Linear Programming — Class 12 MCQs with Answers

Linear Programming — important questions & MCQs with answers (Class 12 Mathematics)

6 solved questions from this chapter's difficulty levels, each with its answer and explanation. Tap an option to check yourself.

0 of 6 answered

1. Q1 of 6Easy

   In LPP, the function to optimise is called:

   AVariableBConstraintCObjective functionDBoundary
   Show answerHide answer

   Answer: Objective function

   Explanation: The expression being maximised or minimised, like Z = 3x + 4y, is the objective function — the constraints only bound where Z is evaluated, they aren't what you optimise.

2. Q2 of 6Easy

   Linear Programming Problem (LPP) involves:

   ASolving differential equationsBMaximising or minimising a linear function under linear constraintsCQuadratic functionsDRandom optimisation
   Show answerHide answer

   Answer: Maximising or minimising a linear function under linear constraints

   Explanation: An LPP means maximising or minimising a linear function subject to linear constraints — swap in a squared term or a derivative and it stops being a standard LPP.

3. Q3 of 6Medium

   Feasible region is the:

   AOriginBSet of points satisfying all constraintsCSingle pointDWhole plane
   Show answerHide answer

   Answer: Set of points satisfying all constraints

   Explanation: The feasible region is the intersection of every constraint half-plane at once, including x ≥ 0 and y ≥ 0 — satisfying just one inequality alone isn't enough.

4. Q4 of 6Medium

   Assertion (A): A dealer who can store at most 60 pieces in all buys x tables and y chairs; his storage constraint is x + y ≤ 60. Reason (R): In a linear programming problem, the objective function is a linear function of the decision variables.

   ABoth A and R are true and R is the correct explanation of ABBoth A and R are true but R is not the correct explanation of ACA is true but R is falseDA is false but R is true
   Show answerHide answer

   Answer: Both A and R are true but R is not the correct explanation of A

   Explanation: A is true: the total number of pieces x + y cannot exceed 60. R is true: Z = ax + by is linear. R describes the objective function, not how the storage constraint is written, so it does not explain A.

5. Q5 of 6Hard

   Maximise Z = 4x + 5y subject to x + y ≤ 6, x + 2y ≤ 8, x, y ≥ 0:

   

   AAt (0, 4)BAt (4, 2)CAt (6, 0)DOrigin
   Show answerHide answer

   Answer: At (4, 2)

   Explanation: Z = 4x + 5y at the vertices (0,0), (6,0), (4,2), (0,4) gives 0, 24, 26, and 20 — the maximum is 26 at (4, 2), beating the larger-looking corner (6,0) which only reaches 24.

6. Q6 of 6Hard

   Assertion (A): Subject to 2x − y ≥ −5, 3x + y ≥ 3, 2x − 3y ≤ 12, x ≥ 0, y ≥ 0, the function Z = −50x + 20y has no minimum value. Reason (R): The feasible region is unbounded, the smallest value of Z at its corner points is −300, and the open half-plane −50x + 20y < −300 has points in common with the feasible region.

   ABoth A and R are true and R is the correct explanation of ABBoth A and R are true but R is not the correct explanation of ACA is true but R is falseDA is false but R is true
   Show answerHide answer

   Answer: Both A and R are true and R is the correct explanation of A

   Explanation: Both are true: the corner values are 100 at (0, 5), 60 at (0, 3), −50 at (1, 0) and −300 at (6, 0), and the feasible point (9, 2) gives Z = −410 < −300. For an unbounded region, the half-plane Z < −300 meeting the region is exactly what means Z has no minimum, so R explains A.

Linear Programming notes: key concepts (Class 12 Mathematics)

Linear programming optimises a linear objective Z = ax + by subject to linear inequality constraints and non-negativity. The chapter builds the feasible region graphically, identifies its corner points, and applies the corner-point method — the optimum of a bounded region lies at a vertex — with manufacturing and diet problems.

Linear programming problem

Optimising (maximising or minimising) a linear objective function of decision variables subject to linear constraints.

Objective function

The quantity to optimise, written Z = ax + by; profit is maximised, cost is minimised.

Decision variables

The unknowns x and y (e.g. units of two products) chosen to optimise Z; usually restricted to be non-negative.

Constraints

Linear inequalities such as x + y ≤ 10 that limit the variables, arising from resources, demand or capacity.

Non-negativity restrictions

x ≥ 0, y ≥ 0 confine the solution to the first quadrant, since quantities cannot be negative.

Feasible region

The set of all points satisfying every constraint simultaneously — the common shaded region of the inequalities.

Feasible vs infeasible

A feasible solution lies in the region; if the constraints share no common point the LPP is infeasible.

Bounded vs unbounded region

A bounded region can be enclosed in a circle; an unbounded one extends infinitely, and then a maximum may not exist.

Corner (vertex) points

The corners of the feasible region, found where boundary lines intersect; the optimum is searched only here.

Corner-point method

Evaluate Z at every vertex; the largest value is the maximum and the smallest the minimum for a bounded region.

Multiple optimal solutions

If the objective line is parallel to a constraint edge, every point on that edge is optimal — infinitely many solutions.

Types of LPP

Manufacturing (maximise profit), diet (minimise cost meeting nutrient minimums) and transportation problems recur in exams.