CWA ICWA Inter - Stage II : Quantitative Methods - December 2009

CWA ICWA Inter - Stage II • Quantitative Methods • December 2009

This Paper has 61 answerable questions with 0 answered.

1—12(QNM) Revised Syllabus
Time Allowed : 3 Hours	Full Marks : 100

The figures in the margin on the right side indicate full marks.
(Notations and symbols used have their usual meanings.)
SECTION I (Mathematical Techniques — 40)
Answer Question No. 1 (Compulsory — 10 marks) and two other questions (15x2=30 marks) from this section.
Marks

The figures in the margin on the right side indicate full marks.

(Notations and symbols used have their usual meanings.)

SECTION I (Mathematical Techniques — 40)

Answer Question No. 1 (Compulsory — 10 marks) and two other questions
(15x2=30 marks) from this section.

Marks

Attempt any five questions:
Choose the correct options showing the proper reasons/calculations.

2x5

(a)

The value of the determinant

a
b
c

–
–
–

b
c
a

–
–
–

c
a
b

–
–
–

a
b
c

(A)

(B)

(C)

a + b + c

(D)

None of these.

(0)

(b)

(

1
3

2
5

4
2

)

(

2
0
4

–1
3
1

)

(A)

(

18
9

14
14

)

(B)

(

18
14

9
14

)

(C)

(

14
18

14
9

)

(D)

None of these.

(0)

(c)

The value of x, for which vectors 2i — j + k, i — 3j — 5k and xi — 4j — 4k are coplanar, is

(A)

(B)

(C)

—1

(D)

None of these.

(0)

(d)

The value of

lim
x→∞

4x⁵ + 2x³ – 5

7x⁸ + x⁴ + 2

(A)

(B)

(C)

(D)

—1

(0)

(e)

For what value of x the function f (x) =

x² — 4

x – 2

is undefined?

(A)

–1

(B)

(C)

(D)

(0)

(f)

If f (x) =

the value of f(c) – f(–c) for positive real value of c is

(A)

(B)

–2

(C)

(D)

None of these.

(0)

(g)

The gradient of the curve y = x + √x² + a² is

(A)

√x² + a²

(B)

√x² + a²

(C)

x + y

√x² + a²

(D)

x – y

√x² + a²

(0)

(h)

The function y = x³ – 3x² + 7 has a minimum value at

(A)

(B)

(C)

(D)

None of these.

(0)

(i)

The value of

∫¹
_–1

x dx is

(A)

(B)

(C)

(D)

None of these.

(0)

(j)

If μ = x² + y² + 2xy then the value of x

∂μ

∂x

+ y

∂μ

∂y

, when x = y = 1, is

(A)

(B)

(C)

(D)

None of these.

(0)

(a)

Show that the vectors i + 2j + k, i + j – 3k and 7i – 4j + k are mutually orthogonal. Determine length of the vectors.

(0)

(b)

If f(x) = x² – 5x + 6 and A =

(

2
2
1

0
1
–1

1
k
0

)

, find k for which f(A) =

(

1
–1
–5

–1
–1
4

–3
–10
4

)

(0)

(c)

Using properties of determinant, show that

a²
b²
c²

–
–
–

bc
ca
ab

a
b
c

1
1
1

= 0

(0)

(a)

Write short note on any one:

(i)

North–West corner method for obtaining initial BFS in transportation problem.

(0)

(ii)

Method of first principle in obtaining first order derivative and its use to get

d√x

(0)

(b)

If f(x) = √ x² + a² + a log_e (x + √ x² + a²), prove that

f’ (0) = 1 and f’’ (0) =

when f’ (0) =

df(x)

at x = 0 and f’’ (0) =

d²f(x)

dx²

at x = 0.

(0)

(c)

Find the extreme value of the function f(x, y) = x² – y² + xy + 5y subject to x + y + 1 = 0.

(0)

(a)

Integrate

∫

e^{x – 1} + x^{e – 1}

e^x + x^e

dx.

(0)

(b)

Evaluate

2
∫
1

x + √x

(0)

(c)

The demand and supply function under perfect competition are y = 16 – x² and y = 2 (x² + 2) respectively. Find the market price, consumer’s surplus and producer’s surplus.

(0)

(a)

Solve the following LPP by using Simplex method.
Maximize z = 4x + 3y subject to 2x + y ≤ 30, x + 2y ≤ 24, x ≥ 0, y ≥ 0.

