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Linear Programming Set-2

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

Linear Programming is crucial for both board and competitive exams because it provides a systematic method to optimize a quantity under multiple linear constraints. Problems based on feasible regions, corner points (vertices), and conditions for unique vs infinite optimal solutions frequently appear in CBSE and JEE/NEET-style questions, making strong concept clarity essential.

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

\begin{aligned} x&\ge0,\\ y&\ge0,\\ x+2y&\le8,\\ 3x+y&\le9. \end{aligned}

Find the maximum value of $Z$ .

(A)

$16$

(B)

$18$

(C)

$12$

(D)

$9$

Q2. For which value of $p$ does the maximum value of $Z=x+py$ subject to

\begin{aligned} x&\ge0,\\ y&\ge0,\\ x+y&\le6,\\ 2x+y&\le8 \end{aligned}

occur at both points $(0,6)$ and $(2,4)$ simultaneously?

(A)

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

(B)

$p=2$

(C)

$p=-1$

(D)

$p=1$

Q3. A factory makes two products P and Q. Each unit of P requires $2$ hours on machine A and $1$ hour on machine B. Each unit of Q requires $1$ hour on machine A and $3$ hours on machine B. Machine A is available for $100$ hours and machine B for $120$ hours. Profit per unit is Rs. $40$ for P and Rs. $50$ for Q. Let $x,y$ denote units of P and Q produced. Maximise $Z=40x+50y$ subject to

\begin{aligned} x&\ge0,\\ y&\ge0,\\ 2x+y&\le100,\\ x+3y&\le120. \end{aligned}

The maximum profit (in rupees) is:

(A)

$2000$

(B)

$2400$

(C)

$2840$

(D)

$3200$

Q4. Maximise $Z=2x+4y$ subject to

\begin{aligned} x&\ge0,\\ y&\ge0,\\ x+2y&\le10,\\ 2x+y&\le10. \end{aligned}

The set of optimal solutions forms a line segment. The length of that segment is:

(A)

$\dfrac{5\sqrt{5}}{3}$

(B)

$\dfrac{10}{3}$

(C)

$\dfrac{4\sqrt{5}}{3}$

(D)

$\dfrac{5\sqrt{2}}{3}$

Q5. Let $p,q>0$ and consider maximise $Z=px+qy$ subject to

\begin{aligned} 2x+y&\le8,\\ x+3y&\le9,\\ x&\ge0,\\ y&\ge0. \end{aligned}

The point $(3,2)$ is a vertex of the feasible region. For which range of $q$ (in terms of $p$ ) will $(3,2)$ be the unique optimal solution?

(A)

$q\le \dfrac{p}{2}$

(B)

$\dfrac{p}{2}<q<3p$

(C)

$q\ge 3p$

(D)

$q=\dfrac{p}{2}\ \text{or}\ q=3p$

Q6. For the feasible region defined by $x+2y\le 8,\ 3x+y\le 12,\ x,y\ge 0$ , the maximum value of $Z=5x+4y$ is:

(A)

$25$

(B)

$\dfrac{128}{5}$

(C)

$24$

(D)

$26$

Q7. Consider the linear programming problem $\max Z=2x+4y$ subject to $x+2y\le 10,\ x\ge 0,\ y\ge 0$ . Which of the following is true?

(A)

The unique maximum occurs at $(10,0)$ .

(B)

The unique maximum occurs at $(0,5)$ .

(C)

$Z$ is unbounded above.

(D)

Every point on the line segment joining $(10,0)$ and $(0,5)$ is optimal (infinitely many optimal solutions).

Q8. Find the minimum value of $Z=3x+4y$ subject to $x+y\ge 6,\ 2x+y\ge 8,\ x,y\ge 0$ .

(A)

Minimum $18$ at $(6,0)$ .

(B)

Minimum $22$ at $(2,4)$ .

(C)

Minimum $21$ at $(3,3)$ .

(D)

No minimum (unbounded below).

Q9. For which value(s) of the parameter $p$ does the linear program $\max Z=px+7y$ subject to $x+2y\le 8,\ 3x+y\le 9,\ x,y\ge 0$ have infinitely many optimal solutions?

(A)

$p>\dfrac{7}{2}$

(B)

$p<\dfrac{7}{2}$

(C)

$p=\dfrac{7}{2}$

(D)

No real value of $p$

Q10. For the problem $\max Z=40x+30y$ subject to $3x+2y\le 18,\ 2x+5y\le 20,\ x,y\ge 0$ , if an additional constraint $x+y\le 6$ is imposed, the new optimal solution is:

(A)

$\left(0,4\right)$

(B)

$(6,0)$

(C)

$\left(\dfrac{10}{3},\dfrac{8}{3}\right)$

(D)

The feasible region becomes empty (no feasible solution)

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