Questions number 19 and 20 are Assertion and Reason based questions. Two statements are given, one labelled Assertion (A) and the other labelled Reason (R). Select the correct answer from the codes (A), (B), (C) and (D) as given below.
(A) Both Assertion (A) and Reason (R) are true and Reason (R) is...
CBSE CLASS 12 • MATHEMATICS • CHAPTER 12
Linear Programming
152 distinct questions across 208 board paper set appearances (2022–2026).
The degree of an objective function of a linear programming problem is
(A) $0$
(B) $1$
(C) $2$
(D) Any natural number
In a linear programming problem, the linear function which has to be maximized or minimized is called
(A) a feasible function
(B) an objective function
(C) an optimal function
(D) a constraint
For the feasible region shown below, the non-trivial constraints of the linear programming problem are
(A) $x + y \le 5, x + 3y \le 9$
(B) $x + y \le 5, x + 3y \ge 9$
(C) $x + y \ge 5, x + 3y \le 9$
(D) $x + y \ge 5, 3x + y \le 9$
In the graph, the feasible region representing the Linear Programming Problem for maximising objective function $Z = px + qy$, $p, q > 0$ is shaded. If all points on segment AB give max (Z), then which of the following is true ?
(A) $p = 2q$
(B) $p = 3q$
(C) $q = 3p$
(D) $q = 2p$
The corner points of the feasible region determined by the system of linear constraints are $(0, 0), (0, 40), (20, 40) (60, 20)$ and $(60, 0)$. If the objective function of an LPP is $Z = 4x + 3y$, then the maximum value is : (A) 200 (B) 300 (C) 240 (D) 120
In an LPP, corner points of the feasible region determined by the system of linear constraints are $(1, 1)$, $(3, 0)$ and $(0, 3)$. If $Z = ax + by$, where $a, b > 0$ is to be minimized, the condition on $a$ and $b$, so that the minimum of $Z$ occurs at $(3, 0)$ and $(1, 1)$, will be :
(A) $a = 2b$
...
The maximum value of $Z = 3x + 4y$ subject to the constraints $x + y \le 1$, $x, y \ge 0$ is :
(A) $3$
(B) $4$
(C) $7$
(D) $0$
The corner points of the feasible region in graphical representation of a L.P.P. are $(2, 72)$, $(15, 20)$ and $(40, 15)$. If $Z = 18x + 9y$ be the objective function, then
(A) $Z$ is maximum at $(2, 72)$, minimum at $(15, 20)$
(B) $Z$ is maximum at $(15, 20)$ minimum at $(40, 15)$
(C) $Z$ is maximu...
If the feasible region of a linear programming problem with objective function $Z = ax + by$, is bounded, then which of the following is correct ?
(A) It will only have a maximum value.
(B) It will only have a minimum value.
(C) It will have both maximum and minimum values.
(D) It will have neither ...
The corner points of the feasible region of a Linear Programming Problem are $(0, 2), (3, 0), (6, 0), (6, 8)$ and $(0, 5)$. If $Z = ax + by; (a, b > 0)$ be the objective function, and maximum value of $Z$ is obtained at $(0, 2)$ and $(3, 0)$, then the relation between $a$ and $b$ is :
(A) $a = b$
(B...
Assertion (A) : In a Linear Programming Problem, if the feasible region is empty, then the Linear Programming Problem has no solution. Reason (R) : A feasible region is defined as the region that satisfies all the constraints.
In a Linear Programming Problem (LPP), the objective function $Z = 2x + 5y$ is to be maximised under the following constraints :
$x + y \le 4$, $3x + 3y \ge 18$, $x, y \ge 0$
Study the graph and select the correct option.
(Note : The figure is not to scale)
The solution of the given LPP :
(A) lies i...
The common region determined by all the constraints of a linear programming problem is called :
(A) an unbounded region
(B) an optimal region
(C) a bounded region
(D) a feasible region
The restrictions imposed on decision variables involved in an objective function of a linear programming problem are called :
(A) feasible solutions
(B) constraints
(C) optimal solutions
(D) infeasible solutions
Of the following, which group of constraints represents the feasible region given below ?
(A) $x + 2y \le 76, 2x + y \ge 104, x, y \ge 0$
(B) $x + 2y \le 76, 2x + y \le 104, x, y \ge 0$
(C) $x + 2y \ge 76, 2x + y \le 104, x, y \ge 0$
(D) $x + 2y \ge 76, 2x + y \ge 104, x, y \ge 0$
The maximum value of $Z = 4x + y$ for a L.P.P. whose feasible region is given below is :
(A) 50
(B) 110
(C) 120
(D) 170
Assertion (A) : The corner points of the bounded feasible region of a L.P.P. are shown below. The maximum value of $Z = x + 2y$ occurs at infinite points.
