In each of the following questions you have to find out which of the following statements is/are sufficient to answer the given questions. Give appropriate answers according to the options.
What is the principal amount?
The simple interest obtained on the principal after 2 years at `8%` rate of interest is ₹450 less than the compound interest obtained on the principal after 2 years at `8%` pa.
The sum becomes double in 10 years at `6%` pa rate of simple interest.
The compound interest obtained on the principal is ₹4540 after 2 years at the rate of `8%` compounded annually.

Options

A. Only I
B. All I, II and 111 together
C. II and III together
D. Only I or III

Answer

To solve the problem of determining the principal amount based on the given statements, we will analyze each statement step by step.

### Step 1: Analyze Statement 1
**Statement 1:** The simple interest obtained on the principal after 2 years at 8% rate of interest is ₹450 less than the compound interest obtained on the principal after 2 years at 8% per annum.

Let the principal amount be \( P \).

- **Simple Interest (SI)** for 2 years at 8%:
\[ SI = \frac{P \times 8 \times 2}{100} = \frac{16P}{100} = 0.16P \]

- **Compound Interest (CI)** for 2 years at 8%:
\[ CI = P \left(1 + \frac{8}{100}\right)^2 - P = P \left(1.08^2 - 1\right) = P \left(1.1664 - 1\right) = 0.1664P \]

According to the statement:
\[ SI = CI - 450 \]
Substituting the values:
\[ 0.16P = 0.1664P - 450 \]

Rearranging gives:
\[ 0.1664P - 0.16P = 450 \]
\[ 0.0064P = 450 \]
\[ P = \frac{450}{0.0064} = 70312.5 \]

Thus, from Statement 1, we can find the principal amount \( P \).

### Step 2: Analyze Statement 2
**Statement 2:** The sum becomes double in 10 years at 6% per annum rate of simple interest.

Let the principal amount be \( P \).

If the sum doubles, then:
\[ \text{Total Amount} = 2P \]
The simple interest for 10 years at 6% is:
\[ SI = \frac{P \times 6 \times 10}{100} = 6P \]

The total amount after 10 years is:
\[ P + SI = P + 6P = 7P \]

Setting this equal to the doubled principal:
\[ 7P = 2P \]
This leads to:
\[ 5P = 0 \]
This means we cannot determine \( P \) from this statement alone.

### Step 3: Analyze Statement 3
**Statement 3:** The compound interest obtained on the principal is ₹4540 after 2 years at the rate of 8% compounded annually.

Using the formula for compound interest:
\[ CI = P \left(1 + \frac{8}{100}\right)^2 - P = 4540 \]
This simplifies to:
\[ P \left(1.1664 - 1\right) = 4540 \]
\[ 0.1664P = 4540 \]
\[ P = \frac{4540}{0.1664} = 27200 \]

Thus, from Statement 3, we can also find the principal amount \( P \).

### Conclusion
From the analysis:
- Statement 1 is sufficient to find \( P \).
- Statement 2 is not sufficient to find \( P \).
- Statement 3 is sufficient to find \( P \).

Therefore, the correct answer is that either Statement 1 or Statement 3 is sufficient to answer the question.