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The image contains an assignment sheet with 10 different mathematical questions. Here’s a breakdown of the topics covered:

1. **Complex Numbers (Evaluation)**
   - \(i^{19}\), \(i^{20}\), \(i^{-65}\)

2. **Proofs Related to Complex Powers**
   - Prove that \(1 + i^{10} + i^{20} + i^{30} = 0\)
   - Prove that \(i^n + i^{n+1} + i^{n+2} + i^{n+3} = 0\)

3. **Complex Number Expressions**
   - Express in the standard form \(a + ib\): 
     \[
     \frac{5 + 4i}{3 - 2i}, \quad \frac{1 + i}{1 - i}
     \]

4. **Conjugate and Modulus**
   - Find the conjugate and modulus of complex numbers like \(4 - 5i\), \(\frac{1}{3 + 5i}\), and \(\frac{1 + 2i}{2 - 3i}\).

5. **Operations on Complex Numbers**
   - Given \(Z_1 = 2 + 3i\) and \(Z_2 = 4 - 5i\):
     - \(Z_1 + Z_2\)
     - \(Z_1 - Z_2\)
     - \(Z_1 \times Z_2\)
     - \(\frac{Z_1}{Z_2}\)

6. **Factorials, Permutations, and Combinations**
   - Evaluations involving factorials, permutations, and combinations like:
     \[
     \frac{10!}{7!}, \quad \frac{9!}{7!3!}, \quad 5P_0, \quad 4P_4, \quad 20C_3, \quad 8C_1
     \]

7. **Binomial Theorem Expansion**
   - Expand using the binomial theorem:
     \((x + 3y)^3\), \((2x + q)^4\)

8. **Specific Terms in Binomial Expansion**
   - Find specific terms in expansions like \((x - 2y)^5\), \((x + \frac{1}{x})^5\)

9. **Binomial Expansions to Specific Terms**
   - Expand up to the third term for expressions like \((1 + x)^{-3}\), \((1 + 3x)^{-2}\)

10. **Solving Equations**
    - Solve equations like:
      \[
      (x + 1)! = 20(x - 1)!, \quad (x - 2)! + (x - 3)! = 5(x - 4)!
      \]

Would you like to work on one specific question or get more details on solving a particular section?

Here are 5 related questions for further study:
1. What is the general form of complex number operations like addition and multiplication?
2. How does the modulus of a complex number help in geometric interpretation?
3. What are some common mistakes when using the binomial theorem for expansions?
4. How does factorial simplification work when solving permutation problems?
5. What methods can be used to prove identities involving powers of \(i\)?

**Tip:** When working with powers of \(i\), remember that \(i^4 = 1\), so powers of \(i\) cycle every four terms.

The original question from the user's assignment sheet (e.g., complex number operations, binomial expansions, etc.).

Explore various mathematical problems involving complex numbers, binomial theorem expansions, and permutations with step-by-step solutions. This assignment is ideal for high school students (Grades 10-12) focusing on algebra and higher mathematics.

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