Reason (R) : The optimal solution of a LPP having bounded feasible region must occur at corner points.
The corner points of the feasible region in the graphical representation of a linear programming problem are $(2, 72)$, $(15, 20)$ and $(40, 15)$. If $z = 18x + 9y$ be the objective function, then :
(a) $z$ is maximum at $(2, 72)$, minimum at $(15, 20)$
(b) $z$ is maximum at $(15, 20)$, minimum at $...
The objective function $Z = ax + by$ of an LPP has maximum value $42$ at $(4, 6)$ and minimum value $19$ at $(3, 2)$. Which of the following is true ?
(A) $a = 9, b = 1$
(B) $a = 5, b = 2$
(C) $a = 3, b = 5$
(D) $a = 5, b = 3$
The corner points of the bounded feasible region of an LPP are O(0, 0), A(250, 0), B(200, 50) and C(0, 175). If the maximum value of the objective function $Z = 2ax + by$ occurs at the points A(250, 0) and B(200, 50), then the relation between a and b is :
(a) $2a = b$
(b) $2a = 3b$
(c) $a = b$
(d) ...
Solve the following linear programming problem graphically :
Minimize $\quad Z = 13x - 15y$
Subject to constraints
$x + y \le 7$,
$2x - 3y + 6 \ge 0$,
$x \ge 0, y \ge 0$
Assertion $(A) :$ Consider a Linear Programming Problem with minimise $Z = x + 2y$ subject to constraints $2x + y \geq 3, x + 2y \geq 6, x, y \geq 0$ which gives minimum $Z$ at infinitely many points. The corner points of feasible region are $(0, 3)$ and $(6, 0)$.
Reason $(R) :$ If two corner points...
Solve the following Linear Programming Problem graphically : Maximize $Z = \frac{2x}{5} + \frac{3y}{10}$ subject to constraints $2x + y \leq 1000$ , $x + y \leq 800$ , $x, y \geq 0$.
Solve the following Linear Programming Problem graphically :
Minimize $Z = 4x + y$
subject to
$$x + y \ge 3$$
$$x + 2y \ge 4$$
$$x, y \ge 0.$$
A linear programming problem deals with the optimization of a/an :
(A) logarithmic function
(B) linear function
(C) quadratic function
(D) exponential function
A factory produces two products X and Y. The profit earned by selling X and Y is represented by the objective function $Z = 5x + 7y$, where $x$ and $y$ are the number of units of X and Y respectively sold. Which of the following statement is correct ?
(A) The objective function maximizes the differe...
Assertion (A) : Every point of the feasible region of a Linear Programming Problem is an optimal solution.
Reason (R) : The optimal solution for a Linear Programming Problem exists only at one or more corner point(s) of the feasible region.
For a Linear Programming Problem (LPP), the given objective function $Z = 3x + 2y$ is subject to constraints :
$x + 2y \le 10$
$3x + y \le 15$
$x, y \ge 0$
The correct feasible region is :
(A) $ ext{ABC}$
(B) $ ext{AOEC}$
(C) $ ext{CED}$
(D) $ ext{Open unbounded region BCD}$
For a Linear Programming Problem (LPP), the given objective function is $Z = x + 2y$. The feasible region PQRS determined by the set of constraints is shown as a shaded region in the graph. $P = \left$ \frac{3}{13}, \frac{24}{13} \right$, Q = \left$ \frac{3}{2}, \frac{15}{4} \right$, R = \left$ \fra...
Consider the Linear Programming Problem, where the objective function $Z = (x + 4y)$ needs to be minimized subject to constraints
$$2x + y \geq 1000$$
$$x + 2y \geq 800$$
$$x, y \geq 0.$$
Draw a neat graph of the feasible region and find the minimum value of Z.
In the Linear Programming Problem for objective function $Z = 18x + 10y$ subject to constraints
$4x + y \geq 20$
$2x + 3y \geq 30$
$x, y \geq 0$
find the minimum value of Z.
Identify and write all the constraints which determine the given feasible region in Figure-2.
The number of corner points of the feasible region determined by the constraints $x - y \geq 0$, $2y \leq x + 2$, $x \geq 0$, $y \geq 0$ is :
(a) $2$
(b) $3$
(c) $4$
(d) $5$
The corner points of the feasible region of a linear programming problem are $(0, 4), (8, 0)$ and $\left$ \frac{20}{3}, \frac{4}{3} \right$$. If $Z = 30x + 24y$ is the objective function, then (maximum value of $Z - $ minimum value of $Z$) is equal to
(A) $40$
(B) $96$
(C) $120$
(D) $136$
The feasible region of a linear programming problem with objective function $Z = 5x + 7y$ is shown below :
(Graph showing feasible region with vertices at $(0, 2)$, $(3, 4)$, and $(7, 0)$)
The maximum value of $Z - \text{minimum value of } Z \text{ is}$
(A) $8$
(B) $29$
(C) $35$
(D) $43$
The feasible region of a linear programming problem with objective function $Z = 5x + 7y$ is shown below :
[Diagram shows a bounded feasible region with vertices at $(0, 2)$, $(3, 4)$, and $(7, 0)$ approximately]
The maximum value of $Z - \text{minimum value of } Z$ is
(A) $8$
(B) $29$
(C) $35$
(D) ...
