Here is the mathematical analysis of the problems based on the image:

Problem 1 (Choose the incorrect statement):

1. True: The definition of a cylinder is correct. It is a three-dimensional figure bounded by a cylindrical surface and two parallel circles that are not in the same plane.
2. True: The generating line of the cylinder indeed represents its height.
3. True: A cylinder can be obtained by rotating a rectangle about one of its sides, as this is a method of generating a cylinder.
4. False: This statement is incorrect. The axis of the cylinder is parallel to the generating lines, not perpendicular to them.
5. True: The lateral surface area of a cylinder is indeed the product of the circumference of the base and the height.

The incorrect statement is #4.

Problem 2 (Diagonal of a cross-section of a cylinder):

Given: Diagonal = 9 cm, height = 5.4 cm. We use the Pythagorean theorem to find the radius $r$.

The diagonal forms a right triangle with the diameter $2r$ as one leg and the height $h = 5.4$ cm as the other: $d^2 = (2r)^2 + h^2$ Substitute the known values: $9^2 = (2r)^2 + 5.4^2$ $81 = 4r^2 + 29.16$ $4r^2 = 81 - 29.16 = 51.84$ $r^2 = \frac{51.84}{4} = 12.96$ $r = \sqrt{12.96} = 3.6 \, \text{cm}$ So, the radius of the cylinder is 3.6 cm (Option 3).

Problem 3 (Cylinder cross-section parallel to its axis):

Given: Distance from the axis = 6 cm, section area = $20\sqrt{7} \, \text{cm}^2$, and height = 5 cm.

We are given the area of the cross-section of the cylinder, which is a rectangle. The area of this section is given by: $A = 2a \cdot h$ where $a$ is the distance from the axis to the section (i.e., half the chord length). Using the formula for a chord in a circle: $a = \sqrt{r^2 - d^2}$ where $r$ is the radius of the cylinder, and $d = 6 \, \text{cm}$ is the given distance. We also know: $A = 20\sqrt{7} = 2a \cdot 5$ $a = \frac{20\sqrt{7}}{10} = 2\sqrt{7}$ Now, use the chord formula: $a = \sqrt{r^2 - d^2} = 2\sqrt{7}$ $4 \cdot 7 = r^2 - 36$ $r^2 = 28 + 36 = 64$ $r = \sqrt{64} = 8 \, \text{cm}$ So, the radius of the cylinder is 8 cm (Option 1).

Problem 4 (Surface area of a cylinder):

Given: Total surface area $S = 392\pi \, \text{cm}^2$, and the height is 3 times the radius.

The formula for the total surface area of a cylinder is: $S = 2\pi r(h + r)$ Substitute $h = 3r$: $S = 2\pi r(3r + r) = 2\pi r(4r) = 8\pi r^2$ Given $S = 392\pi$, we solve for $r$: $8\pi r^2 = 392\pi$ $r^2 = \frac{392}{8} = 49$ $r = \sqrt{49} = 7 \, \text{cm}$ Now, find the height $h = 3r = 3 \cdot 7 = 21 \, \text{cm}$.

So, the radius is 7 cm, and the height is 21 cm.

Would you like further details on any of the problems? Here are some related questions you might be interested in:

1. How do you derive the lateral surface area formula of a cylinder?
2. Can you explain the relationship between radius, height, and diagonal in more depth?
3. What is the geometrical interpretation of a section parallel to the axis of a cylinder?
4. How can you visualize the chord in the third problem using geometry?
5. What other methods can be used to calculate surface area for different shapes?

Tip: For complex geometry problems, drawing a diagram can significantly help visualize the relationships between different parts of the figure.