Which of the following statements is not correct ?

Options

A. For a given radius and height, a right circular cone has the lesser volume among a right circular cone and a right right circular cylinder.
B. If side of a cube is increased by 10%, the volume will increase by 33.1%.
C. If the radius of a sphere is increased by 20%, the surface area will increase by 40%.
D. Cutting a sphere into 2 parts does not change the total volume.

Answer

To determine which of the statements is not correct, we will analyze each statement one by one.

### Step 1: Analyze the First Statement
**Statement 1:** For a given radius and height, a right circular cone has less volume than a right circular cylinder.

- **Volume of Right Circular Cylinder (V_cylinder)**:
\[ V_{cylinder} = \pi r^2 h \]

- **Volume of Right Circular Cone (V_cone)**:
\[ V_{cone} = \frac{1}{3} \pi r^2 h \]

- **Comparison**:
\[ V_{cone} = \frac{1}{3} V_{cylinder} \]
This means the volume of the cone is indeed less than that of the cylinder. **Statement 1 is correct.**

### Step 2: Analyze the Second Statement
**Statement 2:** If a side of a cube is increased by 10%, then the volume is increased by 13.1%.

- **Initial Side Length of Cube (a)**:
\[ V_{initial} = a^3 \]

- **New Side Length after 10% increase**:
\[ New \, side = 1.1a \]

- **New Volume**:
\[ V_{new} = (1.1a)^3 = 1.331a^3 \]

- **Increase in Volume**:
\[ Increase = V_{new} - V_{initial} = 1.331a^3 - a^3 = 0.331a^3 \]

- **Percentage Increase**:
\[ \text{Percentage Increase} = \left(\frac{0.331a^3}{a^3}\right) \times 100 = 33.1\% \]

Thus, the statement claiming a 13.1% increase is incorrect. **Statement 2 is not correct.**

### Step 3: Analyze the Third Statement
**Statement 3:** If the radius of a sphere is increased by 20%, the volume increases by 40%.

- **Volume of Sphere**:
\[ V_{initial} = \frac{4}{3} \pi r^3 \]

- **New Radius after 20% increase**:
\[ New \, radius = 1.2r \]

- **New Volume**:
\[ V_{new} = \frac{4}{3} \pi (1.2r)^3 = \frac{4}{3} \pi (1.728r^3) = 1.728 \cdot \frac{4}{3} \pi r^3 \]

- **Increase in Volume**:
\[ Increase = V_{new} - V_{initial} = (1.728 - 1) \cdot \frac{4}{3} \pi r^3 = 0.728 \cdot \frac{4}{3} \pi r^3 \]

- **Percentage Increase**:
\[ \text{Percentage Increase} = \left(\frac{0.728 \cdot \frac{4}{3} \pi r^3}{\frac{4}{3} \pi r^3}\right) \times 100 = 72.8\% \]

Thus, the statement claiming a 40% increase is incorrect. **Statement 3 is also not correct.**

### Step 4: Analyze the Fourth Statement
**Statement 4:** The total volume of two spheres is equal to the volume of one sphere with double the radius.

- **Volume of Two Spheres**:
\[ V_{two \, spheres} = 2 \cdot \frac{4}{3} \pi r^3 = \frac{8}{3} \pi r^3 \]

- **Volume of Sphere with Double Radius**:
\[ V_{double \, radius} = \frac{4}{3} \pi (2r)^3 = \frac{4}{3} \pi (8r^3) = \frac{32}{3} \pi r^3 \]

Clearly, the total volume of two spheres is not equal to the volume of one sphere with double the radius. **Statement 4 is also not correct.**

### Conclusion
The statements that are not correct are **Statement 2, Statement 3, and Statement 4**. However, the question asks for which is not correct, and since we found Statement 2 to be incorrect first, we can conclude that **the answer is Statement 2**.