In each of the following questions, two equations (I) and (II) are given . Solve the equations and mark the correct options
I. ` x = 3 sqrt( 4096) `
II.` y^(2) + 121 = 377`

Options

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

Answer

Let's solve the equations step by step.

### Given Equations:
1. \( I: x = 3 \sqrt{4096} \)
2. \( II: y^2 + 121 = 377 \)

### Step 1: Solve for \( x \)
We start with the first equation:
\[ x = 3 \sqrt{4096} \]

#### Calculation:
First, we need to find \( \sqrt{4096} \).
- We can simplify \( 4096 \) by recognizing it as \( 64^2 \) or \( 16^4 \).
- Therefore, \( \sqrt{4096} = 64 \).

Now substituting back into the equation for \( x \):
\[ x = 3 \times 64 = 192 \]

### Step 2: Solve for \( y \)
Now, we move to the second equation:
\[ y^2 + 121 = 377 \]

#### Rearranging:
To isolate \( y^2 \), we subtract \( 121 \) from both sides:
\[ y^2 = 377 - 121 \]
\[ y^2 = 256 \]

#### Calculation:
Now, we take the square root of both sides:
\[ y = \pm \sqrt{256} \]
\[ y = \pm 16 \]

### Step 3: Summary of Results
We have:
- \( x = 192 \)
- \( y = 16 \) or \( y = -16 \)

### Step 4: Establish the Relationship
Now we need to establish the relationship between \( x \) and \( y \):
- Comparing \( x \) and \( y \):
- For \( y = 16 \): \( 192 > 16 \)
- For \( y = -16 \): \( 192 > -16 \)

In both cases, \( x \) is greater than \( y \).

### Final Conclusion
Thus, the relationship can be summarized as:
\[ x \geq y \]

### Answer:
The correct option is \( \text{B: } x \geq y \).

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