$3\mathrm{.}1$ Develop a Python script to simulate a basic user authentication system using if$-$else statements and logical operators. The script should prompt the user for a username and password and validate them against predefined credentials.

$3\mathrm{.}2$ Write a Python program using a while loop to calculate and display the factorial of a user$-$input number. Ensure your program handles invalid inputs gracefully.

$3\mathrm{.}3$ Design a Python script that uses a for loop to print a pattern of numbers or characters. For instance, create a script that prints a pyramid pattern using asterisks $(*)$ based on a user$-$input height