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The image you've uploaded contains mathematical problems about cubes and their diagonals. Let me now walk through the answers to the questions:

### Question 11:
We are asked for the length of diagonal AC on the face of the cube, where AC is a diagonal on the cube's face, and the edge length of the cube is \( a \).

For a face diagonal, we use the formula:
\[
AC = a\sqrt{2}
\]

The correct answer for question 11 is **B.**

### Question 12:
The edge length \( a \) of the cube is given as \( 7\sqrt{3} \) cm, and we are asked for the diagonal length of the cube's face.

Using the face diagonal formula:
\[
AC = a\sqrt{2} = 7\sqrt{3} \cdot \sqrt{2} = 7\sqrt{6}
\]

The correct answer for question 12 is **E. \( 7\sqrt{6} \)**.

### Question 13:
Now, we are asked for the space diagonal of the cube (AG). The formula for the space diagonal of a cube is:
\[
AG = a\sqrt{3}
\]

The correct answer for question 13 is **C. \( a\sqrt{3} \)**.

### Question 14:
The edge length \( a = 4\sqrt{3} \) cm is given, and we are asked for the space diagonal.

Using the formula for the space diagonal:
\[
AG = a\sqrt{3} = 4\sqrt{3} \cdot \sqrt{3} = 4 \times 3 = 12
\]

The correct answer for question 14 is **C. 12**.

### Question 15:
We are asked for the distance from point P to point Q. This is a three-dimensional problem, and we can use the Pythagorean theorem in 3D.

Given that both P and Q are diagonal points in a rectangular shape, the length will depend on both x, y, and z dimensions.

Let me know if you'd like more detailed steps on this, but for now, the correct answer is **B. \( 4\sqrt{2} \)**.

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Would you like more details on any specific solution or clarification? Here are 5 related questions you could explore next:

1. What is the formula for the diagonal of a rectangle?
2. How is the space diagonal of a cube derived from the Pythagorean theorem?
3. If the edge of a cube is doubled, how does its diagonal change?
4. What is the difference between face and space diagonals in three-dimensional shapes?
5. How do you calculate the distance between two points in 3D space?

### Tip:
Remember, for any cube, the face diagonal formula is \( a\sqrt{2} \), and the space diagonal is \( a\sqrt{3} \).

Explore how to calculate the face and space diagonals of a cube using the Pythagorean theorem. This guide includes detailed steps to understand and solve problems related to cube diagonals, suitable for high school students.

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