Linear Programming and Simplex Methods - Linear Programming - Mathematics Previous Year Questions

1

2008 · Mathematics · Linear Programming · Linear Programming and Simplex Methods

Mathematics (MA) 2008

The maximum value of \(z = 3x_1 - x_2\) subject to \(2x_1 - x_2 \le 1\), \(x_1 \le 3\) and \(x_1, x_2 \ge 0\) is

A

0

B

4

C

6

D

9

Open complete paper

2

2008 · Mathematics · Linear Programming · Linear Programming and Simplex Methods

Mathematics (MA) 2008

Consider the problem of maximizing \(z = 2x_1 + 3x_2 - 4x_3 + x_4\) subject to \[\begin{align*}\] x_1 + x_2 + x_3 &= 2, \\ x_1 - x_2 + x_3 &= 2, \\ 2x_1 + 3x_2 + 2x_3 - x_4 &= 0, \\ x_1, x_2, x_3, x_4 &\ge 0. \end{align*} Then

A

(1, 0, 1, 4) is a basic feasible solution but (2, 0, 0, 4) is not

B

(1, 0, 1, 4) is not a basic feasible solution but (2, 0, 0, 4) is

C

neither (1, 0, 1, 4) nor (2, 0, 0, 4) is a basic feasible solution

D

both of (1, 0, 1, 4) and (2, 0, 0, 4) are basic feasible solutions

Open complete paper

3

2009 · Mathematics · Linear Programming · Linear Programming and Simplex Methods

Mathematics (MA) 2009

Which one of the following is TRUE?

A

Every linear programming problem has a feasible solution.

B

If a linear programming problem has an optimal solution then it is unique.

C

The union of two convex sets is necessarily convex.

D

Extreme points of the disk x² + y² ≤ 1 are the points on the circle x² + y² = 1.

Open complete paper

4

2009 · Mathematics · Linear Programming · Linear Programming and Simplex Methods

Mathematics (MA) 2009

For a fixed \(t \in \mathbb{R}\), consider the linear programming problem: Maximize \(z = 3x + 4y\) subject to \(x + y \leq 100\) \(x + 3y \leq t\) and \(x \geq 0, y \geq 0\) The maximum value of \(z\) is 400 for \(t =\)

A

50

B

100

C

200

D

300

Open complete paper

5

2009 · Mathematics · Linear Programming · Linear Programming and Simplex Methods

Mathematics (MA) 2009

The minimum value of \(z = 2x_1 - x_2 + x_3 - 5x_4 + 22x_5\) subject to \(x_1 - 2x_4 + x_5 = 6\) \(x_2 + x_4 - 4x_5 = 3\) \(x_3 + 3x_4 + 2x_5 = 10\) \(x_j \geq 0, j = 1, 2, ..., 5\) is

A

28

B

19

C

10

D

9

Open complete paper

6

2010 · Mathematics · Linear Programming · Linear Programming and Simplex Methods

Mathematics (MA) 2010

For the linear programming problem
Minimize \( z = x - y \), subject to \( 2x + 3y \le 6 \), \( 0 \le x \le 3 \), \( 0 \le y \le 3 \),
the number of extreme points of its feasible region and the number of basic feasible solutions respectively, are

A

3 and 3

B

4 and 4

C

3 and 5

D

4 and 5

Open complete paper

7

2010 · Mathematics · Linear Programming · Linear Programming and Simplex Methods

Mathematics (MA) 2010

If \( z^* \) is the optimal value of the linear programming problem
Maximize \( z = 5x_1 + 9x_2 + 4x_3 \)
subject to \( x_1 + x_2 + x_3 = 5 \)
\( 4x_1 + 3x_2 + 2x_3 = 12 \)
\( x_1, x_2, x_3 \ge 0 \),
then

A

0 ≤ z* < 10

B

10 ≤ z* < 20

C

20 ≤ z* < 30

D

30 ≤ z* < 40

Open complete paper

8

2011 · Mathematics · Linear Programming · Linear Programming and Simplex Methods

Mathematics (MA) 2011

The Linear Programming Problem:
Maximize \( z = x_1 + x_2 \)
subject to
\( x_1 + 2x_2 \leq 20 \)
\( x_1 + x_2 \leq 15 \)
\( x_2 \leq 6 \)
\( x_1, x_2 \geq 0 \)

A

has exactly one optimum solution

B

has more than one optimum solutions

C

has unbounded solution

D

has no solution

Open complete paper

9

2012 · Mathematics · Linear Programming · Linear Programming and Simplex Methods

Mathematics (MA) 2012

For the linear programming problem
Maximize \(z = x_1 + 2x_2 + 3x_3 - 4x_4\)
Subject to
\(2x_1 + 3x_2 - x_3 - x_4 = 15\)
\(6x_1 + x_2 + x_3 - 3x_4 = 21\)
\(8x_1 + 2x_2 + 3x_3 - 4x_4 = 30\)
\(x_1, x_2, x_3, x_4 \geq 0\),
\(x_1 = 4, x_2 = 3, x_3 = 0, x_4 = 2\) is

A

an optimal solution

B

a degenerate basic feasible solution

C

a non-degenerate basic feasible solution

D

a non-basic feasible solution

Open complete paper

10

2012 · Mathematics · Linear Programming · Linear Programming and Simplex Methods

Mathematics (MA) 2012

Which one of the following statements is TRUE?

