CWA/ICWA Inter :: Quantitative Methods : June 2004

I-12(QNM)
Revised Syllabus

Quantitative Methods
Time Allowed : 3 Hours Full Marks : 100
SECTION I(Mathematical Techniques — 40 marks)
Answer Question No. 1 (compulsory — 10 marks)
and two other questions from this section (15x2 = 30 marks).
1. Attempt any five of the following: 2x5=10
(a)
If A = ( 2
4
− 3
− 1
) then obtain the matrix B such that AB = BA = I2.
(b)
Show that:   x − y
y − z
z − x
1
1
1
x
y
z
  =   x
y
z
1
1
1
y
z
x
 
(c)

Show that the vectors corresponding to the positions of the points (2, 1, -1) and (1, 1, 3) are at right angles.

(d)Determine f(x — 1) when f(x + 3) = 2x2 — 1.
(e)
Evaluate : Lim
x→ -1
x4 − 1
x + 1
(f)
If y = x2, find
dy
dx
(g) Differentiate x5 W.r.t x2.
(h)
Evaluate :
e3x + ex
ex + e− x
dx
(i)
Evaluate : 1

0
1 − t
1+ t
dt
(j)A function f(x) is defined as follows __
f(x) =
|x − 3|
x − 3
, x ≠ 3
=0, x = 3
Is f(x) continuous at x = 3?
2. (a)

Given that a = 2i + 3j + 6k, b = 3i — 6j + 2k and c = 6i + 2j — 3k, show that

a × b = 7 c
5

Also find a unit vector perpendicular to each of the vectors

a and b
(b)
If y = log (x + l2 + x2 ) prove that (l2 + x2)y2 + xy1 = 0
5
(c)

A shop-keeper stocks four brands of bathsoaps. The costs of the four brands are given by the row vector A = (1-, 15, 20, 25). The beginning inventory of the soaps is given by the vector

5
B = { 40
30
20
0
} and the ending inventory, { 0
10
15
20
}
Assuming the purchase of inventory, determine the cost of goods sold during the period.
3. (a) Discuss the continuity of f(x) at x = — 2, where
f(x) = {x +
x + 2
|x + 2|
}, at x ≠ − 2
=− 1, at x = − 2
5
(b)
Evaluate: 1/z

0
dx
2 − 3x
5
(c)
If u =
x3 + y3
x − y
find the value of x
δu
δx
+
δu
δy
5
 
Please turn over
 

( 2 )

I-12(QNM)
Revised syllabus
Marks
4. (a)
The total cost of daily output of q tons of coal is Rs. (
1
10
q3 − 3q2 + 50q )
4

what is the value of q, when average cost is minimum ? Verify that at this level average cost = marginal cost.

(b) Show that there is a saddle point for function z = 18x2 — 6y2 — 36x — 48y. 5
(c)

Marketing department of a company calculated the pay-offs in terms of yearly net profits for each of the strategies of expected sale prices in following table:

6
States of nature of sale
Strategiesn1n2n3
p1
p2
p3
7000
5000
3000
3000
4500
3000
1500
0
3000
Which strategy should the marketing executive choose on the basis of —
(i)Maximim criterion,
(ii)Minimax criterion, and
(iii)Laplace criterion?
5. (a) Find the area of the region bounded by the curves 6y = x2 and 6x = y2 4
(b)

Obtain a polynomial of degree 3 passing through points (0, 1), (1, 2), (2, 5) and (3, 16).

5
(c)

A Company produces two types of containers K and L. Each Product has resource requirements and profit contribution as follows:
ResourceKLTotal resource available
1. Material (kg./unit)
2. Labour (hr./unit)
1
6
2
6
10 kg.
36 hr.
Profit45 

6

In addition because of demand, a maximum of 4 units of container K will be produced. Obtain by gra0phical method, the optimal production plan that maximises the profit.

 
SECTION II(Statistical Techniques — 30 marks)
Answer Question No. 6 (compulsory — 10 marks) and two
other questions (10x2 = 20 marks) from this section.
 
