Rajasthan Technical University 2011-3rd Sem B.Tech (Back) Computer science & engineering, IT, Mathematics III - Question Paper
Rajasthan tech. University
B.Tech three sem (Back)
Feb 2011
Common for CS, IT
Mathematics III
Roll No. _______[Total No. of Pages :J 4
3E1465 B.Tech. Illrd Semester (Back) Examination, Feb.-2011 Mathematics - III Common for Comp. Engg & ITUTi
Tf
fH
W
Time : 3 Hours Maximum Marks : 80
Min. Passing Marks : 24
Instructions to Candidates:
Attempt overall Five questions in all. Schematic diagrams must be shown wherever necessary. Any data you feel missing may suitable be assumed and stated clearly.
Unit -1
1. a) Show that the volume of the greatest rectangular parallelepiped that can be
X1 y1 z2 8 ,
inscribed in the ellipsoid -+--+=1 is (8)
b) Use Lagranges method of undetermined multipliers to find the minimum value
of x2 + y2 + z2 subject to the conditions x + y + z = l, xyz+l = 0. (8)
a) Obtain the stationary points of x4 + y4 -2x2 + 4 xy -2 y2. Determine their nature also. (8)
b) In a plane triangle, find the maximum value of
cos/4 cosB cosC. (8)
2. a) Solve graphically the following linear programming problem:
Maximize z =5xj +3x2, subject to 3x, + 5x2 < 15,
5x, + 2x7 < 10,
and x, x, >0. (8)
3E1465/20H (I) JContd-..
b) Solve the following transportation problem :
n
D,
D
a.
30
50
20
(8)
|
0, |
1 |
2 |
1 |
4 |
|
O, |
3 |
3 |
2 |
1 |
|
0, |
4 |
2 |
5 |
9 |
|
b. ! |
20 |
40 |
30 OR |
10 |
D,
Solve the following L:P. problem by simplex method :
a)
Max. z=2x,+4x2 subject to 2xl + 3x2 <48 x, + 3x2 <42 x, + x2 < 21 and x,, x2 > 0.
(8)
Use graphical method to solve the L.P. problem given below : Min. z = 1.5x,+2.5x2 subject to x, + 3x2 >3
b)
x, + x2 > 2
and x,,x2 >0.
Unit - III
Prepare a network diagram for the following activities concerning a certain project:
3. a)
|
Activity |
Name of Activity |
Preceding Activity |
Duration (in weeks) |
|
1,2 |
A |
3 | |
|
1,3 |
B |
- |
5 |
|
1,4 |
C |
4 | |
|
2,5 |
D |
A |
2 |
|
3, 5 |
E |
B |
3 |
|
4,6 |
F |
C |
9 |
|
5,7 |
G |
D, E |
8 |
|
3, 6 |
H |
B |
7 |
|
6,7 |
i |
H, F |
9 |
i) Find the critical path.
ii) Find EST, EFT, LSI, LFT.
(2,2,2,2-81
iii) Find float.
(2)
3E146f
w
c
b) A machine operator has to perform two operations on six jobs. The time required to form these operations in minutes for each job is given below. Find the sequence in which the jobs should be processed in order to minimise the total time required. Also, find the time elapsed.
Jobs
|
Timings for |
1 |
2 |
3 |
4 |
5 |
6 |
|
operation I: Mj |
3 |
12 |
5 |
2 |
9 |
11 |
|
operation II: M2 |
8 |
10 |
9 |
6 |
3 |
1 |
a) Five jobs are to be processed in three machines Mp M2 and M3 in that order. Processing times on the three machines are given below. Obtain the optimal sequence of jobs that minimises the total elapsed time. Find also the idle time on the three machines M]5 M2 and Mr
Jobs
|
1 |
2 |
3 |
4 |
5 | |
|
Processing on time |
8 |
10 |
6 |
7 |
11 |
|
Processing on M2 time |
5 |
6 |
2 |
3 |
4 |
|
Processing on M3 time |
4 |
9 |
8 |
6 |
5 |
b) Given the following information of time estimates concerning various activities of a project:
|
Activities |
h |
t p |
t m |
|
1 |
2 |
6 |
4 |
|
2 |
6 |
10 |
8 |
|
3 |
1 |
15 |
5 |
|
4 |
1 |
9 |
5 |
|
5 |
6 |
10 |
8 |
|
6 |
5 |
9 |
7 |
Find t for all the six activities. Also, calculate standard deviation a and
e 7 I
variance v. for each activity. (8)
3E1465 (3) [Contd....
Unit-IV
,1
at
I2 J
