In each question, two equations numbered I and II have been given. you have to solve both the equations and mark the appropiate option.
I `6x^2-5x+1=0`
II `12y^2-23y+10=0`

Options

A. `xley`
B. `xgey`
C. `xlty`
D. `xgty`

Answer

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

### Step 1: Solve Equation I
The first equation is:
\[ 6x^2 - 5x + 1 = 0 \]

To solve this quadratic equation, we can use the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
where \( a = 6 \), \( b = -5 \), and \( c = 1 \).

#### Step 1.1: Calculate the Discriminant
First, we calculate the discriminant \( D = b^2 - 4ac \):
\[ D = (-5)^2 - 4 \cdot 6 \cdot 1 = 25 - 24 = 1 \]

#### Step 1.2: Apply the Quadratic Formula
Now, we can find the values of \( x \):
\[ x = \frac{-(-5) \pm \sqrt{1}}{2 \cdot 6} = \frac{5 \pm 1}{12} \]
This gives us two solutions:
1. \( x_1 = \frac{5 + 1}{12} = \frac{6}{12} = \frac{1}{2} \)
2. \( x_2 = \frac{5 - 1}{12} = \frac{4}{12} = \frac{1}{3} \)

### Step 2: Solve Equation II
The second equation is:
\[ 12y^2 - 23y + 10 = 0 \]

We will again use the quadratic formula where \( a = 12 \), \( b = -23 \), and \( c = 10 \).

#### Step 2.1: Calculate the Discriminant
Calculate the discriminant \( D \):
\[ D = (-23)^2 - 4 \cdot 12 \cdot 10 = 529 - 480 = 49 \]

#### Step 2.2: Apply the Quadratic Formula
Now, we can find the values of \( y \):
\[ y = \frac{-(-23) \pm \sqrt{49}}{2 \cdot 12} = \frac{23 \pm 7}{24} \]
This gives us two solutions:
1. \( y_1 = \frac{23 + 7}{24} = \frac{30}{24} = \frac{5}{4} \)
2. \( y_2 = \frac{23 - 7}{24} = \frac{16}{24} = \frac{2}{3} \)

### Step 3: Compare Values of x and y
Now we have:
- For \( x \): \( \frac{1}{2} \) and \( \frac{1}{3} \)
- For \( y \): \( \frac{5}{4} \) and \( \frac{2}{3} \)

We need to determine the relationship between \( x \) and \( y \):
- The larger value of \( x \) is \( \frac{1}{2} \).
- The larger value of \( y \) is \( \frac{5}{4} \).

Since \( \frac{5}{4} > \frac{1}{2} \), we can conclude that \( y > x \).

### Final Answer
The correct answer is that \( x < y \).

---