equal the sum or difference of the principal values of and , respectively, because the principal values are confined to the interval , and wrapping may occur.

Answer:

Given: Now squaring both sides: Use identity: Subtracting both sides: Therefore, (C) zero, (D) purely imaginary