Direction: In the following questions two equations numbered (I) and (II) are given. You have to solve both equations and Give answer
I. `2p^2-7p-60=0` II. `3q^2+13q+4=0`

Options

A. no relation between p and q can be established
B. If `p ge q`
C. If `q gt p`
D. if `q ge p`

Answer

To solve the given equations step by step, we will follow these procedures:

### Step 1: Solve the first equation \(2p^2 - 7p - 60 = 0\)

1. **Identify the coefficients**:
- Here, \(a = 2\), \(b = -7\), and \(c = -60\).

2. **Use the quadratic formula**:
The quadratic formula is given by:
\[ p = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Substituting the values:
\[ p = \frac{-(-7) \pm \sqrt{(-7)^2 - 4 \cdot 2 \cdot (-60)}}{2 \cdot 2} \]

3. **Calculate the discriminant**:
\[ b^2 - 4ac = 49 + 480 = 529 \]

4. **Substitute back into the formula**:
\[ p = \frac{7 \pm \sqrt{529}}{4} = \frac{7 \pm 23}{4} \]

5. **Calculate the two possible values for \(p\)**:
- \(p_1 = \frac{30}{4} = \frac{15}{2} = 7.5\)
- \(p_2 = \frac{-16}{4} = -4\)

### Step 2: Solve the second equation \(3q^2 + 13q + 4 = 0\)

1. **Identify the coefficients**:
- Here, \(a = 3\), \(b = 13\), and \(c = 4\).

2. **Use the quadratic formula**:
\[ q = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Substituting the values:
\[ q = \frac{-13 \pm \sqrt{13^2 - 4 \cdot 3 \cdot 4}}{2 \cdot 3} \]

3. **Calculate the discriminant**:
\[ b^2 - 4ac = 169 - 48 = 121 \]

4. **Substitute back into the formula**:
\[ q = \frac{-13 \pm \sqrt{121}}{6} = \frac{-13 \pm 11}{6} \]

5. **Calculate the two possible values for \(q\)**:
- \(q_1 = \frac{-2}{6} = -\frac{1}{3}\)
- \(q_2 = \frac{-24}{6} = -4\)

### Step 3: Compare the values of \(p\) and \(q\)

- The values obtained are:
- For \(p\): \(7.5\) and \(-4\)
- For \(q\): \(-\frac{1}{3}\) and \(-4\)

### Step 4: Establish relationships

1. **Comparing \(p = 7.5\) and \(q = -\frac{1}{3}\)**:
- \(7.5 > -\frac{1}{3}\) (p is greater than q)

2. **Comparing \(p = 7.5\) and \(q = -4\)**:
- \(7.5 > -4\) (p is greater than q)

3. **Comparing \(p = -4\) and \(q = -\frac{1}{3}\)**:
- \(-4 < -\frac{1}{3}\) (p is less than q)

4. **Comparing \(p = -4\) and \(q = -4\)**:
- \(p = q\) (they are equal)

### Conclusion

Based on the comparisons:
- In the first case, \(p\) is greater than \(q\).
- In the second case, \(p\) is less than \(q\).
- In the third case, \(p\) is equal to \(q\).

Thus, the relationship between \(p\) and \(q\) cannot be conclusively established.

### Final Answer
The correct option is that there is no relation between \(p\) and \(q\).

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