**Solution:**

To find the value of 'A' that satisfies the given condition, we need to determine the common root of the two equations and then substitute it into both equations to solve for 'A'.

**Step 1: Find the common root**

Let's assume that the common root is 'r'. This means that both equations will have 'r' as one of their roots.

For the first equation: 2x^2 - Ax - 20 = 0

Using the quadratic formula, we can find the roots of this equation:

x = (-b ± √(b^2 - 4ac)) / (2a)

Here, a = 2, b = -A, and c = -20.

The discriminant (b^2 - 4ac) will help us determine if the roots are real and distinct, real and equal, or complex.

For the second equation: 2x^2 - 11x + 12 = 0

Using the quadratic formula again, we can find the roots:

x = (-b ± √(b^2 - 4ac)) / (2a)

Here, a = 2, b = -11, and c = 12.

**Step 2: Substitute the common root into both equations**

Since 'r' is the common root, we can substitute it into both equations:

For the first equation: 2r^2 - Ar - 20 = 0

For the second equation: 2r^2 - 11r + 12 = 0

Both of these equations should have the same root, so we can set them equal to each other:

2r^2 - Ar - 20 = 2r^2 - 11r + 12

Simplifying this equation, we get:

-Ar - 20 = -11r + 12

Rearranging the terms, we have:

11r - Ar = 12 - 20

11r - Ar = -8

Factoring out 'r', we get:

r(11 - A) = -8

**Step 3: Solve for 'A'**

Since we are given that 'A' is a positive value, we can assume that 'A' is not equal to 11. If 'A' were equal to 11, then the common root 'r' would be equal to 0, which would not satisfy the given condition.

Therefore, we can divide both sides of the equation by (11 - A):

r = -8 / (11 - A)

Since 'r' is a common root, it must satisfy both equations.

Substituting this value of 'r' into the first equation, we get:

2(-8 / (11 - A))^2 - A(-8 / (11 - A)) - 20 = 0

Simplifying this equation, we can solve for 'A':

128 / (11 - A)^2 + 8A / (11 - A) - 20 = 0

Multiplying through by (11 - A)^2 to eliminate the denominators:

128 + 8A(11 - A) - 20(11 - A)^2 = 0

Expanding and simplifying the equation, we get:

128 + 88A - 8A^2 - 220 + 40A -