**Solution:**

To solve this problem, we need to understand the concept of buoyancy and Archimedes' principle.

**Buoyancy and Archimedes' principle:**

According to Archimedes' principle, when an object is immersed in a fluid (liquid or gas), it experiences an upward buoyant force equal to the weight of the fluid displaced by the object. This buoyant force acts in the opposite direction to the force of gravity.

Mathematically, the buoyant force (Fb) can be calculated using the formula:

Fb = ρ * V * g

where:
- Fb is the buoyant force,
- ρ is the density of the fluid,
- V is the volume of the fluid displaced by the object,
- g is the acceleration due to gravity.

**Finding the volume of the block inside the water:**

Given that the mass of the block is 12 kg and the density of water is 1000 kg/m³, we can calculate the volume of the block using the formula:

Density = Mass / Volume

Volume = Mass / Density

Volume = 12 kg / 1000 kg/m³

Volume = 0.012 m³

Therefore, the volume of the block inside the water is 0.012 m³.

**Finding the extra weight required for immersion:**

To find the extra weight required for the block to immerse another 75 cm³ in the water, we need to calculate the buoyant force acting on the block.

Buoyant force = ρ * V * g

Given that the density of water is 1000 kg/m³, the volume to be immersed is 75 cm³, and the acceleration due to gravity is 9.8 m/s², we can calculate the required extra weight using the formula:

Buoyant force = Extra weight

ρ * V * g = Extra weight

(1000 kg/m³) * (75 cm³ / 1000000 cm³/m³) * (9.8 m/s²) = Extra weight

Extra weight ≈ 0.0735 N

Therefore, the extra weight that should be placed on the object for it to immerse another 75 cm³ in the water is approximately 0.0735 N.

In summary, the volume of the block inside the water is 0.012 m³, and the extra weight required for immersion is approximately 0.0735 N.