The feasible region of a linear programming problem with objective function $Z = 5x + 7y$ is shown below :
The maximum value of $Z - \text{minimum value of } Z$ is
(A) $8$
(B) $29$
(C) $35$
(D) $43$
Solve the following linear programming problem graphically :<br>Minimize $Z = 13x - 15y$<br>Subject to constraints<br>$x + y \le 7$,<br>$2x - 3y + 6 \ge 0$,<br>$x \ge 0, y \ge 0$
Solve the following linear programming problem graphically :\nMaximize $Z = 10500x + 9000y$\nSubject to constraints\n$x + y \le 50$\n$2x + y \le 80$\n$x, y \ge 0$
Solve the following linear programming problem graphically :
Maximize $Z = 8000x + 12000y$
Subject to constraints
$3x + 4y \le 60$
$x + 3y \le 30$
$x \ge 0$, $y \ge 0$
Solve the following linear programming problem graphically :
Maximize $Z = 4500x + 5000y$
Subject to constraints
$x + y \le 250$
$25x + 40y \le 7000$
$x \ge 0, y \ge 0$
The corner points of the feasible region for an LPP are $(0, 2)$, $(6, 0)$, $(6, 8)$ and $(0, 5)$. Let $Z = 4x + 6y$ be the objective function. $(\text{Maximum of } Z) - (\text{Minimum of } Z)$ is equal to :
(A) $60$
(B) $48$
(C) $42$
(D) $18$
In an LPP, if the objective function has the same minimum value on two corner points of the feasible region, then the number of points at which the minimum value of objective function occurs is :
(A) $0$
(B) $2$
(C) Infinite
(D) Finite
The corner points of the feasible region determined by the system of linear constraints in an LPP are $(0, 10)$, $(5, 5)$, $(15, 15)$ and $(0, 20)$. Let $Z = px + qy$, $p, q > 0$, be the objective function, then find the relation between $p$ and $q$ so that the maximum of $Z$ occurs at both the poin...
Solve the following linear programming problem graphically :
Minimize or Maximize (as per standard context, but here it says maximize): $Z = 600x + 400y$ subject to the constraints
$x + 2y \le 12$
$4x + 5y \ge 20$
$2x + y \le 12$
$x, y \ge 0$
Solve the following Linear Programming Problem graphically :
Maximise $Z = 600x + 400y$
subject to the constraints
$x + 2y \le 12$
$4x + 5y \ge 20$
$2x + y \le 12$
$x, y \ge 0$
Solve the following Linear Programming Problem graphically :
Maximise $Z = 12x + 18y$
subject to the constraints
$x + y \le 1200$
$x - 2y \ge 0$
$x + 3y \ge 600$
$x \ge 0, y \ge 0$
Solve the following Linear Programming Problem graphically :
Maximize $Z = 8x + 8y$
subject to the constraints
$x - 20 \leq 0$
$2x + 3y \leq 120$
$2x + y \leq 60$
$x \geq 0, y \geq 0$
Solve the following Linear Programming Problem graphically :
Maximise $Z = 200x + 120y$
subject to the constraints
$x + y \leq 300$
$3x + y \leq 600$
$x - y \geq -100$
$x, y \geq 0$
Solve the following Linear Programming Problem graphically :
Minimize $Z = 20x + 10y$
subject to
$x + 2y \leq 40$
$3x + y \geq 30$
$4x + 3y \geq 60$
$x, y \geq 0$
Solve the following Linear Programming Problem graphically : Maximize $Z = 20x + 10y$ subject to constraints $x + 2y \le 28$, $3x + y \le 24$, $x \ge 2$, $x, y \ge 0$.
Solve the following Linear Programming Problem graphically :
Maximize $\text{Z} = \frac{2x}{5} + \frac{3y}{10}$
subject to constraints
$2x + y \le 1000$
$x + y \le 800$
$x, y \ge 0$.
The feasible region for an LPP is shown shaded in the following figure. At which point does $ ext{Min. } z = 50x + 70y$ occur ? Also, find the minimum value of $z$.
