The Miller indices (hkl) are a notation system used in crystallography to describe the orientation of crystal planes. These indices are determined based on the intercepts made by the plane on the three crystallographic axes.

To determine the intercepts, we can use the following steps:

1. Identify the crystallographic axes: In this case, we have three axes, denoted as a, b, and c.

2. Determine the intercepts: The Miller indices (hkl) represent the reciprocals of the intercepts made by the plane on the three axes. In other words, if a plane intersects the a-axis at a distance of 1/a, the b-axis at a distance of 1/b, and the c-axis at a distance of 1/c, then the Miller indices of the plane would be (hkl) = (1/a, 1/b, 1/c).

3. Simplify the indices: To simplify the Miller indices, we can multiply all the intercepts by a common factor to eliminate any fractions. In this case, the Miller indices are (3, 2, 6).

Now, let's determine the intercept mode by the plane on the three crystallographic axes using the given Miller indices of (3, 2, 6).

- The intercept on the a-axis is given by 1/a = 1/3, which can be simplified as 2/6 or 2a.
- The intercept on the b-axis is given by 1/b = 1/2, which is already in the simplified form of b.
- The intercept on the c-axis is given by 1/c = 1/6, which can be simplified as 1/6 or c.

Therefore, the intercept mode by the plane on the three crystallographic axes is (2a, b, c), which matches with option A.

In summary, the Miller indices (3, 2, 6) correspond to a plane that intersects the a-axis at 2a, the b-axis at b, and the c-axis at c. Option A, (2a, 3b, c), accurately represents the intercept mode of the plane on the crystallographic axes.