Given:
- Triangle ABC with medians CD and BE intersecting at point O.
- Ratio of area of triangle ODE to triangle ABC is 1:12.

To find:
The correct option.

Solution:
1. Understanding the concept:
- In a triangle, the medians are the line segments joining a vertex to the midpoint of the opposite side.
- The medians of a triangle intersect each other at a point called the centroid.
- The centroid divides each median into two segments. The segment joining the centroid to a vertex is twice as long as the segment joining the midpoint of the opposite side to the centroid.
- The ratio of areas of two triangles with the same base is equal to the ratio of their corresponding heights.

2. Finding the ratio of areas:
- Let the area of triangle ABC be A.
- The ratio of areas of triangle ODE to triangle ABC is given as 1:12.
- Hence, the area of triangle ODE = 1/12 * A.

3. Finding the ratio of heights:
- Since O is the centroid of triangle ABC, the ratio of the lengths of the segments OD and CD is 2:1.
- Similarly, the ratio of the lengths of the segments OE and BE is also 2:1.

4. Finding the ratio of areas using the ratio of heights:
- The ratio of the heights of triangle ODE and triangle ABC is also 2:1, since the heights are measured from the same base.
- Therefore, the ratio of the areas of triangle ODE and triangle ABC is equal to the square of the ratio of their heights.
- Hence, the ratio of the areas of triangle ODE and triangle ABC is (2:1)^2 = 4:1.

5. Comparing the ratio of areas:
- The given ratio of areas is 1:12, and the ratio of areas calculated using the heights is 4:1.
- Since the two ratios are not the same, the correct option is not 4:1.
- To find the correct option, we need to simplify the ratio of areas obtained using the heights.
- The ratio of areas 4:1 can be simplified to 1:1 by dividing both sides by 4.
- Hence, the correct option is 1:12, which matches the given ratio of areas.

Therefore, the correct option is A) 1:12.