a$)$ Limitations and problems of using UML and other structured analysis methods include:

Complexity: UML diagrams can become very complex and difficult to manage, especially for large systems.

Ambiguity: Different interpretations of UML diagrams can lead to misunderstandings among team members.

Overhead: Creating and maintaining UML diagrams can be time$-$consuming and may not always add value.

Tool Dependency: Effective use of UML often requires specialized tools, which can be costly and require training.

Lack of Formal Semantics: UML lacks formal semantics, making it difficult to verify and validate models rigorously.

Scalability Issues: UML may not scale well for very large systems, leading to incomplete or overly simplified models.

Integration with Code: Keeping UML diagrams in sync with the actual code can be challenging.

Learning Curve: Mastering UML and structured analysis methods requires significant effort and time.

b$)$ Software engineers use UML and Design Patterns in different phases of software development as follows:

Requirement Analysis: Use UML diagrams like use case diagrams to capture functional requirements.

System Design: Employ class diagrams, sequence diagrams, and state diagrams to design the system architecture.

Implementation: Apply design patterns to solve common problems and improve code maintainability and reuse.

Testing: Use sequence diagrams to design test cases and ensure all scenarios are covered.

Maintenance: Update UML diagrams to reflect changes in the system and use design patterns to refactor code.

c$)$ Software engineers deal with complexity in large projects by:

Modularization: Breaking down the system into smaller, manageable modules or components.

Abstraction: Using abstraction to hide complex details and expose only necessary functionalities.

Design Patterns: Applying design patterns to solve recurring problems and improve code structure.

Documentation: Maintaining comprehensive documentation to ensure all team members understand the system.

Automated Tools: Using automated tools for code generation, testing, and continuous integration to manage complexity.

Agile Practices: Implementing agile practices like iterative development and continuous feedback to handle changes effectively.