Solve the following Linear Programming Problem (LPP) by graphical method :\nMinimize $Z = 20x + 10y$\nsubject to constraints :\n$x + 2y \le 40$\n$3x + y \ge 30$\n$4x + 3y \ge 60$\n$x \ge 0, y \ge 0$
A manufacturer produces nuts and bolts for industrial machinery. It takes 1 hour of work on machine A and 3 hours on machine B to produce a packet of nuts, while it takes 3 hours on machine A and 1 hour on machine B to produce a packet of bolts. He earns a profit of ₹ 5 per packet of nuts and ₹ 2 pe...
In an LPP, if the objective function $Z = px + y$ is minimum at a corner point $(3, 2)$ of the feasible region and the minimum value is $14$, then the value of $p$ is :
(A) $4$
(B) $\frac{16}{3}$
(C) $\frac{11}{2}$
(D) $\frac{14}{3}$
In an LPP, the optimal value of the objective function is attained at the points :
(A) on x-axis
(B) on y-axis
(C) on corners of the feasible region
(D) inside the feasible region
Determine the maximum and the minimum values of $Z = 4x + 3y$, if the feasible region for an LPP is a bounded region with corner points, A(25, 0), B(16, 16) and C(0, 24).
For an LPP with constraints
$$2x + y \leq 70$$
$$x + y \leq 40$$
$$x, y \geq 0,$$
shade the feasible region on the graph and write the coordinates of its corner points.
Solve the following Linear Programming Problem graphically :
Minimize $Z = 4x + y$
subject to
$x + y \geq 3$
$x + 2y \geq 4$
$x, y \geq 0$.
For the given Linear Programming Problem, shade the feasible region on the graph and write the coordinates of the corner points.
Minimize $Z = 5x + 7y$
subject to constraints
$2x + y \geq 8$
$x + 2y \geq 10$
$x, y \geq 0$.
For the following Linear Programming Problem, shade the feasible region on the graph and find the coordinates of its corner points : \\[0.5em] Minimize $Z = 18x + 10y$ \\[0.5em] subject to constraints \\[0.5em] $4x + y \ge 20$ \\[0.5em] $2x + 3y \ge 30$ \\[0.5em] $x, y \ge 0$.
A student of class XII studying Mathematics comes across an incomplete question in a book.
Maximise $Z = 3x + 2y + 1$
Subject to the constraints $x \ge 0, y \ge 0, 3x + 4y \le 12$,
He/ She notices the below shown graph for the said LPP problem, and finds that a constraint is missing in it:
Help h...
For Visually Impaired:
If $Z = ax + by + c$, where $a, b, c > 0$, attains its maximum value at two of its corner points $(4, 0)$ and $(0, 3)$ of the feasible region determined by the system of linear inequalities, then
(A) $4a = 3b$
(B) $3a = 4b$
(C) $4a + c = 3b$
(D) $3a + c = 4b$
The feasible region of a linear programming problem is bounded but the objective function attains its minimum value at more than one point. One of the points is $(5, 0)$.
Then one of the other possible points at which the objective function attains its minimum value is
(A) $(2, 9)$
(B) $(6, 6)$
(C...
For Visually Impaired:\nThe objective function $Z = 3x + 2y$ of a linear programming problem under some constraints is to be maximized and minimized. The corner points of the feasible region are $A(600,0), B(1200,0), C(800,400)$ and $D(400,200)$. Find the point at which $Z$ is maximum and the point ...
Minimise $Z = 5x + 10y$, subject to the constraints
$x + 2y \le 120$
$x + y \ge 60$
$x - 2y \ge 0$
$x, y \ge 0$
Case Study – 2
The feasible region for an LPP is shown in the graph given below :
Based on the above information, answer the following questions :
(i) Determine the equation of CD.
Determine all the constraints for the LPP.
Find the maximum value of the objective function $Z = 600x + 400y$.
Of all the points of the feasible region of an LPP, for maximum or minimum values of objective function, the points lie :
(A) inside the feasible region
(B) at the boundary line of the feasible region
(C) at the corners of the feasible region
(D) at the points of intersection of the feasible region ...
The corner points of the feasible region determined by the system of linear constraints for an LPP are $(0, 10)$, $(5, 5)$, $(15, 15)$ and $(0, 20)$. Let $z = ax + by$, where $a, b > 0$ be the objective function. Find the condition on $a$ and $b$ so that maximum of $z$ occurs at both the points $(15...
Solve the following LPP graphically :\nMaximize $Z = 2x + 3y$\nsubject to the constraints $x + 4y \le 8$\n$2x + 3y \le 12$\n$3x + y \le 9$\n$x \ge 0, y \ge 0$.
Solve the following LPP graphically :
Maximize $\text{Z} = 2x + 3y$
subject to the constraints $x + 4y \le 8$
$2x + 3y \le 12$
$3x + y \le 9$
$x \ge 0, y \ge 0$.
