CWA ICWA Inter Stage II : Quantitative Methods - June 2009

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

This Paper has 57 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 are as usual)
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 are as usual)

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

1
2
3

2
3
1

4
7
7

(A)

56,

(B)

(C)

–28

(D)

None of these.

(0)

(b)

If A =

)

2
1

3
4

)

, then A^–1 is

(A)

)

4
–1

–3
2

)

(B)

)

–4
1

3
2

)

(C)

)

4
–1

–3
2

)

(D)

None of these.

(0)

(c)

If a = 2i – j + k and b = i – 3j – 5k, then the magnitude of a + b is

(A)

(B)

21,

(C)

√41

(D)

None of these.

(0)

(d)

If f(x) =

2x + 3

3x + 2

(for x ≠ 0), then f(x), f

)

(A)

(B)

)

2x+3

3x+2

)²

(C)

)

3x+2

2x+3

)²

(D)

None of these.

(0)

(e)

If A =

)

1
1

1
–1

)

and B =

)

k
1

–1
–1

)

and AB is a null matrix, then the value of k is

(A)

(B)

(C)

–1,

(D)

None of these.

(0)

(f)

lim
x→0

e^3x – e^2x

(A)

(B)

(C)

–1,

(D)

None of these.

(0)

(g)

If y = x^x then

(A)

x(1 + log x),

(B)

x^x log x,

(C)

x^x (1 + log x),

(D)

None of these.

(0)

(h)

If x^y = e^x–y then the value of y is

(A)

1 + log _e x

(B)

1 + log ₁₀ x

(C)

1 – log _e x

(D)

None of these.

(0)

(i)

2
∫
1

4x²

2x² – 1

dx is

(A)

log 31,

(B)

½ log 31,

(C)

½ log 2

(D)

None of these.

(0)

(j)

A manufacturer produced x unit of items at a cost of 100 – 40x + x² (in rupees). The number of items he produced at which the cost is a minimum is

(A)

20,

(B)

10,

(C)

40,

(D)

None of these.

(0)

(a)

Give three vectorsa = 2i + 3j + 2k, b = 2i + j + 2k and c = i – 2j + 2k, show that a x b = 4c. Also find a unit vector perpendicular to each of a and b.

(0)

(b)

If A =

)

1
a

–1
–1

)

, B =

)

1
4

1
–1

)

, and A² + B² = (A + B)² then find the value of a.

(0)

(c)

Solve by Cramer’s rule; 3x + y + z = 4, x + 2y – z = 3, x – y + 2z = 6.

(0)

(a)

Investigate the existence of the limit of the function f(x) = 2x² – x + 3 as x → 1 and the continuity of f(x) at x = 1.

(0)

(b)

If xy = ax³ +

x²

, x ≠ 0 prove that x

d²y

dx²

+ 2

= 6

(0)

(c)

Write short note on any one:
(i) two–person zero–sum game, (ii) Vogel’s approximation method for obtaining initial BFS.

(0)

(a)

Find the function whose first order derivative with respect to x is

x² – 2x + 7

x² – 4x + 3

(0)

(b)

Evaluate

4
∫
0

(√x +

√x

) dx.

(0)

(c)

The demand and supply function under perfect competition are y = 50 – x² and y = 2 (x² + 1) respectively. Find the price under market equilibrium, consumer’s surplus and producer’s surplus.

(0)

(a)

Find the area of the region lying in first quadrant bounded by the poarabola y² = 16x, the x–axis and the ordinate at focus after drawing a diagram.

(0)

(b)

Using graph paper maximize z = 4x + 3y subject to x + 2y ≤ 6, 2x + y ≤ 6, x ≥ 0, y ≥ 0.

(0)

(c)

In a bank cheques are cashed at a single teller counter. Customers arrive at the counter is a Poisson manner at an average rate of 20 customers per hour. The teller takes on an average a minute and a half to cash the cheque.
The service time had been shown exponentially distributed.

Calculate:

(i)
(ii)
(iii)

The utilisation factor for the system,
The percentage of time the teller is busy,
The average time a customer is expected to wait.

(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:
Choose the correct alternative stating proper reasons/calculations.

2x5

(a)

If P(A) = 0.3, P(B) = 0.4 and P(A/B) = 0.5 then probability that, of two events A and B, only A occurs is

(i)

0.3

(ii)

0.1

(iii)

0.05

(iv)

0.18

(0)

(b)

For two independent events A and B, P(A) = 0.4, P(B) = 0.5. Then P(A ∪ B) is

(i)

0.9

(ii)

0.7

(iii)

0.2

(iv)

0.1.

