(1) (i)The unit digit in 77 is 7

And 7² = 49 whose unit digit is 9.

\(\because\) The ones digit (or unit digit) in the square of 77 is 9.

(ii) \(\because\) Number of non-square numbers between squares of two conscutive integers (or between n² & (n + 1)²) is 2n.

\(\therefore\) Number of non-square numbers between 24² and 25² is 2 x 24 = 48

(iii) \(\because\) 17² = 289 < 300 and 22² = 484 < 500 & 23² = 529 > 500

(iv) If a number has n digits the number of digits in its square root is (i) n/2 if n is even (ii) \(\frac{n+2}2\) if n is odd.

\(\because\) 6/2 = 3 and \(\frac{5+1}2=3\)

Hence, if a number has 5 or 6 digits in it then number of digits in its square root will be 3.

(v) \(\because\) 169 < 180 < 196 i.e., 13² < 180 < 14²

⇒ 13 < \(\sqrt{180}\) < 14

Hence, the value of \(\sqrt{180}\) lies between integers 13 and 14.

(2) (i) \(\because\) 16 is a square of 4 and ends in 6

Also square root of 16 is 4 which does not have 6 in unit's place.

Hence, the statement is false.

(ii) A square number(or perfect square) always have even number of zeros at the end.

Hence, given statement is true.

\(\because\) If a number has n number of zeros then its square will have 2n number of zeros.

(iii) The number of zeroes in the square of 91000 is 2 x 3 = 6

Hence, given statement is false.

(iv) 75² = (70 + 5)² = 70² + 2 x 70 x 5 + 5²

 = 4900 + 700 + 25 = 5625

Hence, given statement is false

(v) \(\sqrt{225}=\sqrt{3\times3\times5\times5}=\sqrt{3^2\times5^2}\) 

\(=\sqrt{(3\times5)^2}\) = 3 x 5 = 15

Hence, given statement is true.