Solve the following LPP graphically :
Maximize $Z = 2x + 3y$
subject to the constraints $x + 4y \le 8$
$2x + 3y \le 12$
$3x + y \le 9$
$x \ge 0, y \ge 0$.
The number of corner points of the feasible region determined by constraints $\text{x} \geq 0, \text{y} \geq 0, \text{x} + \text{y} \geq 4$ is :
(A) $0$
(B) $1$
(C) $2$
(D) $3$
Solve the following linear programming problem graphically :
Maximise $z = 500x + 300y$,
subject to constraints
$$x + 2y \le 12$$
$$2x + y \le 12$$
$$4x + 5y \ge 20$$
$$x \ge 0, y \ge 0$$
Solve the following linear programming problem graphically :
Maximise $Z = x + 2y$
Subject to the constraints :
$x - y \ge 0$
$x - 2y \ge -2$
$x \ge 0, y \ge 0$
Solve the following linear programming problem graphically :
Maximise $Z = 20x + 30y$
Subject to the constraints :
$x + y \le 80$
$2x + 3y \ge 100$
$x \ge 14$
$y \ge 14$
Solve the following linear programming problem graphically :
Maximize $Z = 8x + 9y$
Subject to the constraints :
$2x + 3y \le 6$
$3x - 2y \le 6$
$y \le 1$
$x \ge 0, y \ge 0$
Solve the following linear programming problem graphically:
Minimise $Z = x - 5y$
subject to the constraints:
$x - y \geq 0$
$-x + 2y \geq 2$
$x \geq 3, y \leq 4, y \geq 0$
The feasible region along with corner points for a linear programming problem are shown in the graph. Write all the constraints for the given linear programing problem.
Of all the points of the feasible region, for maximum or minimum values of the objective function, the point lies :
(A) inside the feasible region
(B) at the boundary line of the feasible region
(C) at the corner points of the feasible region
(D) at the coordinate axes
The corner points of the feasible region determined by some system of linear inequations, are $(0, 0), (5, 0), (3, 4)$ and $(0, 5)$. Let $Z = ax + by$, where $a, b > 0$. Find the condition on $a$ and $b$ so that the maximum of $Z$ occurs at both points $(3, 4)$ and $(0, 5)$.
Solve the following Linear Programming Problem using graphical method : Maximise $Z = 100x + 50y$ subject to the constraints $3x + y \leq 600$ $x + y \leq 300$ $y \leq x + 200$ $x \geq 0, y \geq 0$
Solve the following Linear Programming Problem graphically :
Minimise $Z = 3x + 5y$
subject to the constraints
$x + 2y \geq 10$
$x + y \geq 6$
$3x + y \geq 8$
$x, y \geq 0$
For the given graph of a Linear Programming Problem, write all the constraints satisfying the given feasible region.
For a Linear Programming Problem (LPP), the given objective function $Z = 3x + 2y$ is subject to constraints :
$x + 2y \le 10$
$3x + y \le 15$
$x, y \ge 0$
[Image of graph showing feasible region]
The correct feasible region is :
(A) $\text{ABC}$
(B) $\text{AOEC}$
(C) $\text{CED}$
(D) $\text{Open un...
In the Linear Programming Problem (LPP), find the point/points giving maximum value for $Z = 5x + 10y$ \
subject to constraints \
$x + 2y \le 120$ \
$x + y \ge 60$ \
$x - 2y \ge 0$ \
$x, y \ge 0$
Assertion (A) : The shaded portion of the graph represents the feasible region for the given Linear Programming Problem (LPP).
Min $Z = 50x + 70y$
subject to constraints
$2x + y \geq 8$, $x + 2y \geq 10$, $x, y \geq 0$
$Z = 50x + 70y$ has a minimum value $= 380$ at $B(2, 4)$.
Reason (R) : The regi...
In the Linear Programming Problem (LPP), find the point/points giving maximum value for $Z = 5x + 10y$ subject to constraints \
$x + 2y \le 120$ \
$x + y \ge 60$ \
$x - 2y \ge 0$ \
$x, y \ge 0$
In the Linear Programming Problem (LPP), find the point/points giving maximum value for $Z = 5x + 10y$
subject to constraints
$$x + 2y \le 120$$
$$x + y \ge 60$$
$$x - 2y \ge 0$$
$$x, y \ge 0$$
Consider the Linear Programming Problem, where the objective function $Z = (x + 4y)$ needs to be minimized subject to constraints
$2x + y \ge 1000$
$x + 2y \ge 800$
$x, y \ge 0$.
Draw a neat graph of the feasible region and find the minimum value of $Z$.