(0)

(b)

A customer repairing shop has one mechanic only. Customers are appearing at a rate of 20 per hour. The time required to serve a customer has exponential distribution with mean of 144 seconds. Find the number of customers to be expected in the shop. Also find average waiting time of a customer (i) in the system, (ii) in the queue.

(0)

(c)

Solve the following game using the maximin–minimax principle;

		Player B
		B₁	B₂	B₃
Player A	A₁	7	4	1
	A₂	4	2	0
	A₃	3	–1	–2
	A₄	1	5	–3

(0)

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

(a)

If for two events A and B, P(A) =

, P(B) =

and P (A∪B) =

then P (exactly one of A and

B occurs) is

(i)

(ii)

(iii)

(iv)

none of these

(0)

(b)

If for two events A and B, P(A) =

, P(B/A) =

then P(A – B) is

(i)

(ii)

(iii)

(0)

(c)

If for three mutually exclusive and exhaustive events A, B and C P(A) =

, P(B) and P(B) =

P(C)

then P(A) is

(i)

(ii)

(iii)

(iv)

None of these

(0)

(d)

The standard deviation of the probability distribution

x	0	1	2
Prob	¼	½	¼

then P(A) is

(i)

√1

(ii)

0.2

(iii)

(iv)

√ 0.3

(0)

(e)

If for two variables x and y

10
Σ
i = 1

(x_i – x)² = 20,

10
Σ
i = 1

(y_i – y)² = 45,

10
Σ
i = 1

(x_i – x ) (y_i – y ) = 27,

then the correlation coefficient between x and y is

(i)

0.9

(ii)

0.6

(iii)

(iv)

√3

(0)

(f)

For a random variable X following Poisson distribution with mean 2 then P(X ≥ 1) is

(i)

e^–2

(ii)

3e^–2

(iii)

1 – e^–2

(iv)

1 – 3e^–2

(0)

(g)

If X be a standard normal variable with

1.96
∫
–1.96

(2π)^½ e^–x²/2 dx = 0.95 then P(X > 1.96) is

(i)

0.025

(ii)

0.25

(iii)

0.05

(iv)

0.475

(0)

(h)

Proportion of defective items in a large lot of items is p. Taking a random sample of 6 items from the lot and accepting the null hypothesis if the number of defectives in the sample being 5 or less, then the probability of type II error when p = 0.3 is

(i)

1 – 4.2(0.3)⁵

(ii)

1 – 0.3 (4.2)⁵

(iii)

4.2 – (0.3)⁵

(iv)

None of these

(0)

(i)

If two simple random samples each of size 5 are drawn with and without replacement from a finite population of size 21 with variance 25, have standard errors of sample means s₁ and s₂ respectively, then s₁/s₂ is

(i)

√5

(ii)

√5

(iii)

(iv)

(0)

(j)

A simple random sample of size 100 has mean 10, population variance being 25 and

1.96
∫
–1.96

(2π)^–½ e^–x²/2 dx = 0.95 Then the 95% upper confidence limit for population mean is

(i)

9.02

(ii)

10.98

(iii)

9.05

(iv)

10.95

(0)

(a)

The personnel department of a company has records which show the following analysis of its 200 engineers:

Age (in years)	Qualification		Total
	Graduate	Post–Graduate
Under 30 30–40 Over 40	90 20 40	10 30 10	100 50 50

If an engineer is selected at random find the probability that he is (i) a graduate; (ii) a post–graduate given that his age is over 40; and (iii) under 30 and a graduate.

(0)

(b)

A shopkeeper of some highly perishable type of fruit observes that the daily demand of this fruit in his locality has the following probability distribution:

Daily demand (in dozen)
Probability

:
:

6
0.1

7
0.4

8
0.3

9
0.2

He sells for Rs. 10 a dozen while be buys each dozen at Rs. 4. Unsold fruits at the end of the day are sold next day at Rs. 2 per dozen. Assuming he stocks the fruits in dozens, how many dozens should he stock so that his expected profit is maximum?

(0)

(a)

A committee of 4 persons is to be appointed from 3 officers of the production department, 4 officers from the purchase department, 2 officers of the sales department and 1 Chartered Accountant. Find the probability of forming the committee such that there must be (i) one from each category; (ii) at least one from the purchase department; and (iii) the chartered Accountant.

(0)

(b)

Fit a binomial distribution to the following data and find expected frequencies:

x
f

:
:

0
20

1
120

2
140

3
80

4
40

Total
400

Where x = number of heads, f = number of times x heads occur in 400 times of 4 tosses of a coin.

(0)

(a)

Find the correlation coefficient between x and y, the value of x when y = 10 and the value of y when x = 100 from the regression lines 2y = x + 50 and 3y = 2x + 10.

