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Here are the solutions to your questions:

1. Which of the following numbers are not perfect squares?

Step 1: A perfect square number can only end with the digits 0, 1, 4, 5, 6, or 9. Numbers ending with 2, 3, 7, or 8 cannot be perfect squares.

Step 2: Check the last digit of each given number:

• (i) 2032 ends with 2.
• (ii) 2048 ends with 8.
• (iii) 1027 ends with 7.
• (iv) 1089 ends with 9.

Step 3: Based on the rule, numbers ending in 2, 8, or 7 are not perfect squares. Thus, 2032, 2048, and 1027 are not perfect squares. The number 1089 ends in 9, which means it could be a perfect square. Indeed, $33^2 = 1089$.

Step 4: The numbers that are not perfect squares are 2032, 2048, and 1027.

$\boxed{\text{(i) 2032, (ii) 2048, (iii) 1027}}$

2. Which one of $64^2, 108^2, 292^2, 36^2$ has the last digit 4?

Step 1: To find the last digit of a square, we only need to square the last digit of the original number.

Step 2: Determine the last digit for the square of each number:

• For $64^2$: The last digit of 64 is 4. $4^2 = 16$. The last digit of $64^2$ is 6.
• For $108^2$: The last digit of 108 is 8. $8^2 = 64$. The last digit of $108^2$ is 4.
• For $292^2$: The last digit of 292 is 2. $2^2 = 4$. The last digit of $292^2$ is 4.
• For $36^2$: The last digit of 36 is 6. $6^2 = 36$. The last digit of $36^2$ is 6.

Step 3: The numbers whose squares have a last digit of 4 are $108^2$ and $292^2$.

$\boxed{108^2 \text{ and } 292^2}$