For a Linear Programming Problem (LPP), the given objective function is $Z = x + 2y$. The feasible region $ ext{PQRS}$ determined by the set of constraints is shown as a shaded region in the graph.
(Note : The figure is not to scale)
$P = \left(\frac{3}{13}, \frac{24}{13}\right)$, $Q = \left(\frac...
In a Linear Programming Problem, the objective function $Z = 5x + 4y$ needs to be maximised under constraints $3x + y \le 6$, $x \le 1$, $x, y \ge 0$. Express the LPP on the graph and shade the feasible region and mark the corner points.
In the Linear Programming Problem for objective function $Z = 18x + 10y$ subject to constraints
$$4x + y \ge 20$$
$$2x + 3y \ge 30$$
$$x, y \ge 0$$
find the minimum value of $Z$.
In a Linear Programming Program (LPP) for objective function $Z = 14x - 10y$
subject to constraints
$x + y \le 8$
$3x - 2y \ge -6$
$x, y \ge 0$
shade the feasible region and mark the corner points in a neatly drawn graph.
For a Linear Programming Problem, find $\min Z = 5x + 3y$ (where Z is the objective function) for the feasible region shaded in the given figure.
The number of corner points of the feasible region determined by constraints $x \ge 0$, $y \ge 0$, $x + y \ge 4$ is :
(A) 0
(B) 1
(C) 2
(D) 3
Solve the following linear programming problem graphically :
Minimise $z = 5x - 2y$
subject to the constraints
$x + 2y \le 120$
$x + y \ge 60$
$x - 2y \ge 0$
$x, y \ge 0$
For the linear programming problem (LPP), the objective function is $Z = 4x + 3y$ and the feasible region determined by a set of constraints is shown in the graph:
Q.16. A linear programming problem (LPP) along with the graph of its constraints is shown below.
The optimal solution of the above linear programming problem ________.
(A) does not exist as the feasible region is unbounded.
(B) does not exist as the inequality $18x + 10y < 134$ does not have any point in common with the feasible region.
(C) exists as the inequality $18x + 10y > 134$ has infinit...
Consider the following Linear Programming Problem:
Minimise $Z = x + 2y$
Subject to $2x + y \ge 3$, $x + 2y \ge 6$, $x, y \ge 0$.
Show graphically that the minimum of $Z$ occurs at more than two points.
The maximum value of the objective function $z = 3x + 5y$ subject to the constraints $x \ge 0, y \ge 0$ and $4x + 3y \le 12$ is :
(A) 15
(B) 29
(C) 9
(D) 20
The corner points of the feasible region determined by the system of linear constraints are $A(0, 40)$, $B(20, 40)$, $C(60, 20)$ and $D(60, 0)$. The objective function of the L.P.P. is $z = 4x + 3y$. Find the point of the feasible region at which the value of objective function is maximum and the po...
Solve the following linear programming problem graphically :
Maximise $z = 4x + 3y$,
subject to the constraints
$x + y \le 800$
$2x + y \le 1000$
$x \le 400$
$x, y \ge 0$.
Solve the following linear programming problem graphically : Minimize $z = 600x + 400y$, subject to the constraints $x + y \geq 8$, $x + 2y \leq 16$, $4x + y \leq 29$, $x, y \geq 0$.
Solve the following linear programming problem graphically : Maximize $z = x + y$ subject to constraints $2x + 5y \le 100$, $8x + 5y \le 200$, $x \ge 0, y \ge 0$.
Solve the following linear programming problem graphically : Maximise $Z = 2x + 3y$ subject to the constraints : $x + y \leq 6$, $x \geq 2$, $y \leq 3$, $x, y \geq 0$
Solve the following L.P.P. graphically :
Maximise $Z = x + 3y$
subject to the constraints :
$x + 2y \le 200$
$x + y \le 150$
$y \le 75$
$x, y \ge 0$
The corner points of the feasible region determined by the system of linear constraints are as shown in the following figure: (i) If $Z = 3x - 4y$ be the objective function, then find the maximum value of $Z$. (ii) If $Z = px + qy$ where $p, q > 0$ be the objective function. Find the condition on $p...
Solve the following Linear Programming problem graphically :
Maximise $Z = 300x + 600y$
Subject to $x + 2y \le 12$
$2x + y \le 12$
$x + \frac{5}{4}y \ge 5$
$x \ge 0, y \ge 0$.
