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.