Quiz Tweek

Preparing your workspace...

Linear Programming Set-6 MCQ Online Practice Test | Class 12 | Quiz Tweek

Linear Programming Set-6

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

Linear Programming (LP) is one of the most scoring and conceptually important topics for Class 12 Mathematics as it directly trains you to model real-life constraints into inequalities and optimize a linear objective. It also has high relevance in competitive exams because the same core ideas—feasible region, corner points, and conditions for unique vs infinitely many optimal solutions—appear repeatedly in different forms and parameter-based questions.

Minutes

Questions

1 / -0

Marking

Question Bank Preview

Q1. Consider the linear programming problem (LPP):

\max Z=3x+4y

subject to

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

.
The maximum value of $Z$ is:

(A)

$14$

(B)

$16$

(C)

$12$

(D)

$10$

Q2. Maximize

Z=5x+7y

subject to

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

.
The maximum value of $Z$ is:

(A)

$31$

(B)

$28$

(C)

$15$

(D)

$32$

Q3. Consider the LPP

\max Z=2x+4y

subject to

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

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

(A)

Unique optimal solution at $(0,4)$ with $Z=16$

(B)

Unique optimal solution at $(8,0)$ with $Z=16$

(C)

No finite maximum since feasible region is unbounded

(D)

Infinitely many optimal solutions lying on the segment joining $(8,0)$ and $(0,4)$ , each with $Z=16$

Q4. For which value(s) of $k$ does the LPP

\max Z=(4+k)x+6y

subject to

\begin{aligned} x+y &\le 10 \\ 2x+y &\le 14 \\ x,y &\ge 0 \end{aligned}

have infinitely many optimal solutions?

(A)

$k=2$ only

(B)

$k=8$ only

(C)

$k=-4$

(D)

$k=2$ or $k=8$

Q5. Assertion (A): Consider the LPP

\max Z=x-y

subject to

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

. Then the LPP has a finite maximum.
Reason (R): The constraint $x-y\le 2$ implies $Z=x-y\le 2$ for every feasible point, so $Z$ is bounded above by $2$ .

(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

Q6. A linear programming problem is given: maximize $Z=5x+3y$ subject to

\begin{aligned} x+2y &\le 10 \\ 3x+y &\le 9 \\ x,\,y &\ge 0 \end{aligned}

. Find the maximum value of $Z$ .

(A)

$15$

(B)

$20$

(C)

$\dfrac{103}{5}$

(D)

$21$

Q7. Consider the linear programming problem: minimize $Z=4x+6y$ subject to

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

. Determine the minimum value of $Z$ and where it occurs.

(A)

Minimum is $16$ , attained only at $(1,2)$ .

(B)

Minimum is $16$ , attained at every point on the line segment joining $(1,2)$ and $(4,0)$ .

(C)

Minimum is $18$ , attained at $(0,3)$ .

(D)

No minimum; the problem is unbounded below.

Q8. For the linear programming problem maximize $Z=x+ky$ subject to

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

, for which value(s) of $k$ does the maximum value of $Z$ occur at more than one vertex of the feasible region?

(A)

$\left\{\dfrac{1}{3}\right\}$

(B)

$\{2\}$

(C)

$\left\{\dfrac{3}{4}\right\}$

(D)

$\left\{\dfrac{1}{3},\,2\right\}$

Q9. Statement I: The linear program maximize $Z=2x+4y$ subject to

\begin{aligned} x+2y &\le 8 \\ 4x+8y &\le 32 \\ x,\,y &\ge 0 \end{aligned}

has infinitely many optimal solutions.
Statement II: The second constraint is redundant (a multiple of the first) and the objective function is proportional to $x+2y$ , so the maximum is attained on the entire line $x+2y=8$ within the feasible region.

(A)

Both statements are true and Statement II is a correct explanation of Statement I.

(B)

Both statements are true but Statement II is not a correct explanation of Statement I.

(C)

Statement I is true and Statement II is false.

(D)

Statement I is false and Statement II is true.

Q10. Let the linear programming problem be: maximize $Z=3x+my$ subject to

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

. For which values of $m$ is the point $(2,3)$ the unique optimal solution?

(A)

$m\le 1$

(B)

$1<m<6$

(C)

$1\le m\le 6$

(D)

$m\ge 6$

Q11. Maximise $Z=4x+3y$ subject to $x+y\le 5,\;2x+y\le 8,\;x,y\ge 0$ . The maximum value of $Z$ is

(A)

$Z_{\max}=16$ at $(4,0)$

(B)

$Z_{\max}=15$ at $(0,5)$

(C)

$Z_{\max}=18$ at $(3,2)$

(D)

$Z_{\max}=17$ at $(2,3)$

Q12. For which values of $p$ is the point $(3,3)$ an optimal solution of the LP: maximise $Z=px+2y$ subject to $x+y\le 6,\;2x+y\le 9,\;x,y\ge 0$ ?

(A)

$p<2$

(B)

$2\le p\le 4$

(C)

$p>4$

(D)

$2<p<4$

Q13. Consider the LP: maximise $Z=6x+3y$ subject to $2x+y\le 8,\;x+y\le 6,\;x,y\ge 0$ . Which statement is correct about the optimal solution?

(A)

Unique maximum at $(4,0)$ with $Z_{\max}=24$

(B)

Unique maximum at $(2,4)$ with $Z_{\max}=24$

(C)

Maximum attained only at $(0,6)$ with $Z_{\max}=18$

(D)

Infinitely many optimal solutions lie on the line segment joining $(4,0)$ and $(2,4)$ , with $Z_{\max}=24$

Q14. Maximise $Z=5x+7y$ subject to $x+y\le 6,\;2x+2y\ge 12,\;x,y\ge 0$ . Which of the following is true?

(A)

Unique maximum at $(0,6)$ with $Z_{\max}=42$

(B)

Infinitely many optimal solutions along the segment joining $(6,0)$ and $(0,6)$

(C)

The LP is unbounded

(D)

Unique maximum at $(6,0)$ with $Z_{\max}=30$

Q15. For which values of $k$ is the point $(2,4)$ an optimal solution of the LP: maximise $Z=kx+y$ subject to $x+y\le 6,\;2x+y\le 8,\;x,y\ge 0$ ? In addition, when is the optimum at $(2,4)$ unique?

(A)

$(2,4)$ is optimal only for $1<k<2$ and not optimal at $k=1$ or $k=2$

(B)

$(2,4)$ is optimal for $k\in[1,2]$ , and the optimum is unique for $1<k<2$ while for $k=1$ or $k=2$ there are infinitely many optimal solutions

(C)

$(2,4)$ is optimal only at $k=1$ and $k=2$ , and unique there

(D)

$(2,4)$ is never an optimal solution

...and 5 more challenging questions available in the interactive simulator.

More Quizzes for Class 12

Explore other chapters and subjects, or view all quizzes for this class.

Quiz Tweek

Linear Programming Set-6

Question Bank Preview

More Quizzes for Class 12

Chemistry

Mathematics

Physics