Given:
- Neck diameter = 2 cm
- Bottom diameter = 10 cm
- Force applied on the cork in the neck = 1.2 kgf

To find:
- Force exerted on the bottom of the bottle

Assumptions:
- The bottle is cylindrical in shape
- The cork is tightly fitted in the neck of the bottle

Solution:

Step 1: Calculate the radius of the neck and bottom of the bottle:
- The diameter of the neck is given as 2 cm. Therefore, the radius of the neck (r1) can be calculated using the formula: r1 = d/2 = 2/2 = 1 cm.
- The diameter of the bottom is given as 10 cm. Therefore, the radius of the bottom (r2) can be calculated using the formula: r2 = d/2 = 10/2 = 5 cm.

Step 2: Calculate the area of the neck and bottom of the bottle:
- The area of the neck (A1) can be calculated using the formula: A1 = π * r1^2, where π is a constant (approximately 3.14). Substituting the value of r1, we get A1 = 3.14 * 1^2 = 3.14 cm^2.
- The area of the bottom (A2) can be calculated using the formula: A2 = π * r2^2. Substituting the value of r2, we get A2 = 3.14 * 5^2 = 78.5 cm^2.

Step 3: Calculate the pressure on the neck:
- Pressure (P1) can be calculated using the formula: P1 = F1/A1, where F1 is the force applied on the cork and A1 is the area of the neck. Substituting the given values, we get P1 = 1.2 kgf / 3.14 cm^2 ≈ 0.382 kgf/cm^2.

Step 4: Calculate the force on the bottom:
- Force (F2) on the bottom can be calculated using the formula: F2 = P2 * A2, where P2 is the pressure on the bottom and A2 is the area of the bottom. Since pressure is force per unit area, we can rearrange the formula as P2 = F1/A1, where F1 is the force on the neck and A1 is the area of the neck. Substituting the given values, we get P2 = 0.382 kgf/cm^2.
- Substituting the calculated value of P2 and the area of the bottom (A2), we can calculate the force on the bottom as F2 = 0.382 kgf/cm^2 * 78.5 cm^2 ≈ 30 kgf.

Step 5: Answer:
The force exerted on the bottom of the bottle is approximately 30 kgf.