Solve the following Linear Programming problem graphically :
Maximise $Z = 300x + 600y$
Subject to $x + 2y \leq 12$
$2x + y \leq 12$
$x + \frac{5}{4}y \geq 5$
$x \geq 0, y \geq 0.$
Solve the following L.P.P. graphically :
Minimise $Z = 6x + 3y$
Subject to constraints
$4x + y \geq 80$;
$x + 5y \geq 115$;
$3x + 2y \leq 150$
$x, y \geq 0$
The number of solutions of an L.P.P. to minimize $z = 3x + 2y$ under the constraints $x + y \geq 8$, $3x + 5y \leq 15$ and $x, y \geq 0$, is :
(A) 2
(B) 5
(C) infinitely many
(D) zero
Two tailors A and B earn ₹ 1500 and ₹ 2000 per day, respectively. Tailor A can stitch 6 shirts and 4 pants, while tailor B can stitch 10 shirts and 4 pants per day. Form a linear programming problem to minimize the labour cost to produce at least 60 shirts and 32 pants.
Solve the following linear programming problem graphically : Minimise Z = 6x + 7y subject to constraints x + 2y \geq 240, 3x + 4y \leq 620, 2x + y \geq 180, x, y \geq 0.
Solve the following linear programming problem graphically : Minimise Z = 6x + 7y subject to constraints x + 2y $\ge$ 240, 3x + 4y $\le$ 620, 2x + y $\ge$ 180, x, y $\ge$ 0.
Solve graphically the following linear programming problem : Maximise z = 6x + 3y, subject to the constraints 4x + y ≥ 80, 3x + 2y ≤ 150, x + 5y ≥ 115, x ≥ 0, y ≥ 0.
The number of corner points of the feasible region determined by the constraints $x - y \ge 0$, $2y \le x + 2$, $x \ge 0$, $y \ge 0$ is :
(a) $2$
(b) $3$
(c) $4$
(d) $5$
Solve the following linear programming problem graphically : Maximize $P = 100x + 5y$ subject to the constraints $x + y \le 300$, $3x + y \le 600$, $y \le x + 200$, $x, y \ge 0$.
Solve the following linear programming problem graphically : Maximize $z = 600x + 400y$ subject to the constraints : $x + 2y \le 12$, $2x + y \le 12$, $x + 1.25y \ge 5$, $x, y \ge 0$
The corner points of the bounded feasible region determined by a system of linear constraints are $(0,3), (1,1)$ and $(3,0)$. Let $Z = px + qy$, where $p, q > 0$. The condition on $p$ and $q$ so that the minimum of $Z$ occurs at $(3,0)$ and $(1,1)$ is (a) $p = 2q$ (b) $p = \frac{q}{2}$ (c) $p = 3q$ ...
The feasible region corresponding to the linear constraints of a Linear Programming Problem is given below. [Image description: A graph shows a shaded region bounded by lines. The lines are labeled $x-y=1$, $x+2y=10$, and $x+y=2$. The vertices of the shaded region are at $(0,2), (0,5), (2,0), (1,0)$...
Solve the following Linear Programming Problem graphically: Minimize: $z = x+2y$, subject to the constraints: $x+2y \geq 100, 2x-y \leq 0, 2x+y \leq 200, x,y \geq 0$. OR Solve the following Linear Programming Problem graphically: Maximize: $z = -x+2y$, subject to the constraints: $x \geq 3, x+y \geq...
The corner points of the feasible region of a linear programming problem are $(0, 4)$, $(8, 0)$ and $(\frac{20}{3}, \frac{4}{3})$. If $Z = 30x + 24y$ is the objective function, then (maximum value of $Z$ - minimum value of $Z$) is equal to
(A) $40$
(B) $96$
(C) $120$
(D) $136$
Solve the following linear programming problem graphically :
Minimize : $Z = 5x + 10y$
subject to constraints : $x + 2y \leq 120, x + y \geq 60, x - 2y \geq 0,$
$x \geq 0, y \geq 0$
Solve the following linear programming problem graphically :
Maximize : $Z = x + 2y$
subject to constraints : $x + 2y \geq 100$,
$2x - y \leq 0$,
$2x + y \leq 200$,
$x \geq 0, y \geq 0$.
A manufacturing company makes two types of items A and B. Each item of type A requires 2 hours of grinding and 3 hours of polishing, whereas each item of type B requires 3 hours of grinding and 2 hours of polishing. The company has 2 grinding machines and 2 polishing machines, each working for at mo...
Solve the following Linear Programming problem graphically :
Maximize : $Z = 3x + 3.5y$
subject to constraints : $x + 2y \ge 240$,
$3x + 1.5y \ge 270$,
$1.5x + 2y \le 310$,
$x \ge 0, y \ge 0$.
The minimum value of $z = 3x + 8y$ subject to the constraints $x \leq 20$, $y \geq 10$ and $x \geq 0$, $y \geq 0$ is :
(a) 80
(b) 140
(c) 0
(d) 60
The objective function $\text{z} = 4\text{x} + 3\text{y}$ of a linear programming problem under some constraints is to be maximized and minimized. The corner points of the feasible region are A(0, 700), B(100, 700), C(200, 600) and D(400, 200). Find the point at which $\text{z}$ is maximum and the p...
