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Linear Programming Set-4 MCQ Online Practice Test | Class 12 | Quiz Tweek

Linear Programming Set-4

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

Linear Programming is a core topic in Class 12 Mathematics that trains you to model real-world constraints into mathematical form and to solve optimization problems efficiently. It is frequently asked in CBSE board exams and also forms a base for higher-level reasoning in competitive exams (like JEE/NEET) because it develops strong skills in graphical/vertex methods, feasibility checks, and parameter-based solution analysis.

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Q1. Maximise $Z=3x+4y$ subject to

\begin{aligned} x &\ge 0 \\ y &\ge 0 \\ x+2y &\le 8 \\ 3x+y &\le 9 \end{aligned}

The maximum value of $Z$ is:

(A)

$16$

(B)

$9$

(C)

$18$

(D)

$20$

Q2. A factory makes two products A and B. Let $x$ and $y$ denote the number of units of A and B produced. Each unit of A requires $1.5$ hours on machine I and $1$ hour on machine II; each unit of B requires $1$ hour on machine I and $2$ hours on machine II. Machine I is available for at most $120$ hours and machine II for at most $150$ hours. Profit per unit of A is ₹ $40$ and of B is ₹ $50$ . The maximum profit (in ₹) is:

(A)

₹ $4425$

(B)

₹ $3750$

(C)

₹ $3200$

(D)

₹ $4025$

Q3. For which values of $k$ is the point $(4,3)$ an optimal solution of the linear program Maximise $Z=x+ky$ subject to

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

(A)

$k\in\left(\tfrac{1}{3},2\right)$

(B)

$k\in\left[\,\tfrac{1}{3},3\right]$

(C)

$k\in\left[\tfrac{1}{2},2\right]$

(D)

$k\in\left[\tfrac{1}{3},2\right]$

Q4. Consider the system of inequalities

\begin{aligned} x-y &\le 1 \\ 2x+y &\le 8 \\ x+3y &\ge p \\ x,y &\ge 0 \end{aligned}

For which values of the parameter $p$ does this system have no feasible solution?

(A)

$p>18$

(B)

$p>24$

(C)

$p\ge 24$

(D)

$p\ge 25$

Q5. Consider the linear program Maximise $Z=x+y$ subject to

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

where $x,y$ are real numbers (no non-negativity restriction). Which of the following is true about the optimal solutions?

(A)

The problem has a unique optimal solution at $(4,2)$ .

(B)

The system has no feasible solution.

(C)

There are infinitely many optimal solutions; every point $(t,6-t)$ with $t\ge 4$ is optimal.

(D)

The maximum value of $Z$ is $5$ .

Q6. Maximise $Z=3x+2y$ subject to

\begin{aligned} x+y &\le 6 \\ 2x+y &\le 8 \\ x,\;y &\ge 0 \end{aligned}

The maximum value of $Z$ is:

(A)

$12$

(B)

$13$

(C)

$14$

(D)

$10$

Q7. Consider the linear programming problem: Maximise $Z=2x+4y$ subject to

\begin{aligned} x+2y &\le 6 \\ 2x+4y &\le 12 \\ x,\;y &\ge 0 \end{aligned}

The nature of the optimal solution is:

(A)

Unique optimal solution at a single vertex

(B)

Infinitely many optimal solutions (alternate optima)

(C)

No feasible solution

(D)

Unbounded (no maximum)

Q8. For which values of $k$ does the linear programming problem Maximise $Z=x+ky$ subject to

\begin{aligned} x+y &\le 6 \\ 2x+y &\le 8 \\ x,\;y &\ge 0 \end{aligned}

have a unique optimal solution at the vertex $(2,4)$ ?

(A)

$k\in\left(\tfrac{1}{2},1\right)$

(B)

$k\in\left[\tfrac{1}{2},1\right]$

(C)

$k<\tfrac{1}{2}$

(D)

$k>1$

Q9. Assertion (A): The feasible region of the linear programming problem given below is unbounded. Reason (R): The linear program has no minimum. Consider: minimise $Z=3x+4y$ subject to

\begin{aligned} x+2y &\ge 6 \\ x+y &\ge 4 \\ x,\;y &\ge 0 \end{aligned}

Which one of the following is correct?

(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 true but (R) is false

(D)

(A) is false but (R) is true

Q10. Minimise $Z=x+2y$ subject to

\begin{aligned} x+y &\ge 6 \\ 2x+y &\ge 8 \\ x,\;y &\ge 0 \end{aligned}

The minimum value of $Z$ is:

(A)

$8$

(B)

$10$

(C)

$16$

(D)

$6$

Q11. Maximise $Z=3x+2y$ subject to

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

. The maximum value of $Z$ is:

(A)

$9$

(B)

$6$

(C)

$10$

(D)

$8$

Q12. For which values of $k$ does the linear programme Maximise $Z=4x+ky$ subject to

\begin{aligned} x+y &\le 6 \\ 2x+y &\le 8 \\ x,y &\ge 0 \end{aligned}

have $(0,6)$ as an optimal solution (not necessarily unique)?

(A)

$k\ge 4$

(B)

$k>4$

(C)

$k\le 4$

(D)

No real value of $k$ makes $(0,6)$ optimal

Q13. Find the minimum value of $Z=6x+4y$ subject to

\begin{aligned} x+2y &\ge 6 \\ 3x+y &\ge 6 \\ x,y &\ge 0 \end{aligned}

(A)

$12$

(B)

$18$

(C)

No finite minimum

(D)

$\dfrac{84}{5}$

Q14. For what value of $k$ does the linear programme Maximise $Z=kx+4y$ subject to

\begin{aligned} x+2y &\le 8 \\ 3x+y &\le 12 \\ x,y &\ge 0 \end{aligned}

have infinitely many optimal solutions?

(A)

$k=4$

(B)

$k=2$

(C)

$k=-2$

(D)

No real value of $k$ gives infinitely many optima

Q15. Maximise $Z=5x+7y$ subject to

\begin{aligned} x+2y &\le 10 \\ 3x+y &\le 12 \\ x &\le 4 \\ x,y &\ge 0 \end{aligned}

. Which constraint is redundant at the optimal solution?

(A)

Constraint 3 ( $x\le 4$ ) is redundant

(B)

Constraint 1 ( $x+2y\le 10$ ) is redundant

(C)

Constraint 2 ( $3x+y\le 12$ ) is redundant

(D)

No constraint is redundant

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