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Let’s go through each part of question 6 step-by-step.

### Part (a)
We need to describe the line formed on the Argand plane by all points \( z \) where \( \arg(z) = \frac{\pi}{4} \).

1. **Understanding**: The argument (angle) of \( z \) is \( \frac{\pi}{4} \), meaning every point on this line makes an angle of \( \frac{\pi}{4} \) (or 45 degrees) with the positive real axis.
2. **Line Equation**: Since \( \arg(z) = \tan^{-1}\left(\frac{\text{Im}(z)}{\text{Re}(z)}\right) = \frac{\pi}{4} \), we get:
   \[
   \frac{\text{Im}(z)}{\text{Re}(z)} = \tan\left(\frac{\pi}{4}\right) = 1
   \]
   This simplifies to \( \text{Im}(z) = \text{Re}(z) \), which is the equation of a line with a slope of 1 passing through the origin.

So, the line formed is \( y = x \) on the Argand plane.

### Part (b)
We need to find the values of \( t \) at which the curve \( z = 0.5 e^{t \cdot e^{i\pi/4}} \) first crosses:
   - (i) the imaginary axis
   - (ii) the real axis

1. **Imaginary Axis Crossing**: 
   - The imaginary axis is crossed when \( \text{Re}(z) = 0 \).
   - This occurs for specific values of \( t \) that make the real part zero. By solving this expression, you’ll find the first point of intersection on the imaginary axis.

2. **Real Axis Crossing**: 
   - The real axis is crossed when \( \text{Im}(z) = 0 \).
   - Similarly, solve for \( t \) to find when the imaginary part is zero for the first time.

### Part (c)
Convert the complex numbers at points \( A \), \( B \), and \( C \) to Cartesian form.

1. Each point \( A \), \( B \), and \( C \) corresponds to specific values of \( t \) in \( z = 0.5 e^{t \cdot e^{i\pi/4}} \).
2. By substituting the values of \( t \) you found in part (b) into \( z = 0.5 e^{t \cdot e^{i\pi/4}} \), convert each resulting complex number into Cartesian form: \( z = x + iy \).

### Part (d)
Write the exponential forms of \( z_1 \), \( z_2 \), and \( z_3 \) at points \( A \), \( B \), and \( C \) in terms of \( \theta \).

1. Substitute the values of \( t \) corresponding to \( A \), \( B \), and \( C \) into \( z = 0.5 e^{t \cdot e^{i\pi/4}} \) to get \( z_1 \), \( z_2 \), and \( z_3 \) in exponential form.

### Part (e)
Calculate \( |z_2| - |z_1| \) and \( |z_3| - |z_2| \).

1. Find the magnitudes \( |z_1| \), \( |z_2| \), and \( |z_3| \).
2. Compute \( |z_2| - |z_1| \) and \( |z_3| - |z_2| \) to see if there is a pattern and justify if it continues.

Would you like further details on any specific part, or would you like me to work through calculations for a particular section? Here are some additional questions to explore:

1. How is the argument of a complex number determined in general?
2. What does the exponential form of a complex number represent geometrically?
3. How does the magnitude of a complex number relate to its position on the Argand plane?
4. What happens to the argument of \( z \) as \( t \) changes in the exponential expression?
5. Why does \( |z_2| - |z_1| \) and similar terms create a pattern?

**Tip**: For complex numbers in polar form \( z = re^{i\theta} \), the magnitude is \( |z| = r \) and the argument is \( \theta \), making it easy to represent rotations and scaling geometrically.

Describe the line formed on the Argand plane by all the points, z, for which arg z = π/4. Find the values of t at which the curve z = 0.5e^(t * e^(iπ/4)) first crosses the imaginary and real axes. Also, find the Cartesian form of the complex numbers at points A, B, and C, and express the exponential form in terms of θ.

Explore a step-by-step solution to find the line formed by complex numbers with argument π/4 on the Argand plane, including Cartesian and exponential forms. Suitable for high school students in Grades 11-12, this example covers the properties of complex numbers and conversions between forms.

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