In each question two equations numbered I and II are given . you have to solve both the equations and mark the appropriate option
I`x^2-4x-12=0`
II `y^2-5y-14=0`

Options

A. `xley`
B. `xlty`
C. The relation between x and y cannot be established.
D. `xgty`

Answer

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

### Step 1: Solve the first equation \( x^2 - 4x - 12 = 0 \)

1. **Identify the equation**:
\[ x^2 - 4x - 12 = 0 \]

2. **Factor the quadratic equation**:
We need to find two numbers that multiply to \(-12\) (the constant term) and add up to \(-4\) (the coefficient of \(x\)). The numbers are \(2\) and \(-6\).
\[ (x - 6)(x + 2) = 0 \]

3. **Set each factor to zero**:
\[ x - 6 = 0 \quad \text{or} \quad x + 2 = 0 \]

4. **Solve for \(x\)**:
\[ x = 6 \quad \text{or} \quad x = -2 \]

### Step 2: Solve the second equation \( y^2 - 5y - 14 = 0 \)

1. **Identify the equation**:
\[ y^2 - 5y - 14 = 0 \]

2. **Factor the quadratic equation**:
We need to find two numbers that multiply to \(-14\) (the constant term) and add up to \(-5\) (the coefficient of \(y\)). The numbers are \(2\) and \(-7\).
\[ (y - 7)(y + 2) = 0 \]

3. **Set each factor to zero**:
\[ y - 7 = 0 \quad \text{or} \quad y + 2 = 0 \]

4. **Solve for \(y\)**:
\[ y = 7 \quad \text{or} \quad y = -2 \]

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

Now we have the following possible values:
- For \(x\): \(6\) and \(-2\)
- For \(y\): \(7\) and \(-2\)

1. **Case 1**:
- If \(x = 6\) and \(y = 7\):
\[ x < y \quad (6 < 7) \]

2. **Case 2**:
- If \(x = 6\) and \(y = -2\):
\[ x > y \quad (6 > -2) \]

3. **Case 3**:
- If \(x = -2\) and \(y = 7\):
\[ x < y \quad (-2 < 7) \]

4. **Case 4**:
- If \(x = -2\) and \(y = -2\):
\[ x = y \quad (-2 = -2) \]

### Conclusion

From the above comparisons, we can see that:
- In some cases \(x < y\)
- In some cases \(x > y\)
- In one case \(x = y\)

Thus, we cannot establish a consistent relationship between \(x\) and \(y\). Therefore, the correct option is that the relationship between \(x\) and \(y\) cannot be established.