Direction: In the following questions two equations numbered (I) and (II) are given. You have to solve both equations and Give answer
I. `p^2-17p-84=0` II. `q^2+4q-117=0`

Options

A. If `p gt q`
B. If `p ge q`
C. If `q gt p`
D. if relationship cannot be determined

Answer

To solve the given equations, we will follow these steps:

### Step 1: Solve the first equation \( p^2 - 17p - 84 = 0 \)

1. **Identify the coefficients**: Here, \( a = 1 \), \( b = -17 \), and \( c = -84 \).
2. **Factor the quadratic equation**: We need to find two numbers that multiply to \( ac = -84 \) and add to \( b = -17 \). The factors are \( -21 \) and \( 4 \).
3. **Rewrite the equation**:
\[ p^2 - 21p + 4p - 84 = 0 \]
4. **Group the terms**:
\[ (p^2 - 21p) + (4p - 84) = 0 \]
5. **Factor by grouping**:
\[ p(p - 21) + 4(p - 21) = 0 \]
\[ (p - 21)(p + 4) = 0 \]
6. **Set each factor to zero**:
- \( p - 21 = 0 \) → \( p = 21 \)
- \( p + 4 = 0 \) → \( p = -4 \)

### Step 2: Solve the second equation \( q^2 + 4q - 117 = 0 \)

1. **Identify the coefficients**: Here, \( a = 1 \), \( b = 4 \), and \( c = -117 \).
2. **Factor the quadratic equation**: We need to find two numbers that multiply to \( ac = -117 \) and add to \( b = 4 \). The factors are \( 13 \) and \( -9 \).
3. **Rewrite the equation**:
\[ q^2 + 13q - 9q - 117 = 0 \]
4. **Group the terms**:
\[ (q^2 + 13q) + (-9q - 117) = 0 \]
5. **Factor by grouping**:
\[ q(q + 13) - 9(q + 13) = 0 \]
\[ (q + 13)(q - 9) = 0 \]
6. **Set each factor to zero**:
- \( q + 13 = 0 \) → \( q = -13 \)
- \( q - 9 = 0 \) → \( q = 9 \)

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

1. **Possible values of \( p \)**: \( 21, -4 \)
2. **Possible values of \( q \)**: \( -13, 9 \)

### Step 4: Establish relationships

1. **Comparing \( p = 21 \) and \( q = -13 \)**:
- \( 21 > -13 \) → \( p > q \)

2. **Comparing \( p = 21 \) and \( q = 9 \)**:
- \( 21 > 9 \) → \( p > q \)

3. **Comparing \( p = -4 \) and \( q = -13 \)**:
- \( -4 > -13 \) → \( p > q \)

4. **Comparing \( p = -4 \) and \( q = 9 \)**:
- \( -4 < 9 \) → \( p < q \)

### Conclusion

Since we have established that \( p > q \) in some cases and \( p < q \) in others, we cannot determine a consistent relationship between \( p \) and \( q \). Therefore, the answer is that the relationship cannot be established.

### Final Answer
The relationship cannot be determined, so the correct option is 4.