SVKM’s NMIMS UNIVERSITY
SCHOOL OF DISTANCE LEARNING
COURSE:
SUBJECT:
DATE:
TIME:
ADSCM SEM-I
Quantitative Techniques
6-06-2008
MARKS: 100
11 A.M TO 2 P.M
 Attempt any five questions.
 Assume data where ever necessary and state them clearly.
 Use of calculator is permitted.
Q.1
a) Solve the following linear equations using determinants: 2x - y = 5 and 3x + 2y = -3.
b) Write a note on scatter diagram. Draw a scatter diagram for the following data:
X
2
3
5
6
8
9
Y
6
5
7
8
12
11
c) The income of a group of 10000 persons was found to be normally distributed with mean Rs. 750 p.m and
standard deviation Rs.50. Show that of this group about 95% has income exceeding Rs.668 and only 5% has
income exceeding Rs.832. What was the lowest income among the richest 100?
Q.2
a) A factory has 3 departments I,II and III. It manufactures two products X and Y. The labor hours consumed per
unit in each department and the capacities of the departments are shown below. The selling price of each product
is product X Rs.15/unit and product Y is Rs.20/ unit. Recommend an optimum product mix to maximize the
selling price.
Department I
Department II
Department III
Product X
2 hr.
8 hr.
8 hr.
Product Y
4 hr.
6 hr.
30 hr.
Capacity
1600 hr.
4800 hr.
9600 hr.
b) The following table gives the results of matriculation examination held in 1999. Calculate the Karl Pearson’s
coefficient of correlation and comment on the result.
Age of
13.5
14.5
15.2
16.3
17.3
18.5
19.5
20.5
21.5
Candidate
% of
39.2
40.2
43.3
34.2
36.6
39.2
48.9
47.1
54.5
failures
c) A food processing plant has a problem with delivery of goods that must all be processed in a single day. The
following mathematical function y = 12x – 2a – ax2 describes the profit function ( y in ‘ooos Rs.) given
number of processing machines used (x) and the number of deliveries (a). i) Show that the system is
uneconomical if 4 deliveries (a =4) are made in one day. Ii) If 3 deliveries are made in one day, find the number
of processing machines that should be used in order that profit is maximized. In this case what is the maximum
profit?
Q.3
a) State the merits and demerits of Mode. Find the mode for the following data:
Marks
0-10 10-20 20-30 30-40 40-50 50-60 60-70 70-80 80-90 90-100
No. of
3
5
7
10
12
15
12
6
2
8
students
b) Solve by matrix method: 2x + y –z =3 ; x + y +z =1 ; x -2y -3z = 4
c) A manufacturer Knows that if x ( hundred) products are demanded in a particular week: The total cost function
(Rs.000) is 14 + 3x and the total revenue function ( Rs.000) is 19x – 2x2. Derive i) the total profit function
ii) find the profit break – even points. Calculate the level of demand that maximizes profit and the amount of
profit obtained.
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6
7
7
6
7
7
6
7
7
Q.4
a) Write a note on Binomial distribution. The probability that in a particular town a college student will graduate
is 0.4. Determine the probability that out of 5 students i) none will graduate ii) one will graduate.
b) Determine the Spearman’s Rank correlation coefficient for the following data:
X
92
89
87
86
83
77
71
63
53
50
Y
86
83
91
77
68
85
52
82
37
57
c) State the functions of Statistics. Plot a histogram for the data:
C.I
0-20
20-40
40-60
60-80
80-100
100-120
f
7
12
15
10
9
4
Q.5
a) Find mean and median for the following data:
Rainfall
20-25
25-30
30-35
35-40
40-45
45-50
50-55
No. of
2
5
8
12
10
7
6
years
b) Determine the coordinates and the nature of turning points on the curve: y = x3 – 7.5x2+ 18x + 6.
c) Write a note on Poisson distribution. 10% of the tools produced in a certain manufacturing process turn out to be
defective. Find the probability that in a sample of 10 tools chosen at random, exactly 2 will be defective.