6. Answer any five of the following 2x5=10
(a)

If A1, A2, A3, A4 are equally likely, mutually exclusive and exhaustive events, then show that P( A1) is:
(i)0.25 (ii)0.50 (iii)0.75 (iv)1.0

(b) For two equally likely, exhaustive and independent events A and B, P(AB) is:
(i)0 (ii)0.25 (iii)0.50 (iv)1
(c)

Let A and B are two events such that P(A) = 0.4, P(A U B) =0.7 and P(B) = p. For what choice p, are A and B independent?
(i)
1
4
(ii)
1
3
(iii)
1
2
(iv)None of these.

(d)

If the mean and variance of a binomial distribution with parameters (n, p) are 40 and 30 respectively, then the parameters are:
(i)(40, 0.50) (ii)(30, 0.25) (iii)(120, 0.50) (iv)(160, 0.25)

(e) If the expectation of a Poisson variable is 1, then p(x > 1), is
(i)1 — e—1 (ii)1 — 2e—1 (iii)1 — 3e—1 (iv)none of these
 
Please turn over
 

( 3 )

I-12(QNM)
Revised Syllabus

(f) The p.d.f. of a continuous variable is given by
f(x) = kx(x — 1), 0 < x < 1; the value of k is
(i)—6 (ii)—4 (iii)—2 (iv)none of these
(g)

Rank correlation coefficient between the marks in Mathematics and Statistics obtained by a group of students is 2/3 and sum of the squares of the differences in ranks is 55, then number of students in the group is:
(i)9 (ii)10 (iii)11 (iv)none of these

(h) Two lines of regression are given by x + 2y = 5 and 2x + 3y = 8. Then rxy is :
(i)
√3
2
(ii)
− √3
2
(iii)
√3
4
(iv)
− √3
2
(i)

A random smaple of size 5 is drawn without replacement from a finite population consistiong of 41 units. If the standard deviation of the population is 6.25, then standard error of the sample mean is

(i)0.65 (ii)1.65 (iii)2.65 (iv)3.65
(j)

A sample random sample of size 100 has mean 15 and population variance 25. Then the 99% confidence interval for the population mean is:
(i)(13.71, 16.29)(ii)(14.02, 15.98)
(iii)(13.71, 15.98)(iv)(14.02, 16.29)

7. (a)
If P(A) =
3
4
, P(B) =
1
2
and P(A — B) =
2
8
then find the probability that (i)
5

exactly one of A and B occurs and (ii) none of them occurs. Also examine whether the events A and B are independent.

(b)

From 20 tickets marked with the first 20 numerals, one is drawn at random. What is the probability that it is a multiple of 3 or 7 ?

5
8. (a) The table below shows the respective heights in centimetre of 10 fathers and their eldest sons:
Father:
Son
67
68
63
66
66
65
71
70
69
69
65
67
62
64
70
71
61
60
72
63
Find the rank correlation coefficient.
5
(b)

Is it likely that a sample of size 300 whose mean is 12, is a random, sample from a large population with mean 12.8 and s.d. 5.2 ?

5
9. (a)

Between the hours 2 p.m. and 4 p.m. the average number of phone calls per minute coming into the switch board of a company is 2.35. Find the probability that during one particular minute, there will be at most 2 phone calls.

6
[ Assume Poisson distribution.    Given e—2.35 = 0.095374].
 
Please turn over
 

( 4 )

I-12(QNM)
Revised syllabus
Marks
(b)
For a normal distribution N(m, σ2). evaluate P{|x − m| ≤ 3σ}.
4
[ Given : 3

0
1
√2π
e-1/z dt = 0.4986 ]
10. (a)

A company has head office at Kolkata and a branch office at Mumbai. The personal director wanted to know if the workers at the two places would like the introduction of a new plan of work and a survey was conducted for this purpose. Out of sample of 500 workers at Kolkata 62% favoured the new plan. At Mumbai out of a sample of 400 workers 41% were against the new plan. Is there any significant difference between the two groups in their attitude towards the new plan at 5% level?

