In the following questions, two equations numbered I and II are given. You have to solve both the equations and
I. `6x^2-5x+1=0`
II. `4y^2-8y+3=0`

Options

A. `xlty`
B. `xgty`
C. `x le y`
D. `x ge y`

Answer

To solve the given equations step by step, we will start with each quadratic equation separately.

### Step 1: Solve the first equation \(6x^2 - 5x + 1 = 0\)

1. **Identify the coefficients**:
- \(a = 6\), \(b = -5\), \(c = 1\)

2. **Calculate the discriminant**:
\[ D = b^2 - 4ac = (-5)^2 - 4 \cdot 6 \cdot 1 = 25 - 24 = 1 \]

3. **Use the quadratic formula**:
\[ x = \frac{-b \pm \sqrt{D}}{2a} \]
Substituting the values:
\[ x = \frac{-(-5) \pm \sqrt{1}}{2 \cdot 6} = \frac{5 \pm 1}{12} \]

4. **Calculate the two possible values for \(x\)**:
- First value:
\[ x_1 = \frac{5 + 1}{12} = \frac{6}{12} = \frac{1}{2} \]
- Second value:
\[ x_2 = \frac{5 - 1}{12} = \frac{4}{12} = \frac{1}{3} \]

Thus, the solutions for the first equation are:
\[ x = \frac{1}{2} \quad \text{and} \quad x = \frac{1}{3} \]

### Step 2: Solve the second equation \(4y^2 - 8y + 3 = 0\)

1. **Identify the coefficients**:
- \(a = 4\), \(b = -8\), \(c = 3\)

2. **Calculate the discriminant**:
\[ D = b^2 - 4ac = (-8)^2 - 4 \cdot 4 \cdot 3 = 64 - 48 = 16 \]

3. **Use the quadratic formula**:
\[ y = \frac{-b \pm \sqrt{D}}{2a} \]
Substituting the values:
\[ y = \frac{-(-8) \pm \sqrt{16}}{2 \cdot 4} = \frac{8 \pm 4}{8} \]

4. **Calculate the two possible values for \(y\)**:
- First value:
\[ y_1 = \frac{8 + 4}{8} = \frac{12}{8} = \frac{3}{2} \]
- Second value:
\[ y_2 = \frac{8 - 4}{8} = \frac{4}{8} = \frac{1}{2} \]

Thus, the solutions for the second equation are:
\[ y = \frac{3}{2} \quad \text{and} \quad y = \frac{1}{2} \]

### Step 3: Compare the values of \(x\) and \(y\)

From the solutions:
- \(x = \frac{1}{2}, \frac{1}{3}\)
- \(y = \frac{3}{2}, \frac{1}{2}\)

We can see that:
- The maximum value of \(x\) is \(\frac{1}{2}\) and the minimum value of \(y\) is also \(\frac{1}{2}\).
- The maximum value of \(y\) is \(\frac{3}{2}\).

Thus, the relationship between \(x\) and \(y\) is:
\[ x \leq y \]

### Final Answer
The correct relationship is \(x \leq y\).