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In the picture it is Complex Variables and Transforms, now slove the equation as given in picture

The problem in the image asks to define a complex number and its modulus and argument, then find the modulus and principal argument of the expression:

\[
\frac{(2 - 5i)(3 - i)}{(1 - 2i)^2}
\]

### Step 1: Simplify the expression
We will first simplify the expression by multiplying the numerators and expanding the denominator.

#### Numerator:
\[
(2 - 5i)(3 - i) = 2 \times 3 + 2 \times (-i) + (-5i) \times 3 + (-5i) \times (-i)
= 6 - 2i - 15i + 5i^2
\]
Since \(i^2 = -1\), the above becomes:
\[
6 - 17i + 5(-1) = 6 - 17i - 5 = 1 - 17i
\]
So, the numerator simplifies to \(1 - 17i\).

#### Denominator:
\[
(1 - 2i)^2 = (1 - 2i)(1 - 2i)
= 1 \times 1 + 1 \times (-2i) + (-2i) \times 1 + (-2i) \times (-2i)
= 1 - 2i - 2i + 4i^2
\]
Again, using \(i^2 = -1\), we get:
\[
1 - 4i - 4 = -3 - 4i
\]

So, the expression simplifies to:
\[
\frac{1 - 17i}{-3 - 4i}
\]

### Step 2: Convert to polar form
To find the modulus and argument, we need to convert the expression to polar form.

#### Modulus:
The modulus of a complex number \(z = a + bi\) is given by:
\[
|z| = \sqrt{a^2 + b^2}
\]

For the numerator \(1 - 17i\):
\[
|1 - 17i| = \sqrt{1^2 + (-17)^2} = \sqrt{1 + 289} = \sqrt{290}
\]

For the denominator \(-3 - 4i\):
\[
|-3 - 4i| = \sqrt{(-3)^2 + (-4)^2} = \sqrt{9 + 16} = \sqrt{25} = 5
\]

Thus, the modulus of the whole expression is:
\[
\frac{\sqrt{290}}{5}
\]

#### Argument:
The argument of a complex number \(z = a + bi\) is given by:
\[
\arg(z) = \tan^{-1}\left(\frac{b}{a}\right)
\]

For the numerator \(1 - 17i\), we get:
\[
\arg(1 - 17i) = \tan^{-1}\left(\frac{-17}{1}\right) = \tan^{-1}(-17)
\]

For the denominator \(-3 - 4i\), we get:
\[
\arg(-3 - 4i) = \tan^{-1}\left(\frac{-4}{-3}\right) = \tan^{-1}\left(\frac{4}{3}\right)
\]

The total argument is:
\[
\arg\left(\frac{1 - 17i}{-3 - 4i}\right) = \arg(1 - 17i) - \arg(-3 - 4i)
\]

Now, let's calculate these values numerically.The results for the expression \(\frac{(2 - 5i)(3 - i)}{(1 - 2i)^2}\) are:

- **Modulus**: \(\approx 3.41\)
- **Principal argument**: \(\approx 0.702\) radians (or \(\approx 40.24^\circ\))

Would you like more details on any of the steps? Here are some related questions you might explore:

1. How can we use polar form to multiply and divide complex numbers?
2. What are the geometric interpretations of modulus and argument?
3. How can we find the argument of a complex number when the real part is negative?
4. What role does the principal value of the argument play in complex number operations?
5. Can the modulus of a complex number ever be negative?

**Tip:** Always make sure to express the argument in radians or degrees based on the context, and use the principal value between \(-\pi\) and \(\pi\) radians.

Define complex number and its modulus and argument. Find the modulus and principal argument of (2 - 5i)(3 - i) / (1 - 2i)^2.

Learn how to find the modulus and principal argument of the complex expression (2 - 5i)(3 - i) / (1 - 2i)^2. This problem covers key concepts in complex numbers, including modulus, argument, and polar form representation, suitable for Grades 11-12.

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