The given questions is followed by three statements. Study the statements carefully and decide which of the following statements (s) is //are necessary to answer the questions.
What is the total surface area of cone ?
(i) The area of the base of cone is 1386 `cm^(2)` ?
(ii) The curved surface area of the cone is 2310 `cm^(2)`.
(iii) The volume of the cone is 3696 `cm^(3)`.

Options

A. I and either II or III
B. II and either I or III
C. III and either I or II
D. Any two of the three

Answer

To find the total surface area of a cone, we need to use the formula:

\[ \text{Total Surface Area (TSA)} = \pi r (r + l) \]

where:
- \( r \) is the radius of the base of the cone,
- \( l \) is the slant height of the cone.

### Step 1: Analyze the Statements

1. **Statement (i)**: The area of the base of the cone is \( 1386 \, cm^2 \).
- The area of the base of the cone (which is a circle) is given by the formula:
\[ \text{Area} = \pi r^2 \]
- From this statement, we can find \( r \) using:
\[ r^2 = \frac{1386}{\pi} \]

2. **Statement (ii)**: The curved surface area of the cone is \( 2310 \, cm^2 \).
- The curved surface area (CSA) of the cone is given by:
\[ \text{CSA} = \pi r l \]
- This statement allows us to find \( l \) if we know \( r \).

3. **Statement (iii)**: The volume of the cone is \( 3696 \, cm^3 \).
- The volume of the cone is given by:
\[ V = \frac{1}{3} \pi r^2 h \]
- This statement can help us find \( h \) if we know \( r \).

### Step 2: Determine Necessary Statements

- To find the total surface area, we need both \( r \) and \( l \).
- From **Statement (i)**, we can find \( r \).
- From **Statement (ii)**, we can find \( l \) if we have \( r \).
- **Statement (iii)** provides additional information but is not necessary to find \( l \) since we can find it using the first two statements.

### Conclusion

- **Statement (i)** is necessary to find \( r \).
- **Statement (ii)** is necessary to find \( l \).
- **Statement (iii)** is not necessary to find the total surface area.

Thus, the necessary statements to answer the question are **(i) and (ii)**.

### Final Answer:
The correct option is that statements (i) and (ii) are necessary to answer the question about the total surface area of the cone.

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