Linear Programming

Step 1: Graphical Representation of Constraints. To solve the problem graphically, we plot the constraint equations: 1. (line passing through (6,0) and (0,6)) 2. (vertical line at ) 3. (horizontal line at ) 4. (first quadrant restriction) The feasible region is the intersection of these constraints. 

Step 2: Identifying Corner Points of Feasible Region. From the graph, the common feasible region forms a bounded polygon. The corner points of this region are:  

Step 3: Compute Objective Function at Corner Points. Evaluating at each corner:  

Step 4: Determine Maximum Value. The maximum value occurs at point with:  

Conclusion: The maximum value of is at .

Step 1: {Analyze Assertion (A)}
From the graph, the line passes through two corner points and , providing the same maximum value. This indicates that the maximum value occurs at infinite points along this segment. Thus, Assertion (A) is true.

Step 2: {Analyze Reason (R)}
In general, the optimal solution of an LPP occurs at corner points of the feasible region. This is true; however, in this case, the solution lies along a line segment connecting two corner points. Thus, Reason (R) is not the correct explanation of Assertion (A).

Step 3: {Conclusion}
Both Assertion (A) and Reason (R) are true, but Reason (R) does not explain Assertion (A). Hence, the correct answer is option (B).

Step 1: {Identify the corner points of the feasible region}
From the graph, the vertices of the feasible region are:

Step 2: {Substitute corner points into }
Evaluate at each vertex:

Step 3: {Find the maximum value}
The maximum value of occurs at , where .

Step 4: {Verify the options}
The maximum value is , which corresponds to option (C).

Step 1: Convert inequalities to equations

• Line 1:
• Line 2:
• Line 3:

Step 2: Find Intercepts

• Line 1: ,
• Line 2: ,
• Line 3: ,

Step 3: Graph and Feasible Region

Plot all three lines and shade the region that satisfies all constraints including and .

Step 4: Find Corner Points (Intersections)

1. Line 1 & Line 2:
   Solve: Multiply (i) by 2: Subtract from (ii): So point A:
2. Line 2 & Line 3:
   Substitute: So point B:
3. Line 1 & Line 3:
   Substitute: So point C:

Step 5: Evaluate at Each Corner Point

✅ Final Answer:

Maximum value of is at two corner points:

Step 1: Let the variables be Let = number of Toy A units produced
Let = number of Toy B units produced 
Step 2: Write the Objective Function We want to maximize profit:  
Step 3: Translate constraints from machine limits

• M1:
• M2:
• M3:
• Non-negativity:

Step 4: Draw the Feasible Region 

Step 5: Find Corner Points of Feasible Region We solve the equations of intersecting lines to find vertices of the region: 
(i) Intersection of and : (ii) Intersection of and : (iii) Intersection of and :  
Step 6: Evaluate Objective Function at Each Corner Point  
Step 7: Select Optimal Value
Final Answer: