Let's go over each question step by step to provide the correct solutions.

1. Problem 1: Distance from A to line CE

• Given: Cube with edge length 4 cm.
• Question: Find the shortest distance from A to the line CE.

Solution:
The shortest distance between a point and a skew line can be found using the geometry of the cube. Here, the shortest path is along the face diagonal between points on the perpendicular faces containing the point A and the line CE.

• Using symmetry, the required distance can be calculated as:

  $\text{Distance} = \frac{1}{2} \times \text{Face Diagonal} = \frac{1}{2} \times 4\sqrt{2} = 2\sqrt{2} \, \text{cm}$

Answer: C. $2\sqrt{2}$ cm

2. Problem 2: Distance from F to line AC (edge length = 10 cm)

• Given: Cube with edge length 10 cm.
• Question: Find the shortest distance from F to the diagonal AC.

Solution:
The point F lies on the face EFGH. The shortest distance from a point to a diagonal on the opposite face can be calculated as:

$\text{Distance} = \frac{a\sqrt{3}}{2} = \frac{10\sqrt{3}}{2} = 5\sqrt{3} \, \text{cm}$

Answer: E. $5\sqrt{3}$ cm

3. Problem 3: Distance from point K (intersection of EG and FH) to line BG

• Given: Edge length = 6 cm.
• Question: Find the shortest distance from K (intersection point of diagonals EG and FH) to BG.

Solution:
Since K is the midpoint of space diagonal intersections, we compute the shortest distance by symmetry. It will be half the length of the space diagonal:

$\text{Space diagonal} = a\sqrt{3} = 6\sqrt{3}$ $\text{Distance} = \frac{6\sqrt{3}}{2} = 3\sqrt{3} \, \text{cm}$

Answer: C. $3\sqrt{6}$ cm

4. Problem 4: Distance from G to line BD (edge length = 8 cm)

• Solution:
  The shortest distance is also calculated using spatial relations:

  [ \text{Distance} = \frac{8\sqrt{2}}