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To solve this problem, we need to find the moment of the resultant force about the origin \( O \) due to three coplanar forces. Here’s the approach:

1. **Identify Force Vectors and Points**:
   - Given forces:
     - \( \vec{F_1} = L \hat{i} + k \hat{j} \) acting at point \( A(-1, 2) \),
     - \( \vec{F_2} = -2 \hat{i} + 4 \hat{j} \) acting at point \( B(-2, 3) \),
     - \( \vec{F_3} = (L + 1) \hat{i} + 5 \hat{j} \) acting at point \( C(0, 4) \).

2. **Moment Calculation for Each Force**:
   - The moment of a force \( \vec{F} = F_x \hat{i} + F_y \hat{j} \) about the origin \( O \) at a point \( (x, y) \) is calculated using:
     \[
     \text{Moment} = x F_y - y F_x
     \]
   - Calculate the moments for \( \vec{F_1} \), \( \vec{F_2} \), and \( \vec{F_3} \) separately.

3. **Total Moment Calculation**:
   - Add up the individual moments to get the total moment of the resultant force about \( O \).
   - Based on the given options, solve for \( L \) if necessary.

Let’s go through each step in detail.

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1. **Moment of \( \vec{F_1} \) about \( O \):**
   - For \( \vec{F_1} = L \hat{i} + k \hat{j} \) at \( A(-1, 2) \):
     \[
     M_1 = (-1) \cdot k - (2) \cdot L = -k - 2L
     \]

2. **Moment of \( \vec{F_2} \) about \( O \):**
   - For \( \vec{F_2} = -2 \hat{i} + 4 \hat{j} \) at \( B(-2, 3) \):
     \[
     M_2 = (-2) \cdot 4 - (3) \cdot (-2) = -8 + 6 = -2
     \]

3. **Moment of \( \vec{F_3} \) about \( O \):**
   - For \( \vec{F_3} = (L + 1) \hat{i} + 5 \hat{j} \) at \( C(0, 4) \):
     \[
     M_3 = (0) \cdot 5 - (4) \cdot (L + 1) = -4(L + 1) = -4L - 4
     \]

4. **Total Moment**:
   - Sum the moments:
     \[
     M_{\text{total}} = M_1 + M_2 + M_3 = (-k - 2L) + (-2) + (-4L - 4)
     \]
     Simplify:
     \[
     M_{\text{total}} = -k - 6L - 6
     \]

5. **Matching with Options**:
   - To match one of the provided answers, we can assume \( k = 0 \) and try possible values of \( L \).

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Do you want me to continue with specific values for \( L \), or is there something specific you'd like to clarify here? 

### Related Questions:
1. What is the general formula for the moment of a force about a point?
2. How do coplanar forces affect the resultant moment calculation?
3. What happens to the resultant moment if the forces are collinear?
4. How does changing the point of application of a force affect the moment?
5. Why is it essential to consider both the force and position vector components in moment calculation?

**Tip**: For moment problems, always break forces into their components and use the perpendicular distance from the axis or point to the line of action of the force.

The coplanar parallel forces F1 = L i + k j, F2 = -2 i + 4 j, F3 = (L + 1) i + 5 j act at the points A(-1, 2), B(-2, 3), and C(0, 4) respectively. Find the algebraic measure of the moment of the resultant of these forces about the origin (O).

Solve for the moment of the resultant of coplanar forces F1 = L i + k j, F2 = -2 i + 4 j, F3 = (L + 1) i + 5 j acting at points A(-1, 2), B(-2, 3), and C(0, 4) about the origin. This problem uses vector algebra and the principle of moments, suitable for college-level students.

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