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Linear Programming Set-1 MCQ Online Practice Test | Class 12 | Quiz Tweek

Linear Programming Set-1

Practice Important MCQs for Linear Programming — Mathematics, Class 12.

Linear Programming (LP) is a core optimization chapter in CBSE Class 12 and a frequent topic in competitive exams because it develops the skill to model real-world constraints and optimize an objective. Mastering LP helps in solving problems using feasible regions, corner-point (vertex) methods, and conditions for unique vs infinitely many optima—concepts that directly appear in board questions and selection-stage MCQs.

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Q1. Consider the linear programming problem: Maximise $Z=3x+2y$ subject to

\begin{aligned} x+2y &\le 8 \\ 2x+y &\le 10 \\ x,y &\ge 0 \end{aligned}

Find the maximum value of $Z$ .

(A)

$16$

(B)

$20$

(C)

$24$

(D)

$12$

Q2. For which real values of $k$ does the maximum of $Z=kx+4y$ subject to

\begin{aligned} x+2y &\le 10 \\ 3x+y &\le 15 \\ x,y &\ge 0 \end{aligned}

occur uniquely at the intersection point of the two constraint lines?

(A)

$2<k<12$

(B)

$2\le k\le 12$

(C)

$k\le 2\ \text{or}\ k\ge 12$

(D)

$k<2$

Q3. For the linear programming problem maximize $Z=px+6y$ subject to

\begin{aligned} x+2y &\le 8 \\ 2x+y &\le 8 \\ x,y &\ge 0 \end{aligned}

find all real values of $p$ for which the problem has infinitely many optimal solutions.

(A)

$3\le p\le 12$

(B)

$p=3\ \text{or}\ p=12$

(C)

$p=6$

(D)

No real value of $p$ gives infinitely many optimal solutions

Q4. Statement A: "If the objective function is parallel to one of the bounding constraints of a bounded feasible region, then the linear programming problem necessarily has infinitely many optimal solutions."
Statement R: "If the objective function is parallel to a boundary segment and that segment lies where the objective achieves its maximum (or minimum), then every point on that segment is optimal."

(A)

Both A and R are true and R is the correct explanation of A.

(B)

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

(C)

A is true and R is false.

(D)

A is false and R is true.

Q5. Consider maximize $Z=mx+ny$ subject to

\begin{aligned} x+2y &\le 6 \\ 2x+y &\le 6 \\ x,y &\ge 0 \end{aligned}

where $m,n>0$ . The maximum is uniquely attained at the vertex $(2,2)$ if and only if which condition on the ratio $\dfrac{m}{n}$ holds?

(A)

$\dfrac{m}{n}\in\left(\tfrac{1}{2},2\right)$

(B)

$\dfrac{m}{n}\le \tfrac{1}{2}\ \text{or}\ \dfrac{m}{n}\ge 2$

(C)

$\dfrac{m}{n}=1$

(D)

$\dfrac{m}{n}=\tfrac{1}{2}\ \text{or}\ \dfrac{m}{n}=2$

Q6. Consider the linear programming problem: Maximize $Z=3x+4y$ subject to $x+2y\le 6,\; x\le 4,\; x\ge 0,\; y\ge 0$ . The maximum value of $Z$ is

(A)

Maximum $Z=12$ at $(0,3)$

(B)

Maximum $Z=12$ at $(4,0)$

(C)

Maximum $Z=16$ at $(4,1)$

(D)

Maximum $Z=15$ at $(3,\tfrac{3}{2})$

Q7. Minimize $Z=5x+3y$ subject to $x+y\ge 4,\; 2x+y\ge 5,\; x\le 3,\; x\ge 0,\; y\ge 0$ . The minimum value of $Z$ and the point at which it occurs is

(A)

Minimum $Z=14$ at $(1,3)$

(B)

Minimum $Z=15$ at $(0,5)$

(C)

Minimum $Z=18$ at $(3,1)$

(D)

Minimum $Z=16$ at $(2,2)$

Q8. For the linear programming problem Maximize $Z=x+ky$ subject to $x+2y\le 8,\; 3x+y\le 12,\; x\ge 0,\; y\ge 0$ , the point of intersection of the two constraints is $\left(\dfrac{16}{5},\dfrac{12}{5}\right)$ . For which values of $k$ is the maximum of $Z$ attained at this point?

(A)

$k<\dfrac{1}{3}$

(B)

$k>2$

(C)

$\dfrac{1}{3}<k<2$

(D)

$\dfrac{1}{3}\le k\le 2$

Q9. Consider Maximize $Z=x+ky$ subject to $x+2y\le 8,\; 2x+3y\le 12,\; x\ge 0,\; y\ge 0$ . For which value of $k$ does this LPP have infinitely many optimal solutions (i.e., the entire edge of the feasible polygon is optimal)?

(A)

$k=\dfrac{2}{3}$

(B)

$k=\dfrac{3}{2}$

(C)

$k=\dfrac{1}{2}$

(D)

No real $k$ gives infinitely many optima

Q10. Assertion (A): In a two‑variable linear programming problem, if at an optimal vertex three or more constraints (including bound constraints) are active, then the LP must have infinitely many optimal solutions.
Reason (R): If three or more constraints are active at a vertex it produces degeneracy of the basic feasible solution; degeneracy alone does not imply multiple optima — multiple optimal solutions occur only when the objective function is parallel to a face (edge) of the feasible region.

(A)

Both A and R are true and R is a correct explanation of A

(B)

Both A and R are true but R is not a correct explanation of A

(C)

A is false and R is true

(D)

A is true and R is false

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