Two equations (I) and (II) are given in each questions . On the basis of these equations you have to decide the relation between 'x' and 'y' and give answer
I. `x^(2) - 82 x + 781 = 0 `
II. ` y^(2) - 5041 = 0 `

Options

A. If ` x gt y `
B. If ` x ge y `
C. If ` x lt y `
D. If x = y or Relation cannot be established

Answer

To solve the problem and find the relation between \( x \) and \( y \) based on the given equations, we will follow these steps:

### Step 1: Solve the first equation for \( x \)

The first equation is:

\[ x^2 - 82x + 781 = 0 \]

This is a quadratic equation, and we will use the quadratic formula:

\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

Here, \( a = 1 \), \( b = -82 \), and \( c = 781 \).

### Step 2: Calculate the discriminant

First, we need to calculate the discriminant \( b^2 - 4ac \):

\[ b^2 = (-82)^2 = 6724 \]
\[ 4ac = 4 \times 1 \times 781 = 3124 \]
\[ b^2 - 4ac = 6724 - 3124 = 3600 \]

### Step 3: Substitute values into the quadratic formula

Now substituting the values into the quadratic formula:

\[ x = \frac{82 \pm \sqrt{3600}}{2} \]

### Step 4: Calculate the square root

The square root of 3600 is:

\[ \sqrt{3600} = 60 \]

### Step 5: Find the values of \( x \)

Now we can find the values of \( x \):

\[ x = \frac{82 + 60}{2} = \frac{142}{2} = 71 \]
\[ x = \frac{82 - 60}{2} = \frac{22}{2} = 11 \]

So, the possible values of \( x \) are \( 71 \) and \( 11 \).

### Step 6: Solve the second equation for \( y \)

The second equation is:

\[ y^2 - 5041 = 0 \]

Rearranging gives:

\[ y^2 = 5041 \]

### Step 7: Calculate the square root for \( y \)

Taking the square root of both sides gives:

\[ y = \pm \sqrt{5041} \]

Calculating the square root:

\[ \sqrt{5041} = 71 \]

Thus, the possible values of \( y \) are \( 71 \) and \( -71 \).

### Step 8: Compare values of \( x \) and \( y \)

Now we compare the values of \( x \) and \( y \):

- When \( y = 71 \):
- \( x = 71 \) gives \( y = x \)
- \( x = 11 \) gives \( y > x \)

- When \( y = -71 \):
- \( x = 71 \) gives \( x > y \)
- \( x = 11 \) gives \( x > y \)

### Conclusion

From the comparisons, we find that:
- If \( y = 71 \), then \( y \geq x \) when \( x = 71 \) and \( y > x \) when \( x = 11 \).
- If \( y = -71 \), then \( x > y \) for both values of \( x \).

Thus, the relationship between \( x \) and \( y \) cannot be established definitively in one way, leading us to conclude that the relation is not consistent.

### Final Answer

The relation between \( x \) and \( y \) cannot be established definitively.

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