Find the values of m so that the quadratic equation 3x² - 5x - 2m = 0 has two distinct real roots.

Find the value(s) of k for which each of the following quadratic equation has equal roots:

(i) 3kx² = 4(kx - 1)

(ii) (k + 4)x² + (k + 1)x + 1 = 0

Also, find the roots for that value(s) of k in each case.

Divide 16 into two parts such that twice the square of the larger part exceeds the square of the smaller part by 164.

Two natural numbers are in the ratio 3 : 4 . Find the numbers if the difference between their squares is 175.