CBSE - NCERT 8 - Squares and Square roots

It is correct.
4 * 4 = 16

Yes, squares of all the above numbers are odd numbers.

There are couple of clues as follows:

1. As the other options are not perfect squares, this has higher chances to be the perfect square.
2. As the one's digit is 5, there is a chance for it to be a perfect square.
3. Rewriting 2025 as 5 * 405 = 5 * 5 * 81 = 5² * 9²
   Therefore, this option is correct.

Yes, it is correct.
17²  =  (17² - 1) / 2 + (17² + 1) / 2

Yes, it is correct.

Square root(2809) = 53
2809 has 4 digits in it and its square root 53 has only 2 digits in it. 
As per this option, number of digits should be 4/2 = 2, which is correct.

Square root(27225) = 165
27225 has 5 digits in it and its square root 165 has only 3 digits in it.
As per this option, number of digits should be (5+1)/2 = 3, which is correct.

This option is correct.

Yes, it is the correct option. For this problem, you don't need to calculate the full value of the square of the given number. Just pick the digit in units place in each number and multiply it with itself.
Lets do it for all the given numbers:
123²: 3 x 3 = 9          So, it will not end with 1
57² : 7 x 7 = 49        So, it will not end with 1
92² : 2 x 2 = 4           So, it will not end with 1
61²   :  1 x 1 = 1        Yes, it will end with 1
209² : 9 x 9 = 81       Yes, it will end with 1

Yes, this is the correct representation of the pythagorean triplets.

You can cross check substituting values (n² - 1)² + (2n)² = (n² + 1)²
Ex: 3, 4 and 5   (2² - 1)² + (2x2)² = (2² + 1)²
8, 15 and 17     (4² - 1)² + (2x4)² = (4² + 1)²