Determine graphically the minimum value of the following objective function :
z = 500x + 400y
subject to constraints
x + y ≤ 200,
x ≥ 20,
y ≥ 4x,
y ≥ 0.
The number of feasible solutions of the linear programming problem given as
Maximize $z = 15x + 30y$ subject to constraints :
$3x + y \le 12$, $x + 2y \le 10$, $x \ge 0$, $y \ge 0$ is
(a) 1
(b) 2
(c) 3
(d) infinite
The feasible region of a linear programming problem is shown in the figure below :
[Figure shows a coordinate plane with shaded feasible region]
Which of the following are the possible constraints ?
(a) $x + 2y \ge 4$, $x + y \le 3$, $x \ge 0$, $y \ge 0$
(b) $x + 2y \le 4$, $x + y \le 3$, $x \ge 0$,...
The number of feasible solutions of the linear programming problem given as
Maximize $z = 15x + 30y$ subject to constraints :
$3x + y \leq 12, x + 2y \leq 10, x \geq 0, y \geq 0$ is
(a) 1
(b) 2
(c) 3
(d) infinite
The feasible region of a linear programming problem is shown in the figure below :
[Figure shows a graph with constraints x+2y=4 and x+y=3, and a shaded region.]
Which of the following are the possible constraints ?
(a) x+2y \geq 4, x+y \leq 3, x \geq 0, y \geq 0
(b) x+2y \leq 4, x+y \leq 3, x \ge...
Solve the following linear programming problem graphically :
Maximize $z = 3x + 9y$
subject to the constraints
$x + y \geq 10$,
$x + 3y \leq 60$,
$x \leq y$,
$x \geq 0, y \geq 0$.
Solve the following linear programming problem graphically :
Minimise : $z = -3x + 4y$
subject to the constraints
$x + 2y \le 8$,
$3x + 2y \le 12$,
$x, y \ge 0$.
Solve the following linear programming problem graphically :
Maximise $z = 5x + 3y$
subject to the constraints
$$3x + 5y \le 15,$$
$$5x + 2y \le 10,$$
$$x, y \ge 0.$$
Solve the following Linear Programming Problem graphically :
Maximize : $P = 70x + 40y$
subject to : $3x + 2y \le 9$,
$3x + y \le 9$,
$x \ge 0, y \ge 0$
Solve the following Linear Programming Problem graphically :
Maximize : $P = 70x + 40y$
subject to : $3x + 2y \leq 9$,
$3x + y \leq 9$,
$x \geq 0, y \geq 0$
The maximum value of the function z = 7x + 5y, subject to constraints x \leq 3, y \leq 2, x \geq 0, y \geq 0, is :
(a) 21
(b) 10
(c) 31
(d) 37
Formulate the following problem as an LPP : A small firm manufactures gold rings and chains. The total number of rings and chains manufactured per day is at most 24. It takes 1 hour to make a ring and 30 minutes to make a chain. The maximum number of hours available per day is 16. If the profit on a...
Solve the following Linear Programming Problem graphically : Minimise z = 3x + 8y subject to the constraints 3x + 4y $\ge$ 8 5x + 2y $\ge$ 11 x $\ge$ 0, y $\ge$ 0
Solve the following Linear Programming Problem graphically: Maximise z = 10x + 15y subject to the constraints : 3x + 2y ≤ 50, x + 4y ≥ 20, x ≥ 8, y ≥ 0
Solve the following Linear Programming Problem graphically:
Minimise $\text{z} = 6x + 7y$
subject to the constraints
$2x + y \ge 8$
$x + 2y \ge 10$
$x, y \ge 0$
While solving the linear programming problem Minimise and Maximise Z = 3x + 9y, subject to the constraints x + 3y $\le$ 60, x + y $\ge$ 10, x $\le$ y and x $\ge$ 0, y $\ge$ 0 graphically, the corner points of the feasible region ABCD are A(0, 10), B(5, 5), C(15, 15) and D(0, 20). Find the minimum va...
The corner points of the shaded unbounded feasible region of an LPP are $(0, 4)$, $(0.6, 1.6)$ and $(3, 0)$ as shown in the figure. The minimum value of the objective function $Z = 4x + 6y$ occurs at
(a) $(0.6, 1.6)$ only
(b) $(3, 0)$ only
(c) $(0.6, 1.6)$ and $(3, 0)$ only
(d) at every point of the...
Solve the following Linear Programming Problem graphically:
Maximize $Z = 400x + 300y$ subject to $x + y \le 200, x \le 40, x \ge 20, y \ge 0$