Class CBSE Class 12 Mathematics Linear Programming Q #1432
KNOWLEDGE BASED
REMEMBER
3 Marks 2025 AISSCE(Board Exam) SA
In the Linear Programming Problem (LPP), find the point/points giving maximum value for $Z=5x+10y$ subject to constraints $x+2y\le120$, $x+y\ge60$, $x-2y\ge0$, $x, y\ge0$.
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Detailed Solution

Step 1: Graph the Constraints

First, we need to graph the constraints to find the feasible region. The constraints are: $x+2y\le120$ $x+y\ge60$ $x-2y\ge0$ $x, y\ge0$

Step 2: Convert Inequalities to Equations

Convert the inequalities to equations to find the boundary lines: $x+2y=120$ $x+y=60$ $x-2y=0$

Step 3: Find Intersection Points

Find the intersection points of these lines: Intersection of $x+2y=120$ and $x+y=60$: Subtract the second equation from the first: $y = 60$. Then $x = 60 - y = 60 - 60 = 0$. So the intersection point is $(0, 60)$. Intersection of $x+2y=120$ and $x-2y=0$: Add the two equations: $2x = 120$, so $x = 60$. Then $60 - 2y = 0$, so $2y = 60$ and $y = 30$. So the intersection point is $(60, 30)$. Intersection of $x+y=60$ and $x-2y=0$: From the second equation, $x = 2y$. Substitute into the first equation: $2y + y = 60$, so $3y = 60$ and $y = 20$. Then $x = 2y = 2(20) = 40$. So the intersection point is $(40, 20)$.

Step 4: Identify the Feasible Region

The feasible region is bounded by the points $(40, 20)$, $(60, 30)$, and $(0, 60)$.

Step 5: Evaluate Z at Corner Points

Evaluate the objective function $Z = 5x + 10y$ at each corner point: At $(40, 20)$: $Z = 5(40) + 10(20) = 200 + 200 = 400$ At $(60, 30)$: $Z = 5(60) + 10(30) = 300 + 300 = 600$ At $(0, 60)$: $Z = 5(0) + 10(60) = 0 + 600 = 600$

Step 6: Determine Maximum Value

The maximum value of $Z$ is 600, which occurs at the points $(60, 30)$ and $(0, 60)$. Since these are two points, all the points on the line segment joining $(60, 30)$ and $(0, 60)$ will give the maximum value.

Final Answer: The maximum value of Z is 600, which occurs at all points on the line segment joining (60, 30) and (0, 60).

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Pedagogical Audit
Bloom's Analysis: This is an REMEMBER question because the student needs to recall the steps involved in solving a linear programming problem, including graphing constraints, finding intersection points, and evaluating the objective function at corner points.
Knowledge Dimension: PROCEDURAL
Justification: The question requires the student to apply a specific algorithm or method (linear programming) to find the optimal solution. It involves a sequence of steps to solve the problem.
Syllabus Audit: In the context of CBSE Class 12, this is classified as KNOWLEDGE. The question directly tests the student's understanding of the linear programming concepts and their ability to apply the standard procedure to solve such problems, as covered in the textbook.

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