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).

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

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

(g)

Differentiate x⁵ W.r.t x².

(h)

Evaluate :

∫

e^3x + e^x

e^x + e^{− x}

(i)

Evaluate :

1
∫
0

1 − t

1+ t

(j)

A function f(x) is defined as follows __

f(x)

|x − 3|

x − 3

, x ≠ 3

, x = 3

Is f(x) continuous at x = 3?

(a)

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

= 7

Also find a unit vector perpendicular to each of the vectors

and

(b)

If y = log (x +

√

l² + x²

) prove that (l² + x²)y₂ + xy₁ = 0

(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

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.

(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

(b)

Evaluate:

1/z
∫
0

√

2 − 3x

(c)

If u =

x³ + y³

x − y

find the value of x

δu

δx

δu

δy

Please turn over

( 2 )

I-12(QNM)
Revised syllabus

Marks

(a)

The total cost of daily output of q tons of coal is Rs. (

q³ − 3q² + 50q )

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.

(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:

States of nature of sale
Strategies	n₁	n₂	n₃
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?

(a)

Find the area of the region bounded by the curves 6y = x² and 6x = y²

(b)

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

(c)

A Company produces two types of containers K and L. Each Product has resource requirements and profit contribution as follows:

Resource K L Total resource available

1. Material (kg./unit)
2. Labour (hr./unit) 1
6 2
6 10 kg.
36 hr.

Profit 4 5

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.

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)

(ii)

0.25

(iii)

0.50

(iv)

(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

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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 r_xy is :

(i)

√3

(ii)

− √3

(iii)

√3

(iv)

− √3

(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)

(a)

If P(A) =

, P(B) =

and P(A — B) =

then find the probability that (i)

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 ?

(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.

(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 ?

(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.

[ Assume Poisson distribution. Given e^—2.35 = 0.095374].

Please turn over

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I-12(QNM)
Revised syllabus

Marks

(b)

For a normal distribution

N(m, σ²). evaluate P{|x − m| ≤ 3σ}.

[ Given :

3
∫
0

√2π

e^-1/zdt = 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?

(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:

Liked Disliked

Males 402 193

Females 245 160

Is the film director's claim supported by the data?

Given: χ²_0.05	=	3.8	5.99	7.81
d.f.	=	1	2	3

11.

(a)

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

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 t_0.01,9 = 2.82. and t_0.01,10 = 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 of	Acts
nature	A₁	A₂	A₃
S₁ S₃	25 400 650	—10 440 740	—125 400 750

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?

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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:

Y_c = 18 + 0.10X, origin = 1990, X unit = 1 year, Y_c 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.)
Commodity	Weight	1990	1995
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 sector Purchasing sector

Agriculture Industry 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.

__________

CWA/ICWA Inter :: Quantitative Methods : June 2004