If $$2 + 3i$$ is one of the roots of the equation $$2x^3 - 9x^2 + kx - 13 = 0$$, $$k \in R$$, then the real root of this equation (where $$i^2 = -1$$):

Given that $$2 + 3i$$ is a root of the equation $$2x^3 - 9x^2 + kx - 13 = 0$$ and $$k$$ is real, we know that complex roots occur in conjugate pairs for polynomials with real coefficients. Therefore, the conjugate $$2 - 3i$$ must also be a root.

Let the roots be $$\alpha = 2 + 3i$$, $$\beta = 2 - 3i$$, and $$\gamma$$ (the real root). The cubic equation can be expressed as $$2(x - \alpha)(x - \beta)(x - \gamma) = 0$$. First, compute the quadratic factor from the complex roots:

$$(x - \alpha)(x - \beta) = (x - 2 - 3i)(x - 2 + 3i) = [(x - 2) - 3i][(x - 2) + 3i] = (x - 2)^2 - (3i)^2$$

Since $$i^2 = -1$$, we have:

$$(x - 2)^2 - 9i^2 = (x^2 - 4x + 4) - 9(-1) = x^2 - 4x + 4 + 9 = x^2 - 4x + 13$$

So the equation becomes:

$$2(x^2 - 4x + 13)(x - \gamma) = 2x^3 - 9x^2 + kx - 13$$

Expand the left side:

$$2(x^2 - 4x + 13)(x - \gamma) = 2\left[ x^2(x - \gamma) - 4x(x - \gamma) + 13(x - \gamma) \right] = 2\left[ x^3 - \gamma x^2 - 4x^2 + 4\gamma x + 13x - 13\gamma \right]$$

Combine like terms:

$$= 2\left[ x^3 - (4 + \gamma)x^2 + (4\gamma + 13)x - 13\gamma \right] = 2x^3 - 2(4 + \gamma)x^2 + 2(4\gamma + 13)x - 2(13\gamma)$$

Simplify:

$$= 2x^3 - (8 + 2\gamma)x^2 + (8\gamma + 26)x - 26\gamma$$

Equate this to the given cubic $$2x^3 - 9x^2 + kx - 13$$ and compare coefficients:

For $$x^2$$: $$- (8 + 2\gamma) = -9$$

For $$x$$: $$8\gamma + 26 = k$$

For constant term: $$-26\gamma = -13$$

Solve the constant term equation for $$\gamma$$:

$$-26\gamma = -13$$

Divide both sides by $$-13$$:

$$\frac{-26\gamma}{-13} = \frac{-13}{-13} \implies 2\gamma = 1 \implies \gamma = \frac{1}{2}$$

Verify with the $$x^2$$ coefficient equation:

$$- (8 + 2\gamma) = -9$$

Substitute $$\gamma = \frac{1}{2}$$:

$$- \left(8 + 2 \times \frac{1}{2}\right) = - (8 + 1) = -9$$

This holds true. The $$x$$ coefficient gives $$k$$, but it is not needed for finding the real root.

Thus, the real root $$\gamma$$ is $$\frac{1}{2}$$. Checking the options:

A. Exists and is equal to $$\frac{1}{2}$$

B. Does not exist

C. Exists and is equal to 1

D. Exists and is equal to $$-\frac{1}{2}$$

Hence, the correct answer is Option A.