In each of the following questions, two equations (I) and (II) are given . Solve the equation and mark the correct option :
I. ` x^(2) - 29 x + 208 = 0 `
II. ` y^(2) - 21 y + 108 = 0 `

Options

A. If ` x gt y `
B. If ` x ge y `
C. if ` x lt y `
D. If ` x le y`

Answer

To solve the given equations step by step, we will first tackle each equation separately and then compare the results.

### Step 1: Solve Equation I
The first equation is:
\[ x^2 - 29x + 208 = 0 \]

To factor this quadratic equation, we need to find two numbers that multiply to \( 208 \) and add up to \( -29 \).

1. **Finding the factors**:
- The factors of \( 208 \) that add up to \( 29 \) are \( 16 \) and \( 13 \) (since \( 16 + 13 = 29 \)).

2. **Factoring the equation**:
\[ x^2 - 29x + 208 = (x - 16)(x - 13) = 0 \]

3. **Finding the values of \( x \)**:
- Set each factor to zero:
\[ x - 16 = 0 \quad \Rightarrow \quad x = 16 \]
\[ x - 13 = 0 \quad \Rightarrow \quad x = 13 \]

So, the solutions for the first equation are:
\[ x = 16 \quad \text{and} \quad x = 13 \]

### Step 2: Solve Equation II
The second equation is:
\[ y^2 - 21y + 108 = 0 \]

Similarly, we need to find two numbers that multiply to \( 108 \) and add up to \( -21 \).

1. **Finding the factors**:
- The factors of \( 108 \) that add up to \( 21 \) are \( 12 \) and \( 9 \) (since \( 12 + 9 = 21 \)).

2. **Factoring the equation**:
\[ y^2 - 21y + 108 = (y - 12)(y - 9) = 0 \]

3. **Finding the values of \( y \)**:
- Set each factor to zero:
\[ y - 12 = 0 \quad \Rightarrow \quad y = 12 \]
\[ y - 9 = 0 \quad \Rightarrow \quad y = 9 \]

So, the solutions for the second equation are:
\[ y = 12 \quad \text{and} \quad y = 9 \]

### Step 3: Compare the Values of \( x \) and \( y \)
Now we have the values:
- For \( x \): \( 16 \) and \( 13 \)
- For \( y \): \( 12 \) and \( 9 \)

We will compare each value of \( x \) with each value of \( y \):
1. Compare \( x = 16 \) with \( y = 12 \):
- \( 16 > 12 \)

2. Compare \( x = 16 \) with \( y = 9 \):
- \( 16 > 9 \)

3. Compare \( x = 13 \) with \( y = 12 \):
- \( 13 > 12 \)

4. Compare \( x = 13 \) with \( y = 9 \):
- \( 13 > 9 \)

### Conclusion
In all comparisons, \( x \) is greater than \( y \). Therefore, the final relation is:
\[ x > y \]

### Final Answer
The correct option is:
**Option A: \( x > y \)**