Given: 2s : 3t is the duplicate ratio of 2s p : 3t p

To find: The value of p

Solution:

Duplicate ratio means the ratio of the squares of the terms of the original ratio.

The original ratio is 2s : 3t, so its square is (2s)^2 : (3t)^2 = 4s^2 : 9t^2

The duplicate ratio is 2s p : 3t p, so its square is (2s p)^2 : (3t p)^2 = 4s^2 p^2 : 9t^2 p^2

We know that the duplicate ratio is equal to the ratio of the squares of the terms of the original ratio, so:

4s^2 p^2 : 9t^2 p^2 = 4s^2 : 9t^2

Divide both sides by p^2:

4s^2 : 9t^2 = 4s^2 : 9t^2

This equation is true for all values of s and t, so we cannot solve for p.

However, we can check which option is true by substituting some values of s, t, and p.

Let's assume s = 1, t = 2, and p = 3:

Original ratio: 2s : 3t = 2(1) : 3(2) = 2 : 6 = 1 : 3

Square of original ratio: 1^2 : 3^2 = 1 : 9

Duplicate ratio: 2s p : 3t p = 2(1)(3) : 3(2)(3) = 6 : 18 = 1 : 3

Square of duplicate ratio: 1^2 : 3^2 = 1 : 9

So, the given condition is satisfied for these values of s, t, and p.

Now, let's check which option is true:

a) p^2 = 6st

Substitute s = 1, t = 2, and p = 3:

p^2 = 6(1)(2) = 12, which is not true.

b) p = 6st

Substitute s = 1, t = 2, and p = 3:

p = 6(1)(2) = 12, which is not true.

c) 2p = 3t

Substitute s = 1, t = 2, and p = 3:

2p = 3(2), which is not true.

d) None of these

This option seems to be true, as options a, b, and c are not true for the given values of s, t, and p.

Therefore, the correct answer is option d) none of these.