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 $a\sqrt{3}$.