( given: e-1 =0.368)
Q.6
a) Calculate the Karl Pearson’s Coefficient of Skew ness for the following data:
Profit (in
10-12
12-14
14-16
16-18
18-20
20-22
22-24
‘000)
No. of
7
15
18
20
25
10
5
companies
b) The following data represents the demand X and supply Y both in thousands of units of a certain commodity
during the first seven months of 1997. Estimate the supply when the demand is 8000.
Demand
1
2
3
4
5
6
7
Supply
2
4
7
6
5
6
5
c) The time taken to complete jobs of a particular type is known to be normally distributed with mean 6.4 hr. and
the standard deviation 1.2 hr. What is the probability that a randomly selected job of this type takes i) less than
7 hours ii) between 6 hours & 7 hours
Q.7
a)
1 2
3 7 
A
 , B =
 Verify (AB)’ = B’A’
3 4
 4 1
b) A firm is independently working on two separate jobs. There is a probability of only 0.3 that either of the jobs
will be finished on time. Find the probability that: i) both ii) neither iii) just one iv) at least one of the jobs is
finished on time.
c) Find the coefficient of variation for the following data:
Duration
0-30
30-60
60-90
90-120
120-150
150-180
180-210
No. of
9
17
43
82
81
44
24
Calls
Q.8
a) A refrigerator manufacturer can sell all the refrigerators of a particular type that he can produce. The total cost
of producing ‘q’ refrigerators per week is given by 300q + 2000. The demand function is estimated as 500 -2q. i)
find the revenue function ii) obtain total profit function iii) Find the output so that the profit is maximized and
the maximum profit.
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6
7
7
6
7
7
6
7
7
6
7
7
6
b) The probabilities of X, Y and Z becoming managers are 4/9, 2/9 and 1/3 respectively. The probabilities that a
Bonus Scheme will be introduced if X, Y and Z become managers are 3/10,1/2 and 4/5 respectively. i) What is
the probability that the bonus scheme will be introduced? ii) If the bonus scheme was introduced, what is the
probability that the manager appointed was ‘X’?
c) The total revenue obtained in( Rs.000) from selling x ( hundred items ) in a particular day is given by R which
is the function of variable x. Given that dR/dx = 20 – 4x. Determine the total revenue function and find the
number of items sold in one day that will maximize total revenue and evaluate this total revenue.
Q.9
a) 1000 light bulbs with a mean life of 120 days are installed in a new factory; their length of life is normally
distributed with standard deviation 20 days. How many bulbs will expire in less than 90 days? If it is decided to
replace all the bulbs together, what interval should be allowed between replacements if not more than 10% should
expire before replacement?
b) Explain the procedure to solve standard assignment problem. A company has 3 jobs to be done. The following
matrix shows the cost of assigning each job W1, W2, W3 to each machine M1, M2 and M3 respectively. Assign 3
jobs to 3 machines so as to minimize total cost
job
machine
W1
W2
W3
M1
120
100
80
M2
80
90
110
M3
110
140
120
c) A farm is engaged in breeding of pigs. The pigs are fed on various products grown on the farm. In view of the
need to ensure certain nutrient constituents it is necessary to buy products say A and B in addition. The content
of various products per unit, in nutrient constituents are given in the following table .If the product A cost Rs.20
and B cost Rs.40 per unit , how much each of these two products should be bought so that the total cost is
minimized.
Nutrient content in the product
Minimum amount of
nutrient required
nutrient
A
B
M1
36
6
108
M2
3
12
36
M3
20
10
100
Q.10
a) State the merits and demerits of arithmetic mean. Find mean median and mode for the data: 2,2,4,6,7,10,12.
b)
 4 6 1 


A   1 3 6  Find the inverse of matrix A.
5 7 9


c) A manufacturing organization has two factories F1 and F2 located at two different cities. The centralized
planning cell has to decide on allocation of orders from 3 markets to the factories with a view of minimizing the
overall cost to the organization. The demand, capacity and cost are given below: Determine the optimum
schedule for the transportation problem using Vogel’s Approximation Method.
Factory
Market
Capacity
M1
M2
M3
F1
25
17
22
300
F2
15
12
19
500
Demand
300
300
500
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