Read the following statements carefully and write 'T' for true and 'F' for false.
(i) Volume of a cylinder is three times the volume of a cone on the same base and of same height
(ii) Volume of biggest sphere in cube of edge 6 cm is ` 30 pi pi cm^(3)`
(iii) Total surface area of a cone of radius r and stant height r is `pi r l + pi r ^(2)`

Options

A. `{:(" (i) (ii) (iii)"),(" T F T"):}`
B. `{:(" (i) (ii) (iii)"),(" T T T"):}`
C. `{:(" (i) (ii) (iii)"),(" F T F"):}`
D. `{:(" (i) (ii) (iii)"),(" F T T"):}`

Answer

To solve the question, we will analyze each statement one by one and determine whether they are true (T) or false (F).

### Statement (i):
**Volume of a cylinder is three times the volume of a cone on the same base and of the same height.**

1. **Formula for Volume of Cylinder**:
\[ V_{cylinder} = \pi r^2 h \]
2. **Formula for Volume of Cone**:
\[ V_{cone} = \frac{1}{3} \pi r^2 h \]
3. **Comparison**:
- If we compare both volumes:
\[ V_{cylinder} = 3 \times V_{cone} \]
- This means:
\[ \pi r^2 h = 3 \times \left(\frac{1}{3} \pi r^2 h\right) \]
- Therefore, this statement is **True (T)**.

### Statement (ii):
**Volume of the biggest sphere in a cube of edge 6 cm is \(30 \pi \, \text{cm}^3\)**.

1. **Finding the Radius of the Sphere**:
- The diameter of the sphere is equal to the edge of the cube, which is 6 cm.
- Therefore, the radius \(r\) of the sphere is:
\[ r = \frac{6}{2} = 3 \, \text{cm} \]
2. **Formula for Volume of Sphere**:
\[ V_{sphere} = \frac{4}{3} \pi r^3 \]
3. **Calculating the Volume**:
- Substituting the radius into the formula:
\[ V_{sphere} = \frac{4}{3} \pi (3)^3 = \frac{4}{3} \pi \times 27 = 36 \pi \, \text{cm}^3 \]
- Since \(36 \pi \, \text{cm}^3\) is not equal to \(30 \pi \, \text{cm}^3\), this statement is **False (F)**.

### Statement (iii):
**Total surface area of a cone of radius \(r\) and slant height \(l\) is \(\pi r l + \pi r^2\)**.

1. **Formula for Total Surface Area of Cone**:
- The total surface area (TSA) of a cone is given by:
\[ TSA = \pi r l + \pi r^2 \]
- Where \(\pi r l\) is the lateral surface area and \(\pi r^2\) is the area of the base.
2. **Conclusion**:
- Since the formula matches the statement, this statement is **True (T)**.

### Final Answers:
- (i) T
- (ii) F
- (iii) T