MTE-12 Solved Assignment 2023

IGNOU MTE-12 Solved Assignment 2023 | MTE

Solved By – Narendra Kr. Sharma – M.Sc (Mathematics Honors) – Delhi University

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IGNOU MTE-12 Assignment Question Paper 2023

Course Code: MTE-12
Assignment Code: MTE-12/TMA/2023
Maximum Marks: 100

1. Which of the following statements are true and which are false? Give a short proof or a counter-example in support of your answer.

a) In a two-dimensional LPP solution, the objective function can assume the same value at two distinct extreme points.

b) Both the primal and dual of an LPP can be infeasible.

c) An unrestricted primal variable converts into an equality dual constraint.

d) In a two-person zero-sum game, if the optimal solution requires one player to use a pure strategy, the other player must do the same.

e) If 10 is added to each entry of a row in the cost matrix of an assignment problem, then the total cost of an optimal assignment for the changed cost matrix will also increase by 10 .

2. a) Solve the following linear programming problem using simple method:

Maximize \(\mathrm{z}=3 \mathrm{x}_{1}+5 \mathrm{x}_{2}+4 \mathrm{x}_{3}\)

Subject to \(2 \mathrm{x}_{1}+3 \mathrm{x}_{2} \leq 8\)

\(2 \mathrm{x}_{2}+5 \mathrm{x}_{3} \leq 10\)

\(3 \mathrm{x}_{1}+2 \mathrm{x}_{2}+4 \mathrm{x}_{3} \leq 15\)

\(\mathrm{x}_{1}, \mathrm{x}_{2}, \mathrm{x}_{3} \geq 0\)

b) Using the principle of dominance, reduce the size of the following game:

\[
\left[\begin{array}{ccc}
-1 & -2 & 8 \\
7 & 5 & -1 \\
6 & 0 & 12
\end{array}\right]
\]

Hence solve the game.

3. a) Find all basic feasible solutions for the following set of equations:

\[
\begin{aligned}
& 2 x_{1}+6 x_{2}+2 x_{3}+x_{4}=3 \\
& 6 x_{1}+4 x_{2}+4 x_{3}+6 x_{4}=2 \\
& x_{1}, x_{2}, x_{3}, x 4 \geq 0
\end{aligned}
\]

b) Examine convexity of the following sets:

i) \(\mathrm{S}_{1}=\left\{\left(\mathrm{x}_{1}, \mathrm{x}_{2}\right) \in \mathbb{R}^{2} \mid 4 \mathrm{x}_{1}+3 \mathrm{x}_{2} \leq 6, \mathrm{x}_{1}+\mathrm{x}_{2} \geq 1\right\}\)

ii) \(\mathrm{S}_{2}=\left\{(\mathrm{x}, \mathrm{y}) \in \mathbb{R}^{2} \mid \mathrm{x}^{2}+\mathrm{y}^{2} \geq 1\right\}\).

4. a) Solve the following linear programming problem by graphical method:

\[
\text { Maximize } \mathrm{z}=5 \mathrm{x}_{1}+7 \mathrm{x}_{2}
\]

Subject to \(\quad \mathrm{x}_{1}+\mathrm{x}_{2} \leq 4\)

\[
\begin{aligned}
& 3 \mathrm{x}_{1}+8 \mathrm{x}_{2} \leq 24 \\
& 10 \mathrm{x}_{1}+7 \mathrm{x}_{2} \leq 35 \\
& \mathrm{x}_{1}, \mathrm{x}_{2} \geq 0
\end{aligned}
\]

b) Find the dual of the following LPP:

\[
\begin{array}{ll}
\text { Maximize } & z=x_{1}+x_{2}+x_{3} \\
\text { Subject to } & x_{1}-3 x_{2}+4 x_{3}=5 \\
& x_{1}-2 x_{2} \leq 3 \\
& 2 x_{2}-x_{3} \geq 4
\end{array}
\]

\(\mathrm{x}_{1}, \mathrm{x}_{2} \geq 0\) and \(\mathrm{x}_{3}\) is unrestricted in sign.

