**Solution:**

To find the value of p for which the given pair of lines have one line in common, we need to solve the two equations simultaneously and determine the condition on p for their common line.

Let's solve the equations step by step:

**Step 1: Rewrite the equations in standard form**

The given pair of equations are:

1) 3x^2 - 5xy + py^2 = 0
2) 6x^2 - xy - 5y^2 = 0

To make it easier to analyze, we will rewrite these equations in standard form.

1) 3x^2 - 5xy + py^2 = 0
Divide the equation by y^2 to get:
3(x^2/y^2) - 5(x/y) + p = 0

2) 6x^2 - xy - 5y^2 = 0
Divide the equation by y^2 to get:
6(x^2/y^2) - (x/y) - 5 = 0

**Step 2: Equate the coefficients of x^2/y^2**

From the two standard form equations, we can equate the coefficients of x^2/y^2:

3(x^2/y^2) - 5(x/y) + p = 6(x^2/y^2) - (x/y) - 5

Simplifying this equation, we obtain:
3x^2 - 5xy + py^2 = 6x^2 - xy - 5y^2

**Step 3: Equate the coefficients of x/y**

Next, we equate the coefficients of x/y:

-5(x/y) = -(x/y)

This equation implies that the coefficient of x/y is the same for both equations.

**Step 4: Evaluate the coefficient of x/y**

From the previous step, we have:
-5 = -1

This equation is satisfied for any value of p.

**Step 5: Determine the condition on p for the common line**

Since the equation -5 = -1 is always true, there is no specific condition on p for the given pair of lines to have one line in common. Hence, p can be any real number.

Therefore, the value of p is not restricted and can take any real value.

In conclusion, the value of p can be any real number.