In each question, two equations numbered I and II have been given. You have to solve both the equations and mark the appropriate option.
I.`2x^2-13x+18=0`
II. `y^2-7y+12=0`

Options

A. `x lt y`
B. `x ge y`
C. `x le y`
D. no relatioship can be established

Answer

To solve the given equations step by step, we will start with the first equation and then move on to the second equation.

### Step 1: Solve the first equation \(2x^2 - 13x + 18 = 0\)

1. **Identify the coefficients**: Here, \(a = 2\), \(b = -13\), and \(c = 18\).
2. **Use the quadratic formula**: The quadratic formula is given by:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
3. **Calculate the discriminant**:
\[ b^2 - 4ac = (-13)^2 - 4 \cdot 2 \cdot 18 = 169 - 144 = 25 \]
4. **Calculate the roots**:
\[ x = \frac{13 \pm \sqrt{25}}{2 \cdot 2} = \frac{13 \pm 5}{4} \]
- First root:
\[ x_1 = \frac{13 + 5}{4} = \frac{18}{4} = 4.5 \]
- Second root:
\[ x_2 = \frac{13 - 5}{4} = \frac{8}{4} = 2 \]

### Step 2: Solve the second equation \(y^2 - 7y + 12 = 0\)

1. **Identify the coefficients**: Here, \(a = 1\), \(b = -7\), and \(c = 12\).
2. **Use the quadratic formula**:
\[ y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
3. **Calculate the discriminant**:
\[ b^2 - 4ac = (-7)^2 - 4 \cdot 1 \cdot 12 = 49 - 48 = 1 \]
4. **Calculate the roots**:
\[ y = \frac{7 \pm \sqrt{1}}{2 \cdot 1} = \frac{7 \pm 1}{2} \]
- First root:
\[ y_1 = \frac{7 + 1}{2} = \frac{8}{2} = 4 \]
- Second root:
\[ y_2 = \frac{7 - 1}{2} = \frac{6}{2} = 3 \]

### Summary of Solutions:
- For the first equation, the solutions are \(x = 2\) and \(x = 4.5\).
- For the second equation, the solutions are \(y = 3\) and \(y = 4\).

### Step 3: Determine the relationship between \(x\) and \(y\)

- The values of \(x\) are \(2\) and \(4.5\).
- The values of \(y\) are \(3\) and \(4\).

### Conclusion:
- Comparing the values:
- \(x = 2\) is less than \(y = 3\).
- \(x = 4.5\) is greater than \(y = 4\).

Since we have one value of \(x\) that is less than \(y\) and another value of \(x\) that is greater than \(y\), we cannot establish a consistent relationship between \(x\) and \(y\).

Thus, the answer is that **no relationship can be established**.