Each of the following questions is provided with 2 statements i.e. Statement I & Statement II. You have the solve them and mark the correct option
I. ` x^(2) - 45 x + 506 = 0 `
II. ` y^(2) - 9y - 360 = 0 `

Options

A. ` x gt y`
B. ` x lt y `
C. ` x le y `
D. x = y or no relation can be established

Answer

To solve the given problem, we will analyze the two statements separately and find the values for \( x \) and \( y \). After that, we will compare the values to determine the relationship between them.

### Step-by-Step Solution:

**Statement I: Solve the equation \( x^2 - 45x + 506 = 0 \)**

1. **Identify the coefficients**:
The equation is in the standard form \( ax^2 + bx + c = 0 \), where \( a = 1 \), \( b = -45 \), and \( c = 506 \).

2. **Calculate the discriminant**:
\[ D = b^2 - 4ac = (-45)^2 - 4 \cdot 1 \cdot 506 = 2025 - 2024 = 1 \]

3. **Find the roots using the quadratic formula**:
The roots are given by:
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{45 \pm 1}{2} \]

4. **Calculate the two possible values for \( x \)**:
\[ x_1 = \frac{46}{2} = 23, \quad x_2 = \frac{44}{2} = 22 \]

So, the values of \( x \) are \( 22 \) and \( 23 \).

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**Statement II: Solve the equation \( y^2 - 9y - 360 = 0 \)**

1. **Identify the coefficients**:
The equation is in the standard form \( ay^2 + by + c = 0 \), where \( a = 1 \), \( b = -9 \), and \( c = -360 \).

2. **Calculate the discriminant**:
\[ D = b^2 - 4ac = (-9)^2 - 4 \cdot 1 \cdot (-360) = 81 + 1440 = 1521 \]

3. **Find the roots using the quadratic formula**:
The roots are given by:
\[ y = \frac{-b \pm \sqrt{D}}{2a} = \frac{9 \pm \sqrt{1521}}{2} \]

4. **Calculate the two possible values for \( y \)**:
First, find \( \sqrt{1521} = 39 \):
\[ y_1 = \frac{9 + 39}{2} = 24, \quad y_2 = \frac{9 - 39}{2} = -15 \]

So, the values of \( y \) are \( 24 \) and \( -15 \).

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### Comparison of Values:

Now we have:
- Values of \( x \): \( 22 \) and \( 23 \)
- Values of \( y \): \( 24 \) and \( -15 \)

1. **Compare \( x = 22 \) with \( y = 24 \)**:
\[ 22 < 24 \quad \text{(True)} \]

2. **Compare \( x = 22 \) with \( y = -15 \)**:
\[ 22 > -15 \quad \text{(True)} \]

3. **Compare \( x = 23 \) with \( y = 24 \)**:
\[ 23 < 24 \quad \text{(True)} \]

4. **Compare \( x = 23 \) with \( y = -15 \)**:
\[ 23 > -15 \quad \text{(True)} \]

### Conclusion:

From the comparisons:
- \( x \) is less than \( y \) when \( y = 24 \).
- \( x \) is greater than \( y \) when \( y = -15 \).

Since we have both \( x < y \) and \( x > y \), we conclude that **no relation can be established** between \( x \) and \( y \).

### Final Answer:
The correct option is **D: No relation can be established**.

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