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Linear Programming Set-5

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

Linear Programming (LP) is crucial because it models real-world constraints with a linear objective and helps decide the best (maximum or minimum) outcome efficiently. It forms the backbone of many CBSE Board and competitive exam questions through graph-based solution, corner-point method, and parameter-based reasoning about optimality and degeneracy (unique/infinite/multiple optima).

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Q1. For the feasible region defined by

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

find the maximum value of $Z=4x+3y$ .

(A)

$12$

(B)

$15$

(C)

$17$

(D)

$9$

Q2. For the linear programme with

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

let $Z=kx+y$ . For which values of $k$ is the point $(2,4)$ an optimal solution (not necessarily unique)?

(A)

$k<1$

(B)

$1\le k\le 2$

(C)

$k>2$

(D)

$k=1\text{ or }k=2$

Q3. A workshop makes two types of chairs S and D. Each S needs $2$ hours of carpentry and $1$ hour of polishing; each D needs $1$ hour of carpentry and $2$ hours of polishing. Daily available hours are $16$ (carpentry) and $12$ (polishing). Profit per S is $40$ and per D is $50$ . Let $x,y$ be numbers of S and D produced. Maximise $Z=40x+50y$ subject to

\begin{aligned} 2x+y &\le 16 \\ x+2y &\le 12 \\ x,y &\ge 0. \end{aligned}

What is the maximum profit and at which $(x,y)$ is it attained?

(A)

$Z_{\max}=400$ at $\left(\dfrac{20}{3},\dfrac{8}{3}\right)$

(B)

$Z_{\max}=320$ at $(8,0)$

(C)

$Z_{\max}=300$ at $(0,6)$

(D)

$Z_{\max}=360$ at $(4,4)$

Q4. Assertion (A): If the feasible region of a linear programming problem is unbounded and non-empty, then the maximum of any linear objective function over that region does not exist (is unbounded above).
Reason (R): If there exists a feasible point $x_0$ and a direction $d$ such that $x_0+td$ is feasible for all $t\ge 0$ , and the objective coefficient vector $c$ satisfies $c\!\cdot\!d>0$ , then $c\!\cdot\!(x_0+td)\to\infty$ as $t\to\infty$ , so the maximum does not exist.

(A)

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

(B)

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

(C)

A is true but R is false.

(D)

A is false but R is true.

Q5. For the LP maximize $Z=x+ky$ subject to

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

determine all real values of $k$ for which the LP has more than one optimal solution.

(A)

No real value of $k$

(B)

$k=\dfrac{1}{3}\ \text{or}\ k=2$

(C)

All $k<0$

(D)

$k=1$ only

Q6. Maximise $Z=3x+5y$ subject to

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

(A)

$20$ at $(0,4)$

(B)

$21$ at $(2,3)$

(C)

$9$ at $(3,0)$

(D)

$19$ at $\left(\dfrac{13}{6},\dfrac{5}{2}\right)$

Q7. Maximise $Z=6x+5y$ subject to

\begin{aligned} 2x+3y &\le 18 \\ x+y &\le 7 \\ x-y &\ge -2 \\ x,y &\ge 0 \end{aligned}

(A)

$38$ at $(3,4)$

(B)

$\dfrac{182}{5}$ at $\left(\dfrac{12}{5},\dfrac{22}{5}\right)$

(C)

$10$ at $(0,2)$

(D)

$42$ at $(7,0)$

Q8. Minimise $Z=4x+3y$ subject to

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

(A)

$12$ at $\left(\dfrac{6}{5},\dfrac{12}{5}\right)$

(B)

$18$ at $(0,6)$

(C)

$24$ at $(6,0)$

(D)

$15$ at $(3,3)$

Q9. For which values of $k$ does the LPP have infinitely many optimal solutions? Maximise $Z=kx+6y$ subject to

\begin{aligned} x+2y &\le 6 \\ 3x+2y &\le 12 \\ x,y &\ge 0 \end{aligned}

(A)

$k=3$

(B)

$k=9$

(C)

$k=3\ \text{or}\ k=9$

(D)

No real $k$

Q10. Consider the LPP: Maximise $Z=x+ky$ subject to

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

For which real value(s) of $k$ does the maximum value of $Z$ occur uniquely at the point $(2,3)$ ?

(A)

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

(B)

No real value of $k$

(C)

$k=\dfrac{5}{2}$

(D)

$k=3$

Q11. (Maximise the objective function $Z=3x+2y$ subject to the constraints

\begin{aligned} x+y &\le 6 \\ 2x+y &\le 9 \\ x,\;y &\ge 0 \end{aligned}

Find the maximum value of $Z$ and the point where it occurs.)

(A)

$\displaystyle \max Z=15\text{ at }(3,3)$

(B)

$\displaystyle \max Z=12\text{ at }(0,6)$

(C)

$\displaystyle \max Z=\frac{27}{2}\text{ at }\left(\frac{9}{2},0\right)$

(D)

$\displaystyle \max Z=14\text{ at }(2,4)$

Q12. (Minimise $Z=5x+3y$ subject to

\begin{aligned} x+2y &\ge 14 \\ 3x+y &\ge 12 \\ x,\;y &\ge 0 \end{aligned}

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

(A)

$\displaystyle \min Z=36\text{ at }(0,12)$

(B)

$\displaystyle \min Z=28\text{ at }(2,6)$

(C)

$\displaystyle \min Z=70\text{ at }(14,0)$

(D)

$\displaystyle \min Z=\frac{63}{2}\text{ at }\left(3,\frac{11}{2}\right)$

Q13. (Maximise $Z=x+3y$ subject to

\begin{aligned} 2x+3y &\le 18 \\ x+4y &\le 16 \\ x,\;y &\ge 0 \end{aligned}

Determine the maximum value of $Z$ and the point where it occurs.)

(A)

$\displaystyle \max Z=\frac{66}{5}\text{ at }\left(\frac{24}{5},\frac{14}{5}\right)$

(B)

$\displaystyle \max Z=12\text{ at }(0,4)$

(C)

$\displaystyle \max Z=9\text{ at }(9,0)$

(D)

$\displaystyle \max Z=13\text{ at }(4,3)$

Q14. (Consider the LPP: maximise $Z=kx+y$ subject to

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

For which values of the parameter $k$ does this LPP have infinitely many optimal solutions?)

(A)

$k=1\ \text{only}$

(B)

$k=2\ \text{only}$

(C)

$1<k<2$

(D)

$k=1\ \text{or}\ k=2$

Q15. (Maximise $Z=kx+y$ subject to

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

For which values of the parameter $k$ is the point $(4,2)$ an optimal solution of the LPP? (Include cases where $(4,2)$ is an optimal solution but not unique.))

(A)

$\displaystyle \frac{1}{2}\le k\le 1$

(B)

$\displaystyle k>1$

(C)

$\displaystyle k<\frac{1}{2}$

(D)

$\displaystyle k=1\ \text{only}$

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