5
(b)

A Mumbai film director claims that the his films are liked equally by males and females. An opinion survey of a random samples of 1,000 film-goers revealed the following results:
LikedDisliked
Males402193
Females245160

5
Is the film director's claim supported by the data?
Given: χ20.05=3.85.997.81
d.f.=123
11.(a)

An I.Q. test was administered on 10 candidates before and after they were trained. The results are given below:

5
Candidates
I.Q. before training:
I.Q. after training:
1
167
170
2
124
138
3
157
158
4
155
158
5
163
156
6
154
157
7
156
167
8
168
172
9
133
142
10
143
138

Test whether there is any improvement in I.Q. after the training.
[ Given that t0.01,9 = 2.82. and t0.01,10 = 2.76]

(b) You are given the following payoffs of three acts A1, A2 and A3, and the states of nature S1, S2, S3.
State ofActs
natureA1A2A3
S1
S3
25
400
650
—10
440
740
—125
400
750
5

The probability of the three states of nature are respectively o.1, 0.7 and 0.2. Calculate and tabulate E.M.V. and conclude which of the acts can be chosen as the best.

 
SECTION III(Economic Techniques — 30 marks)
 
12. Attempt any five of the following: 2x5=10
(a) Express MR in terms of price elasticity and price.
(b)

Show the elasticity of demand with respect to price for the demand function x = 3p—2 is constant, p and x being the price and quantity in demand respectively.

(c)

The daily cost of production c for x units of an assembly is given b c(x) = Rs. 12.5x + 6400; and selling price of each unit Rs. 25. If now the selling price is reduced by Rs. 2.5 per unit, what will be the break even point?

 
Please turn over
 

( 5 )

I-12(QNM)
Revised Syllabus

(d) If 10% fall in price causes a 15% rise in demand, then find price elasticity of demand and its nature.
(e) The trend of annual sales of a company is as follows:
Yc = 18 + 0.10X, origin = 1990, X unit = 1 year, Yc unit is annual production.
Convert the equation to a monthly trend equation.
(f)

Justify the statement: "the purchasing power of money decreases as the wholesale price index increases".

(g) For a trivariate distribution r12 = 0.7 and r23 = r13 = 0.6. Calculate r123.
(h) Determine the new indices for following 3 years after shifting base year from 1970 to 1988:

1987
1988
1989
Old index No.
324
351
379
13. Answer any four of the following 5x4=20
(a)

A demand law is given by x = 100 — 2p where x is the quantity demanded and p is the price. If the price elasticity of demand at p = 10 is increased by 50% then obtain the percentage variation in demand.

(b) Calculate the index number of prices for 1995 on the basis of 1990 from the data given below:
Price per unit (Rs.)
CommodityWeight19901995
A
B
C
D
40
25
20
15
16
40
12
2
20
50
15
3

If the weights of commodities A, B, C, D are increased in the ratio 1 : 2 : 3 : 4, then what will be the increase/decrease in Index number?

(c) Calculate the seasonal from the following data using the average method:
Year
1992
1993
1994
1995
1996
1st quarter
76
75
72
78
75
2nd quarter
68
66
70
74
74
3rd quarter
80
82
86
84
84
4th quarter
70
74
78
80
82
(d)calculate the 4-yearly moving average from the following data:
Year
Annual values
(Rs. 'oooo)
year
Annual values
(Rs.'ooo0)
1990
1991
1992
1993
1994
52.7
79.4
76.3
66.0
68.6
1995
1996
1997
1998
1999
93.8
104.7
87.2
79.3
85.0
(e)

Given the following transaction matrix, find the gross output to meet the new final demand of 200 units of agriculture and 800 units of Industry:
Purchasing sectorPurchasing sector
 AgricultureIndustry Final demand
Agriculture
Industry
300
400
600
1200
100
400

(f) Write short notes on any one of the following :
(i)Scatter diagram,
(ii)Multiple correlation,
(iii)Emperical Production Function Analysis.

__________

 

© Krishbhavara ♣