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Linear Programming Set-3 MCQ Online Practice Test | Class 12 | Quiz Tweek

Linear Programming Set-3

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

Linear Programming (LP) is one of the most scoring and conceptually important topics in Class 12 Mathematics because it trains you to model real-world constraints using linear inequalities and then optimize an objective function. Board exams and competitive tests both ask you to form LP models, find feasible regions, determine optimal solutions (unique or infinite), and analyze how parameters affect the optimum—so mastering LP directly improves both accuracy and speed.

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Q1. Maximise the objective function $Z=3x+5y$ subject to the constraints

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

Find the maximum value of $Z$ and the point at which it occurs.

(A)

Maximum value is $20$ at $(0,4)$

(B)

Maximum value is $21$ at $(2,3)$

(C)

Maximum value is $9$ at $(3,0)$

(D)

Maximum value is $18$ at $(1,3)$

Q2. A shopkeeper mixes two types of grains A and B. Cost per kg of A is ₹ $3$ and of B is ₹ $2$ . Each kg of A contains $4$ units of protein and $1$ unit of fibre; each kg of B contains $2$ units of protein and $3$ units of fibre. He needs at least $10$ units of protein and $9$ units of fibre. Let $x,y$ denote kg of A and B respectively. Formulate and solve the LP to minimise cost. Which of the following gives the minimum-cost purchase?

(A)

Buy $(1,3)$ kg of $(A,B)$ with total cost ₹ $9$

(B)

Buy $(0,5)$ kg of $(A,B)$ with total cost ₹ $10$

(C)

Buy $\left(\tfrac{6}{5},\tfrac{13}{5}\right)$ kg of $(A,B)$ with total cost ₹ $\tfrac{44}{5}$

(D)

Buy $(9,0)$ kg of $(A,B)$ with total cost ₹ $27$

Q3. Consider the LP: Maximise $Z=2x+4y$ subject to

\begin{aligned} x+2y &\le 8<br> 3x+6y &\le 24<br> x,y &\ge 0 \end{aligned}

Which of the following statements is correct about optimality?

(A)

Feasible region is non-empty and there are infinitely many optimal solutions; every point on the line segment joining $(0,4)$ and $(8,0)$ is optimal

(B)

There is a unique optimal solution at $(2,3)$ with maximum $Z=12$

(C)

Feasible region is unbounded so $Z$ is unbounded

(D)

Feasible region is empty

Q4. Assertion (A): "If the feasible region of a linear programming problem is non-empty and bounded, then the optimum of any linear objective function over it is always unique."
Reason (R): "If the objective function is parallel to an edge of the feasible polygon, then all points on that edge yield the same objective value and therefore there are infinitely many optimal solutions."

(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

Q5. For the LP Maximise $Z=px+y$ subject to

\begin{aligned} x+y &\le 4<br> x &\le 2<br> x,y &\ge 0 \end{aligned}

determine for which values of the parameter $p$ the problem has infinitely many optimal solutions.

(A)

For all $p>1$

(B)

For $p=1$

(C)

For all $p<1$

(D)

There is no value of $p$ that yields infinitely many optimal solutions

Q6. Consider the linear programming problem: Maximise $Z=3x+2y$ subject to

\begin{aligned} x+y &\le 4 <br> 2x+y &\le 6 <br> x,y &\ge 0 \end{aligned}

The maximum value of $Z$ is

(A)

$9$

(B)

$10$

(C)

$8$

(D)

$12$

Q7. For which values of $k$ does the LP Maximise $Z=kx+y$ subject to

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

have a unique optimal solution at the point $\left(\tfrac{8}{3},\tfrac{8}{3}\right)$ ?

(A)

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

(B)

$k \ge 2$

(C)

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

(D)

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

Q8. Consider the LP Maximise $Z=6x+5y$ subject to

\begin{aligned} x+y &\le 6 <br> 2x+y &\le 8 <br> x+2y &\le 12 <br> x,y &\ge 0 \end{aligned}

Which of the following constraints is redundant (can be removed without changing the feasible region)?

(A)

$x+2y \le 12$

(B)

$x+y \le 6$

(C)

$2x+y \le 8$

(D)

None of the above — all are necessary

Q9. Consider the LP Maximise $Z=2x+2y$ subject to

\begin{aligned} x+y &\le 4 <br> 2x+y &\le 6 <br> x,y &\ge 0 \end{aligned}

Assertion (A): The LP has infinitely many optimal solutions.
Reason (R): The objective function's level lines are parallel to the side of the feasible polygon joining $(0,4)$ and $(2,2)$ .

(A)

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

(B)

A is true and R is false.

(C)

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

(D)

A is false and R is true.

Q10. Consider the primal LP Maximise $Z=6x+5y$ subject to

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

Its dual minimisation problem (with variables $u,v\ge0$ ) is $W=6u+8v$ subject to $u+2v\ge6,\;u+v\ge5$ . If the primal optimum is attained at $(2,4)$ , which pair is a dual optimal solution by complementary slackness?

(A)

$(u,v)=(4,1)$

(B)

$(u,v)=(3,2)$

(C)

$(u,v)=(2,2)$

(D)

$(u,v)=(4,0)$

Q11. A firm wants to maximize profit given by $Z=5x+4y$ where $x,y$ are quantities of two products. Maximise $Z$ subject to the constraints:

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

What is the maximum value of $Z$ ?

(A)

$16$

(B)

$22$

(C)

$15$

(D)

$0$

Q12. For the LPP maximize $Z=kx+y$ subject to

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

the point of intersection of $x+y=8$ and $2x+y=10$ is $(2,6)$ . For which values of the parameter $k$ does the point $(2,6)$ give the maximum value of $Z$ ?

(A)

$k>2$

(B)

$k<1$

(C)

$1<k<2$

(D)

$1\le k\le 2$

Q13. Consider the LPP maximize $Z=x+2y$ subject to

\begin{aligned} x+2y &\le 8 \\ x &\le 4 \\ x,y &\ge 0 \end{aligned}

Which of the following correctly describes the set of optimal solutions?

(A)

Infinitely many points on the line segment joining $(0,4)$ and $(4,2)$

(B)

Unique optimal solution at $(4,2)$

(C)

No feasible solution

(D)

Unique optimal solution at $(0,4)$

Q14. Consider the LPP maximize $Z=3x+6y$ subject to

\begin{aligned} x+2y &\le 8 \\ 2x+4y &\le 16 \\ x,y &\ge 0 \end{aligned}

Assertion (A): The LPP has infinitely many optimal solutions. Reason (R): One of the constraints is redundant and the objective function is a scalar multiple of the other constraint.

(A)

A is true but R is false

(B)

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

(C)

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

(D)

A is false but R is true

Q15. For the LPP maximize $Z=4x+y$ subject to

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

where $t\in\mathbb{R}$ . When $t=0$ the optimal solution is $(5,0)$ . For which values of $t$ does $(5,0)$ remain an optimal solution?

(A)

$t>0$

(B)

$t\ge -7$

(C)

$-7<t<0$

(D)

$t\le -7$

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