Rajasthan Technical University 2008 B.E Information Technology mathematics-iii - Question Paper

? B.Tech. (SEM.III) , January- 2008

Mathematics -III

Time :3 Hours] Total Marks:80

Attempting 5 ques. in all selecting 1 ques. form every unit. Schematic diagram should be shown wherever necessary.


UNIT- I

Q.1
(a) Use Kuhn –Tucker condition to
min ƒ(x,y,z) =x² +y² +z² +20x + 10y
constraints are :
(i) x = 40
(ii) x + y = 80
(iii) x + y +z =120 (10)

(b) Write 12 application of optimization in Engineering fields ?
(6)
Or

(a) obtain the extreme points of the function
ƒ(x, y) = x³ + y³ +2x² +4²x+6 (8)

(b) Minimize : ƒ (x)= ½ (x² + y² +z²)

Constraints are :
g(x,ys)=x - y =0
g2(x,y,z)=x + y +z -1=0
by Lagrange multipliers method.


UNIT – II

Q.2
(a) Solve the subsequent LPP by graphical method :
Min. Z =2x +3y
Constraints are :
x + y = 4
6x + 2y = 8
x + 5y = 4
x = 3,y=3 and x =0, y=0 (16)

Or
Q.2
Use VAM method to solve the subsequent transportation issue :

D D1 D2 D3 Amount Available

O O1 1 2 1 30
O2 3 3 2 50
O3 4 2 5 20
Amount Required 30 40 30 100
obtain the optimal solution ? (16)


UNIT-3





Q.3
(a) A project consists of the subsequent activities and various time estimates :

Activity t. t? t?
1-2 3 four 8
1-3 2 8
1-4 6 8 12
2-5 5 9 9
4-6 3 6 10
5-6 2 4 8

(i) Draw the network.
(ii) Determine the expected time and variance for every activity.
(iii) obtain the critical path and the project variance.
(iv) What is the probability that the project will be completed by 22 days ?
(Given : P(z = 2.1)=0.9821)

(a) Five jobs are to be processed on 2 machines M, N in that order. Processing time for the jobs on the 2 machines is detailed beneath. Determine a sequence for the 5 jobs that minimizes the elapsed time :

Job 1 2 3 four
M one nine 3 10
N 2 6 7 8

Or

Q.3
(a) Use Laplace transform theory to solve :
(D² +1)x = t cos2t , x(0) = x'(0) =0 (8)
(b) obtain the Fourier transform of : ƒ(x) = 1- x² , ?x? <1 (8)
UNIT-IV

Q.4(a)Apply convolution theorem to evaluate 1/L[ s/(s² + a²)² (6)

(b) Use Fourier transform theory to solve : ??/?t = C² (?²?/?x²)
Subject to the condition :
(i) ?=0 when t=0, x=0
(ii) ??/?x = -u , a constant ,when x=0 and t>0.
presume that ?(x,t) and ??/?x both tend to zero as x?8 (10)

Or

Q.4
(a) The subsequent data are available for a project :

ACTIVITY DURATION (DAYS) PRECEDING ACTIVITY
A 5 -
B 6 A
C 5 B
D 4 A
E 3 D
F 4 C,E

(i) Prepare the network diagram for the project.

(ii) Compute the earliest event time and recent event time.

(iii) Determine the critical path and total project duration .
(iv) calculate total float for every activity. (16)

UNIT-V

Q.5
(a) obtain by Newton-Raphson method a real root of the formula :

x² -3x-5=0 (16)

Or

Q.5
(a) If dy/dx = x + y², use Runge-Kutta method to obtain an approximate value of y for x=0.2 provided that y=1 when x=0 (for h=0.1) (16)

END

Earning:   Approval pending.