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b) Set up the Gaussi-Jacobi iteration scheme in matrix form for the linear system of equations
4 3
4 2
4 3
2 3
1 2 3
1 2
− + =
− + − =
− =
x x
x x x
x x
4 3
4 2
4 3
2 3
1 2 3
1 2
− + =
− + − =
− =
x x
x x x
x x
c) Solve the IVP
y 1 y , )0(y 0
2
′ = + =
using classical R-K method of h(0 )
4
. Find )4.0(y taking h = 2.0 . Compare the solution
obtained with the exact solution and find the error.
y 1 y , )0(y 0
2
′ = + =
using classical R-K method of h(0 )
4
. Find )4.0(y taking h = 2.0 . Compare the solution
obtained with the exact solution and find the error.
b) Find a root of the equation 0 3x 10x 10x 7
3 2
+ + + = which is close to − 0.2 using the
Birge-Vieta method. Perform two iterations of the method
3 2
+ + + = which is close to − 0.2 using the
Birge-Vieta method. Perform two iterations of the method
1. a) A negative root of smallest magnitude of the equation x 5x 20 0
3
+ + = is to be determined
i) Find an interval of unit length which contains this root
ii) Perform two iterations of the bisection method
iii) Taking the end points of the last interval as initial approximations perform one iteration
of the secant method.
3
+ + = is to be determined
i) Find an interval of unit length which contains this root
ii) Perform two iterations of the bisection method
iii) Taking the end points of the last interval as initial approximations perform one iteration
of the secant method.
b) Solve the following game using dominance principle. (5)
Player B
B1 B2 B3 B4
A1
3 2 4 0
Player A A2
3 4 2 4
A3
4 2 4 0
A4
0 4 0 8
6. For the following payoff matrix, transform the zero-sum game into an equivalent linear
programming problem and solve it by using simplex method. (10)
Player B
B1 B2 B3
A1
1 −1 3
Player A A2
3 5 − 3
A3
6 2 − 2 6
Player B
B1 B2 B3 B4
A1
3 2 4 0
Player A A2
3 4 2 4
A3
4 2 4 0
A4
0 4 0 8
6. For the following payoff matrix, transform the zero-sum game into an equivalent linear
programming problem and solve it by using simplex method. (10)
Player B
B1 B2 B3
A1
1 −1 3
Player A A2
3 5 − 3
A3
6 2 − 2 6
7. a) Solve the following LPP using two phase method:
1 2 3 max Z = 2x + 3x +10x
Subject to:
0 2 x1 + x3 =
1 x2 + x3 =
0 , , x1
x2
x3 ≥ (6)
b) Which of the following sets are convex? Give reason.
i) }0 {( , :) ;1 , A = x1
x2
x1
x2 ≤ x1
x2 ≥
ii) }0 {( , :) 3 ; ,
1 2
2 B = x1
x2
x2 − ≥ x1
1 2 3 max Z = 2x + 3x +10x
Subject to:
0 2 x1 + x3 =
1 x2 + x3 =
0 , , x1
x2
x3 ≥ (6)
b) Which of the following sets are convex? Give reason.
i) }0 {( , :) ;1 , A = x1
x2
x1
x2 ≤ x1
x2 ≥
ii) }0 {( , :) 3 ; ,
1 2
2 B = x1
x2
x2 − ≥ x1
8. a) Without sketching the region, check whether ) P ,0( 0 is in the convex hull of the
points )0 A(− ,1 − ),1 B ,1( and )1,0( C . If it is in the region, write P as convex
combination of A, and B C . (4)
b) Use simplex method to solve the following LP problem:
Maximize 1 2 3 Z = 3x + 5x + 4x
Subject to:
8 2 3 x1 + x2 ≤
10 2 5 x2 + x3 ≤
15 3 2 4 x1 + x2 + x3 ≤
0
points )0 A(− ,1 − ),1 B ,1( and )1,0( C . If it is in the region, write P as convex
combination of A, and B C . (4)
b) Use simplex method to solve the following LP problem:
Maximize 1 2 3 Z = 3x + 5x + 4x
Subject to:
8 2 3 x1 + x2 ≤
10 2 5 x2 + x3 ≤
15 3 2 4 x1 + x2 + x3 ≤
0
9. a) Let 4 ,2 x1 = x2 = and 1 x3 = be a feasible solution to the system of equations
2 2 2 x1 − x2 + x3 =
18 4 x1 + x2 =
Reduce the given feasible solution to a basic feasible solution. (5)
b) Compute all the basic feasible solutions to the L.P. problem
Maximize 1 2 3 Z = 3x + 2x + x
Subject to
8 3 2 2 x1 + x2 + x3 + x4 =
7 3 4 x1 + x2 + x3 + x5 =
and
0 , , , , x1
x2
x3
x4
x5 ≥
2 2 2 x1 − x2 + x3 =
18 4 x1 + x2 =
Reduce the given feasible solution to a basic feasible solution. (5)
b) Compute all the basic feasible solutions to the L.P. problem
Maximize 1 2 3 Z = 3x + 2x + x
Subject to
8 3 2 2 x1 + x2 + x3 + x4 =
7 3 4 x1 + x2 + x3 + x5 =
and
0 , , , , x1
x2
x3
x4
x5 ≥
b) Find the range of values of p and q which will render the entry )2,2( a saddle
point for the game (4)
5. a) Use graphical method to solve the following game and find the value of the game:
Player B
B1 B2 B3
A1
2 4 5
Player A A2
10 7 q
A3
4 p 6
Player B
B1 B2 B3 B4
A1
2 2 3 − 2
Player A A2
4 3 2 6
point for the game (4)
5. a) Use graphical method to solve the following game and find the value of the game:
Player B
B1 B2 B3
A1
2 4 5
Player A A2
10 7 q
A3
4 p 6
Player B
B1 B2 B3 B4
A1
2 2 3 − 2
Player A A2
4 3 2 6
4. a) A marketing manager have five salesmen and five sales districts. Considering the
capabilities of the salesmen and the nature of districts, the marketing manager
estimates that sales per month (in hundred rupees) for each salesman in each district
would be as follows:
Districts
A B C D E
1 32 38 40 28 40
2 40 24 28 21 36
Salesman 3 41 27 33 30 37
4 22 38 41 36 36
5 29 33 40 35 39
Find the assignment of salesmen to districts that will result in maximum sales.
capabilities of the salesmen and the nature of districts, the marketing manager
estimates that sales per month (in hundred rupees) for each salesman in each district
would be as follows:
Districts
A B C D E
1 32 38 40 28 40
2 40 24 28 21 36
Salesman 3 41 27 33 30 37
4 22 38 41 36 36
5 29 33 40 35 39
Find the assignment of salesmen to districts that will result in maximum sales.
