LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034
P.G. DEGREE EXAMINATION – COMMON PAPER
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THIRD SEMESTER – NOV 2006
MT 3925 – MATHEMATICAL SOCIAL SCIENCES
Date & Time : 08-11-2006/9.00-12.00 Dept. No. Max. : 100 Marks
Answer ALL Questions.
- (a) Calculate the median for the following distribution.
class interval : 131-135 135-140 140-145 145-150 150-155 155-160 160-165 165-170 170-175
Frequency : 3 11 17 19 27 22 14 8 4
(or)
- What is a scatter diagram? Explain its different types. (5 marks)
- Find the mean , standard deviation and variance for the following :
X : 29 33 37 38 39 40 42 43 45 47 50 59 60
F : 1 1 3 4 2 3 2 2 3 1 1 1 1
(or)
(d) Ten competitors in a musical test were ranked by three judges A, B and C in the following order .
Ranks by A : 1 6 5 10 3 2 4 9 7 8
Ranks by B : 3 5 8 4 7 10 2 1 6 9
Ranks by C : 6 4 9 8 1 2 3 10 5 7
Using rank Correlation method , find which pair of judges has been nearest approach to common
likings in music. (15 marks)
- (a) Define a directed graph, an undirected graph and a multigraph with examples.
(or)
(b) Explain Konigsberg Bridge problem. Is there any solution to this problem?. If not, why?
(5 marks)
(c) (i) Write the incidence matrix and adjacency matrix for the following graph.
(ii) Define a walk, a path and an Eulerian graph with examples. (10 + 5 marks)
(or)
- (i) In a colony there are 5 families represented by F1, F2,..F5. Draw a graph with edges which denote the friendship between the families with the condition that all families are friends to the other.
(ii) The following graph represents the map of a certain area where vertices represent junction and lines represent roads. Help your postman to start from a vertex (which represents the post office) and go through all the streets exactly once and return to the post office (Vertices may be repeated).
(iii) An electrical circuit board has to be prepared with the following requirements. There are 7 terminals T1, T2, …, T7. Wires have to be laid between the terminals as per the table given below.
T1 – T5, T7, T4
T2 – T5, T7, T4
T3 – T5, T7, T4
T4 – T2, T6, T3, T1
T5 – T1, T2, T3
T6 – T4
T7 – T1, T2, T3
Draw a graph to represent this board. As it is always good to avoid crossing of wires (which will cause short circuit), give the best possible layout of the graph.
(4 + 5 + 6 marks)
III. (a) What are the objectives of an investigation.
(or)
(b) What is sampling? State the reasons for sampling. (5 marks)
(c) What are the limitations of sampling?
(or)
(d) Explain the subject matter of a survey. (15 marks)
- (a) The manager of an oil refinery must decide on the optimum mix of two possible blending processes of which the input and output production runs are as follows:
Process Input Output
Crude A Crude B Gasoline X Gasoline Y
1 6 4 6 9
2 5 6 5 5
The maximum amounts available of crudes A and B are 250 units and 200 units respectively. Market demand shows that at least 150 units of gasoline X and 130 units of gasoline Y must be produced. The profits per production run from process 1 and process 2 are Rs.4 and Rs.5 respectively. Formulate the problem for maximizing the profit.
(or)
(b) Construct a network diagram for the activities B, C, ….Q and N satisfying the following constraints:
B<E,F; C < G,L; E,G<H; L,H<I; L<M; H<N; H<J; I,J<P; P<Q.
The notation X<Y means that the activity X must be completed before Y can begin.
(5 marks)
(c) (i) A company makes two kinds of leather belts. Belt A is a high quality belt, and belt B is of lower quality. The respective profits are Rs.4 and Rs.3 per belt. Each belt of type A requires twice as much time as a belt of type B, and if all belts were of type B, the company could make 1000 per day. The supply of leather is sufficient for only 800 belts per day. Belt A requires a fancy buckle and only 400 per day are available. There are only 700 buckles a day available for belt B. Determine the optimum product mix by graphical method.
(ii) What are the rules to be followed for the construction of a network? (10 + 5 marks)
(or)
(d) Use simplex method to solve Maximize z = 2x1 + 4x2 + x3 + x4 subject to constraints
x1 + 22 + 2x3 + 4x4 80, 2x1 + 2x3 + x4 60, 3x1 + 3x2 + x3 + x4 80,
x10, x2 0, x3 0, x4 0. (15 marks)
- (a) Explain the fuzzy association of the intelligent control of a traffic light.
(or)
(b) Solve the following assignment problem.
A B C D
I 1 4 6 3
II 9 7 10 9
III 4 5 11 7
IV 8 7 8 5 (5 marks)
(c) Solve the following transportation problem.
D1 D2 D3 D4 Availability
O1 5 3 6 2 19
O2 4 7 9 1 37
O3 3 4 7 5 34
Demand 16 18 31 25 (15 marks)
(or)
(d) (i) Describe FAMs as mappings.
(ii) Define Fuzzy Hebb Matrix.
(iii) Construct a fuzzy Hebb matrix M given the input A = (.3 .4 .8 1) recalled fit vector B = (.2 .6 .5) upon max-min composition: AoM = B. (5 + 4 + 6 marks)
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