5. a) Find the initial basic feasible solution of the following transportation problem using matrix-minima method:

 DestinationsSupply
 IIIIII 
A2745
Sources B3318
C5477
D16214
Demand791834

Also, find the optimal solution.

b) Solve the following game graphically:

 

Player B

 

Player A

B1

B2

A1

2

7

A2

3

5

A3

11

2

6. a) A firm manufactures two types of products, A and B, and sells them at a profit of ₹ 2 on type \(A\) and \(₹ 3\) on type B. Each product is processed on two machines \(M_{1}\) and \(M_{2}\). Type a requires one minute of processing time on \(M_{1}\) and two minutes on \(M_{2}\); type \(B\) requires one minute on \(M_{1}\) and one minute on \(M_{2}\). The machine \(M_{1}\) is available for not more than 6 hours 40 minutes while machine \(M_{2}\) is available for 10 hours during any working day.

Formulate the problem at LPP.

b) Solve the following assignment problem:

I29271
II68761
III46531
IV42731
V53951
     

7. a) The following table is obtained in the intermediate state while solving an LPP by simplex method:

30232900 
BCBx1x2x3 S1 S2R.H.S.
S1002−9/21−3/231/2
X13011/25/401/47/4

Check whether an optimal solution of the LPP will exist or not.

b) Write the LPP model of the following transportation problem:

576470
283150
174590
50405070

c) Find the range of values of \(\mathrm{p}\) and \(\mathrm{q}\) which will render the entry \((2,2)\), a saddle point for the following game:

Player B
Player A45
107q
4p6

8. a) Test the convexity of the following sets:

\[
\begin{aligned}
& S_{1}=\left\{(x, y) \mid x^{2}+y^{2} \geq 1, y \geq x, y \geq-x\right\} \\
& S_{2}=\left\{(x, y) \mid x^{2}+y^{2} \leq 16, x \leq 2, y \geq 2\right\}
\end{aligned}
\]

b) Determine all the basic feasible solutions to the equations

\[
\begin{aligned}
& x_{1}+x_{2}+2 x_{3}=4 \\
& 2 x_{1}-x_{2}+x_{3}=2 .
\end{aligned}
\]

Identify the degenerate basic feasible solutions.

9. a) Let \(A=\left[\begin{array}{lll}2 & 5 & 1 \\ 3 & 2 & 1\end{array}\right], \mathrm{B}=\left[\begin{array}{ll}1 & 2 \\ 3 & 5\end{array}\right]\) and \(\mathrm{C}=\left[\begin{array}{ll}1 & 3 \\ 2 & 2 \\ 5 & 1\end{array}\right]\). Compute \(\mathrm{AB}, \mathrm{BC}, \mathrm{AC}\), if they exists, otherwise, give reason for their non-existence.

b) Solve the following LPP:

\[
\begin{array}{ll}
\text { Minimize } & \mathrm{z}=\mathrm{x}_{1}-2 \mathrm{x}_{2}-3 \mathrm{x}_{3} \\
\text { Subject to } & -2 \mathrm{x}_{1}+3 x_{2}+3 \mathrm{x}_{3}=2 \\
& 2 \mathrm{x}_{1}+3 \mathrm{x}_{2}+4 \mathrm{x}_{3}=1 \\
& \mathrm{x}_{1}, \mathrm{x}_{2}, \mathrm{x}_{3} \geq 0
\end{array}
\]

c) Find the saddle point (if exists) in the following pay-off matrix:

Player B 
Player A-1-17
313 
6-1-3

Also, find the value of the game.

10. a) Determine an initial basic feasible solution to the following transportation problem and hence find an optimal solution to the problem:

 
O112346
O21210578
O391181310
 4686

b) Find all values of \(\mathrm{k}\) for which the vectors \(\left[\begin{array}{l}1 \\ 0 \\ 1\end{array}\right],\left[\begin{array}{c}1 \\ -1 \\ 0\end{array}\right]\) and \(\left[\begin{array}{c}\mathrm{k} \\ -\mathrm{k} \\ 2\end{array}\right]\) are linearly independent.

\(2\:sin\:\theta \:sin\:\phi =-cos\:\left(\theta +\phi \right)+cos\:\left(\theta -\phi \right)\)

MTE-12 Sample Solution 2023

 

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\(2\:cos\:\theta \:sin\:\phi =sin\:\left(\theta +\phi \right)-sin\:\left(\theta -\phi \right)\)

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