A

A convex set cannot have infinite many extreme points.

B

A linear programming problem can have infinite many extreme points.

C

A linear programming problem can have exactly two different optimal solutions.

D

A linear programming problem can have a non-basic optimal solution.

Open complete paper

11

2012 · Mathematics · Linear Programming · Linear Programming and Simplex Methods

Mathematics (MA) 2012

If the right hand side of the second constraint is changed from 8 to 20, then in the optimal solution of the primal problem, the basic variables will be

A

\(x_{1} \) and \(x_{2} \)

B

\(x_{1} \) and \(x_{3} \)

C

\(x_{2} \) and \(x_{3} \)

D

\(x_{2} \) and \(x_{4} \)

Open complete paper

12

2013 · Mathematics · Linear Programming · Linear Programming and Simplex Methods

Mathematics (MA) 2013

Consider the linear programming problem:
Maximize \(x + \frac{3}{2} y\)
subject to \(2x + 3y \le 16\), \(x + 4y \le 18\), \(x \ge 0, y \ge 0\).
If \(S\) denotes the set of all solutions of the above problem, then

A

\(S\) is empty

B

\(S\) is a singleton

C

\(S\) is a line segment

D

\(S\) has positive area

Open complete paper

13

2013 · Mathematics · Linear Programming · Linear Programming and Simplex Methods

Mathematics (MA) 2013

Consider the following linear programming problem:
Maximize \(x + 3y + 6z - w\)
subject to \(5x + y + 6z + 7w \le 20\), \(6x + 2y + 2z + 9w \le 40\), \(x \ge 0, y \ge 0, z \ge 0, w \ge 0\).
Then the optimal value is ______

Check answer

Open complete paper

14

2013 · Mathematics · Linear Programming · Linear Programming and Simplex Methods

Mathematics (MA) 2013

Let \(X\) be a convex region in the plane bounded by straight lines. Let \(X\) have 7 vertices. Suppose \(f(x, y) = ax + by + c\) has maximum value \(M\) and minimum value \(N\) on \(X\) and \(N < M\). Let \(S = \{P : P \text{ is a vertex of } X \text{ and } N < f(P) < M\}\). If \(S\) has \(n\) elements, then which of the following statements is TRUE?

A

\(n\) cannot be 5

B

\(n\) can be 2

C

\(n\) cannot be 3

D

\(n\) can be 4

Open complete paper

15

2013 · Mathematics · Linear Programming · Linear Programming and Simplex Methods

Mathematics (MA) 2013

X and Y are two positive real numbers such that 2X + Y ≤ 6 and X + 2Y ≤ 8. For which of the following values of (X, Y) the function f(X, Y) = 3X + 6Y will give maximum value?

A

(4/3, 10/3)

B

(8/3, 20/3)

C

(8/3, 10/3)

D

(4/3, 20/3)

Open complete paper

16

2014 · Mathematics · Linear Programming · Linear Programming and Simplex Methods

Mathematics (MA) 2014

Consider the following linear programming problem:
Minimize x₁ + x₂
Subject to:
2x₁ + x₂ ≥ 8
2x₁ + 5x₂ ≥ 10
x₁, x₂ ≥ 0
The optimal value to this problem is __________

Check answer

Open complete paper

17

2016 · Mathematics · Linear Programming · Linear Programming and Simplex Methods

Mathematics (MA) 2016

Minimize \( w = x + 2y \) subject to \( 2x + y \ge 3 \\ x + y \ge 2 \\ x \ge 0, y \ge 0 \). Then, the minimum value of \( w \) is equal to ______________

Check answer

Open complete paper

18

2016 · Mathematics · Linear Programming · Linear Programming and Simplex Methods

Mathematics (MA) 2016

Maximize \( w = 11x - z \) subject to \( 10x + y - z \le 1 \\ 2x - 2y + z \le 2 \\ x,y,z \ge 0 \). Then, the maximum value of \( w \) is equal to ______________

Check answer

Open complete paper

19

2017 · Mathematics · Linear Programming · Linear Programming and Simplex Methods

Mathematics (MA) 2017

Consider the linear programming problem (LPP): Maximize \(4x_1 + 6x_2\) Subject to \(x_1 + x_2 \le 8\), \(2x_1 + 3x_2 \ge 18\), \(x_1 \ge 6\), \(x_2\) is unrestricted in sign. Then the LPP has

A

no optimal solution

B

only one basic feasible solution and that is optimal

C

more than one basic feasible solution and a unique optimal solution

D

infinitely many optimal solutions

Open complete paper

20

2017 · Mathematics · Linear Programming · Linear Programming and Simplex Methods

Mathematics (MA) 2017

Consider the linear programming problem (LPP):
Maximize \( k x_1 + 5 x_2 \)
Subject to \( x_1 + x_2 \le 1, \)
\( 2x_1 + 3x_2 \le 1, \)
\( x_1, x_2 \ge 0. \)
If \( x^* = (x_1^*, x_2^*) \) is an optimal solution of the above LPP with \( k = 2 \), then the largest value of \( k \) (rounded to 2 decimal places) for which \( x^* \) remains optimal equals ______.

Check answer

Open complete paper