Selec the CORRECT statement.

Options

A. If `x=(sqrt(3)+1)/(sqrt(3)-1)+(sqrt(3)-1)/(sqrt(3)+1)+(sqrt(3)-2)/(sqrt(3)+2)`, then the value of `x^(2)+(39/x)^(2)` is 110.
B. Every integer is a whole number
C. Between two rational numbers there exists infinite number of integers.
D. None of these

Answer

To solve the question of selecting the correct statement, we will analyze each option step by step.

### Step 1: Analyze Option 1
The first option states a mathematical expression involving square roots and rationalization.

1. **Expression**:
\[ \frac{\sqrt{3} + 1}{\sqrt{3} - 1} + \frac{\sqrt{3} - 1}{\sqrt{3} + 1} + \frac{\sqrt{3} - 2}{\sqrt{3} + 2} \]

2. **Rationalize** the first term:
\[ \frac{(\sqrt{3} + 1)(\sqrt{3} + 1)}{(\sqrt{3} - 1)(\sqrt{3} + 1)} = \frac{(\sqrt{3} + 1)^2}{3 - 1} = \frac{3 + 2\sqrt{3} + 1}{2} = \frac{4 + 2\sqrt{3}}{2} = 2 + \sqrt{3} \]

3. **Rationalize** the second term:
\[ \frac{(\sqrt{3} - 1)(\sqrt{3} - 1)}{(\sqrt{3} + 1)(\sqrt{3} - 1)} = \frac{(\sqrt{3} - 1)^2}{2} = \frac{3 - 2\sqrt{3} + 1}{2} = \frac{4 - 2\sqrt{3}}{2} = 2 - \sqrt{3} \]

4. **Rationalize** the third term:
\[ \frac{(\sqrt{3} - 2)(\sqrt{3} - 2)}{(\sqrt{3} + 2)(\sqrt{3} - 2)} = \frac{(\sqrt{3} - 2)^2}{3 - 4} = -(\sqrt{3} - 2)^2 = -(\sqrt{3}^2 - 4\sqrt{3} + 4) = -(-1 + 4\sqrt{3} - 4) = 4 - 4\sqrt{3} \]

5. **Combine all terms**:
\[ (2 + \sqrt{3}) + (2 - \sqrt{3}) + (4 - 4\sqrt{3}) = 4 + 4 - 4\sqrt{3} = 8 - 4\sqrt{3} \]

This does not equal \( \frac{39}{x} \) as stated. Therefore, **Option 1 is incorrect**.

### Step 2: Analyze Option 2
The second option states that "Every integer is a whole number."

- **Definition**: Whole numbers include 0 and all positive integers (1, 2, 3,...). Integers include negative numbers as well (-1, -2, -3,...).
- **Conclusion**: Since integers can be negative, not every integer is a whole number. Thus, **Option 2 is incorrect**.

### Step 3: Analyze Option 3
The third option states that "Between two rational numbers, there exists an infinite number of integers."

- **Example**: Consider two rational numbers, say 1 and 2. The integers between them are just 1 and 2, which are finite.
- **Conclusion**: Therefore, there cannot be an infinite number of integers between two rational numbers. Hence, **Option 3 is incorrect**.

### Step 4: Analyze Option 4
The fourth option states "None of these are correct."

- Since all previous options (1, 2, and 3) have been shown to be incorrect, **Option 4 must be correct**.

### Final Conclusion
The correct statement is **Option 4: None of these are correct**.

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