Linear Programming – Class 12 Mathematics notes, important questions and MCQs

Chapter overview

Linear programming finds the maximum or minimum of a linear objective function subject to linear inequalities (constraints), using the corner-point method.

Formulation

• Decision variables (x, y ≥ 0), objective function Z = ax + by, and constraints (linear inequalities).
• The feasible region is the common region satisfying all constraints.

Corner-point method

• The optimal value occurs at a corner (vertex) of the feasible region.
• Evaluate Z at each corner; pick the largest (max) or smallest (min).
• For an unbounded region, check with an extra inequality that the optimum exists.

Key terms

Feasible region

set of points satisfying all constraints

Objective function

linear function to be maximised or minimised

Corner point (vertex)

a point of the feasible region where two boundary lines meet

Formulas

• Z = ax + by

Important questions with answers

Quick revision

• Optimum lies at a corner point.
• Feasible region is convex.
• Check boundedness.

Exam tips

• Always include non-negativity constraints x ≥ 0, y ≥ 0.
• For an unbounded region, if the max is M at a corner, check that ax + by > M has no common point with the region before accepting it.

Practice MCQs

1. In an LPP, the function to optimise is the

   1. constraint
   2. slack
   3. feasible region
   4. objective function
   Show answer

   Answer: (d) objective function. Objective.

2. Optimal value of Z occurs at

   1. a corner point of the feasible region
   2. any interior point
   3. the origin only
   4. infinity
   Show answer

   Answer: (a) a corner point of the feasible region. Corner-point theorem.

3. Max Z = 3x + 2y at corners (0, 0), (4, 0), (2, 3), (0, 4) is

   1. 13
   2. 12
   3. 8
   4. 10
   Show answer

   Answer: (b) 12. 12 at (2, 3) and (4, 0); 8 at (0, 4); so max = 12.

4. The region satisfying x ≥ 0, y ≥ 0 lies in

   1. quadrant I
   2. quadrant II
   3. quadrant III
   4. quadrant IV
   Show answer

   Answer: (a) quadrant I. Both non-negative.

5. Which point lies in the region x + y ≤ 4, x ≥ 0, y ≥ 0

   1. (3, 2)
   2. (0, 5)
   3. (5, 0)
   4. (1, 2)
   Show answer

   Answer: (d) (1, 2). 1 + 2 = 3 ≤ 4.

6. Min Z = 2x + 5y at corners (0, 4), (2, 1), (5, 0) is

   1. 9
   2. 10
   3. 20
   4. 7
   Show answer

   Answer: (a) 9. At (2, 1): 4 + 5 = 9.

7. If Z has the same maximum at two corners, then

   1. there is no solution
   2. every point on the segment joining them is optimal
   3. the LPP is unbounded
   4. only one corner counts
   Show answer

   Answer: (b) every point on the segment joining them is optimal. Multiple optimal solutions.

8. Maximise Z = 5x + 3y subject to 3x + 5y ≤ 15, 5x + 2y ≤ 10, x, y ≥ 0. The maximum is

   1. 235/19
   2. 10
   3. 9
   4. 15
   Show answer

   Answer: (a) 235/19. At (20/19, 45/19).