Linear Programming – important questions for Boards 2027
CBSE Class 12 Maths
For a feasible region with corner points (0,0),(2,0),(0,3) and objective function Z=2x+y, evaluate Z at the corner points and give the point where its maximum occurs.
Answer
At (0, 0): Z = 2(0) + 0 = 0 At (2, 0): Z = 2(2) + 0 = 4 At (0, 3): Z = 2(0) + 3 = 3 ∴ Maximum value is 4 at (2, 0).
For the constraints x+y≤5, x≥0, y≥0, which point is an infeasible solution? (A) (1,2) (B) (0,5) (C) (3,3) (D) (2,2)
Answer
(C) (3, 3) — it violates x + y ≤ 5, so it is an infeasible solution.
For the constraints x+y≤6, x≥0, y≥0, determine whether (2,3) and (4,3) belong to the feasible region, checking all the constraints.
Answer
(2, 3): 2 + 3 = 5 ≤ 6 2 ≥ 0, 3 ≥ 0 ∴ (2, 3) belongs to the feasible region. (4, 3): 4 + 3 = 7 > 6 4 ≥ 0, 3 ≥ 0 ∴ (4, 3) does not belong to the feasible region.
The feasible region has corner points (0,0),(4,0),(0,3), and the objective function is Z=2x+3y. Evaluate Z at each corner point and find the optimal feasible solution when Z is to be maximised.
Answer
Z = 2x + 3y At (0, 0): Z = 2(0) + 3(0) = 0 At (4, 0): Z = 2(4) + 3(0) = 8 At (0, 3): Z = 2(0) + 3(3) = 9 ∴ Maximum value of Z is 9 at (0, 3).
Define constraints in a linear programming problem and distinguish non-negative restrictions from other constraints. Define an optimisation problem and state how a linear programming problem is related to it. (5 marks)
Answer
1. Meaning — Constraints: linear inequalities, equations or restrictions on the variables of a linear programming problem; non-negative restrictions: constraints requiring the variables to be non-negative. 2. Form — Other constraints may be linear inequalities or equations, such as 5x + y ≤ 100; non-negative restrictions are expressed as x ≥ 0 and y ≥ 0. 3. Purpose — Other constraints limit the permitted values of the variables according to the problem; non-negative restrictions ensure that the variables cannot take negative values. 4. Optimisation problem — An optimisation problem seeks to maximise or minimise a linear function subject to constraints determined by a set of linear inequalities. 5. Relation — An optimisation problem may involve maximisation or minimisation; a linear programming problem is a special type of optimisation problem in which a linear objective function is subject to linear constraints and non-negative variables.
Find the intersection point of the boundary lines x+y=5 and x-y=1; this point is a corner point of the feasible region when both boundaries are active.
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For an unbounded feasible region, which statement about the objective function is correct? (A) A maximum and minimum must both exist. (B) An optimum may not exist; if it does, it occurs at a corner point. (C) An optimum can occur only away from corner points. (D) The feasible region must be empty.
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A bounded feasible region has corner points (0,0),(4,0),(0,3). Evaluate Z=2x-y at all three points and find its maximum and minimum values.
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For the system x≥0, y≥0, x+y≤8, find the intercepts of the boundary line and use them to determine whether the feasible region is bounded.
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Explain how the corner-point method determines the maximum and minimum of an objective function Z=ax+by when the feasible region is bounded. For an unbounded feasible region, state the half-plane tests used to confirm a candidate maximum M and a candidate minimum m. (5 marks)
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12 more probable questions in this chapter, every one with a full answer.
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