(0)

(b)

If X be a Poisson variable such that P(X = 0) + P(X = 1) = 4P(X = 2), find the probability that (i) X is positive, (ii) X is atmost one.

(0)

10.

(a)

A random sample of size 10, drawn from a normal population with variance 4, has mean 48. Test at 5% level of significance the hypothesis that the population mean is 50.

It is given that

1.96
∫
–∞

(2π)^–½ e^–t²/2 dt = 0.975.

(0)

(b)

In an infantile paralysis epidemic, 500 persons contracted the disease. 200 received no serum treatment and of these 75 became paralysed. Of those who received serum treatment, 65 became paralysed. Was the serum treatment effective?

[Given

χ²
0.01

= 6.64 at df = 1]

(0)

11.

(a)

A population consists of 5 height 60, 61, 62, 64, 62 in feet. Drawing all possible samples without replacement of size 2 from the population find the sampling distribution of the sample mean. Also find mean of the sampling distribution.

(0)

(b)

Write short note on any one:

(i)

Normal distribution and its uses;

(0)

(ii)

Scatter diagram and its uses.

(0)

SECTION III (Economic Techniques — 30 marks)

12.

Attempt any five of the following:

2x5

(a)

Find the elasticity with respect to price for the demand function x =

where p and x are the

price and quantity of demand respectively.

(0)

(b)

The marginal cost function when output is x units is given by MC = x² – x + 4. Find the total cost function C and average cost function AC if fixed cost is 30.

(0)

(c)

For the functions x₁ =

P₁

P₂

and x₂ =

P₁³

P₂

examine the nature of the commodities where P₁, P₂, x₁, x₂

are the prices and quantities demanded of the two commodities.

(0)

(d)

If the seasonal index is 90 and the deseasonalised value of production is 30 find the production there.

(0)

(e)

If the trend equation of production (y) is y = 81 + 2i with origin at 1992 and time unit = t unit = 1 year.Unit of y is 1,000 tonnes. What is monthly increase in production?

(0)

(f)

If the Paasche’s price index number is written a weighted harmonic mean of price relatives what will be the weights?

(0)

(g)

Net monthly income of an employee was Rs. 8,000/- per month in 2000. It rises to Rs. 10,000/- per month in 2008 by giving additional dearness allowance for compensating rightly. If the price index number in 2000 is 160, what was it in 2008?

(0)

(h)

If r₁₂, r₁₃, r₂₃ are equal to r (# 1) then find the value of multiple correlation coefficient R_1.23.

(0)

(i)

The total daily cost for x articles is TC = 3x + 2,000 rupees. If each article is sold at Rs. 5 what is the break even point?

(0)

13.

Answer any four of the following:

5x4

(a)

The average cost function (AC) (in Rs.) for a certain commodity is given by AC = 3x + 1 +

in terms of output x. Find the output for which AC is minimum. Find the total cost (C) and marginal cost (MC) for that output.

(0)

(b)

A consumer demand curve is x = 100 – 2p where p and x denote price and quantity demanded respectively. If the price elasticity of demand when the price at p = 10 is increased by 40% then obtain the percentage decrease in demand.

(0)

(c)

Calculate the trend values by the method of 4–yearly moving averages with weights 1 : 2 : 2 : 1 from the following data:

Year
Output (lakh tons)

:
:

1
21

2
22

3
24

4
23

5
26

6
25

7
28

8
27

(0)

(d)

Quarterly sales in thousand rupees of a company is given below:

Quarters	Year
Quarters	1996	1997	1998
I II III IV	8.4 6.2 8.9 10.3	8.5 7.9 8.5 10.1	9.5 7.2 7.2 8.7

Calculate seasonal indices if there is no appreciable trend.

(0)

(e)

The price relatives and weights of a set of commodities are given below:

Commodity	:	A	B	C	D
Price relative	:	120	127	—	119
Weight	:	2w₁	w₁ + 1	w₁	w₁ + 4

If the index for the set is 122 and the sum of weights is 40 find w₁ and price relative of commodity C.

(0)

(f)

On the basis of observations on some trivariate case r₁₂ = 0.8, r₁₃ = 0.7 and r₂₃ = 0.6 prove that

1 = R²_1.23

= (1 – r²₁₂) (1 – r²_13.2)

where R_1.23 and r_13.2 are multiple correlation and partial correlation respectively.

(0)

(g)

Write short note on any one:

(i)

Least square theory in fitting a straight line;

(0)

(ii)

Leontief’s input output model.

(0)

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