(0)

(c)

If two unbiased dice are thrown once the probability that sum of the points on the upper most faces of the dice fallen is at least 11 is

(i)

(ii)

(iii)

(iv)

(0)

(d)

If pdf of a continuous r, v, X be
f(x) = k (constant), 1 < x < 2.
= 0, otherwise: then E(X) is

(i)

(ii)

(iii)

(iv)

(0)

(e)

For a binomial distribution with parameters n = 4 and p = ½, standard deviation is

(i)

–1

(ii)

√2

(iii)

–√2

(iv)

(0)

(f)

For a r, v, X following a Poisson distribution with parameter 4, P (X is atmost 1) is

(i)

e^–4

(ii)

e⁴

(iii)

5e^–4

(iv)

5e⁴

(0)

(g)

If X is a standard normal variable with

1
∫
0

(2π)^–½ e^x²/2 dx = 0.34, then P (|X| < 1) is

(i)

0.16

(ii)

0.32

(iii)

0.34

(iv)

0.68

(0)

(h)

If the two regression lines are 10y = 9x + 13 and 10x = 9y – 6 then mean (x, y) is

(i)

(3, 4)

(ii)

(4, 3)

(iii)

(3, –6)

(iv)

(–6, 3)

(0)

(i)

A simple random sample of size 10 is drawn without replacement from a finite population of size 50. Variance of the population is 49. Then standard error of the sample mean is

(i)

√10

(ii)

(iii)

(iv)

√10

(0)

(j)

In order to test unbiasedness of a die it is tossed twice. The null hypothesis of unbiasedness is rejected if and only if the sum of the numbers on the uppermost faces of the die is 2 or 12. The probability of type 1 error of the test is

(i)

(ii)

(iii)

(iv)

(0)

(a)

The incidence of a certain disease in an industry is such that on an average, 20% of workers suffer from it. If 7 workers are selected at random, what is the probability that 5 or more have got the disease? Also obtain the mean and s.d. of the distribution.

(0)

(b)

A problem in statistics is given to Asoke, Amal and Abdul and their probabilities of solving it are 1/5, 2/5 and 3/5 respectively. If all of them try independently, find to probability that the problem will be solved.

(0)

(a)

Packets of a certain washing powder are filled with an automatic machine with an average weight of 6 kg and s.d. of 50 gm. If the weight of a packet is normally distributed then find the percentage of packets weighing above 6.1 kg.

(Given

2
∫
0

[Φ (z)dz = 0.4772 where z is a N (0, 1) variable]

(0)

(b)

A company serveyed employees to see whether they prefer a large increase in retirement benefits or a smaller increase in monthly salary. From a group of 1,000 male employees, 850 supported the retirement benefits. Of 500 female employees, 400 supported the retirement benefit.
Test the null hypothesis that the proportions of man and women supporting retirement benefits are equal.

[Given	d.f	=	1	2	3
	X_0.05	=	3.84	5.99	2.81]

(0)

(a)

Find the mean and standard deviation of a normal distribution of marks of students appearing in an examination if 16% of the students got marks below 52 and 16% of the students got marks below 52 and 16% of the students got marks above 62.

(Given

1
∫
0

e^–r²/2

√2π

dt = 0.34]

(0)

(b)

From the following data determine the regression lines of x on y and of y on x, and estimate the value of x when y = 75 and the value of y when x = 35.

Variable
Arithmetic mean
Standard deviation

x
30
2

y
90
3

Correlation co–efficient between x and y is 0.4.

(0)

10.

(a)

A random sample of size 10 is drawn from a normal population with mean μ and variation σ² = 4 and the sample observations are 62, 63, 64, 65, 65, 66, 66, 68, 70, 71. Test whether population mean μ = 67 at 5% level of significance against alternative μ ≠ 67. Determine 95% confidence interval of μ.
[Value of τ for α = 0.025 is 1.96 where τ is N (0, 1)]

(0)

(b)

A die is thrown 120 times with the following results:

Face:
Frequency

1
20

2
18

3
22

4
30

5
16

6
14

Total
120

Test the fairness of the die in 5% level of significance.
Given: Upper 5% points of chi–square distribution with 5 and 6 degrees of freedoms are 11.07 and 12.59 respectively.

(0)

11.

(a)

Write short note on any one:
(i) Rank correlation, (ii) Frequency chi–square test.

(0)

(b)

A bakery keeps stock of a popular brand of cake. Previous experience shows the daily demand pattern for the items with associated probabilities as given below:

Daily demand (nos):
Probability:

0
0.01

10
0.21

20
0.10

30
0.51

40
0.14

50
0.05

Using the following sequence of random numbers to simulate the demand for next 12 days:

Random numbers:

49
76

19
65

89
39

73
25

05
31

12
68

Also estimate the daily average demand for the cakes on the basis of simulated data and find out the stock situation at the end of 12th day, if the owner of the bakery decides to make 30 cakes every day.

(0)

SECTION III (Economic Techniques — 30 marks)

12.

Attempt any five of the following:

2x5

(a)

If 20% price rise causes a fall in demand by 35% then is demand elastic?

(0)

(b)

For output q, cost c = 200q – 5q² +

q³. Determine the output when marginal cost is minimum.

(0)

(c)

For the demand curve q =

√p

with p and q being the price and quantity respectively, show that

elasticity of demand is constant.

(0)

(d)

For the following two demand functions x₁ =

P₁³ P₂²

and x₂ =

P₁² P₂³

examine whether the commodities are competitive or complimentary.

(0)

(e)

If r₁₂ = 0.65, r₂₃ = 0.60 and r₃₁ = 0.40, find r_13.2

(0)

(f)

Calculate the Paasche’s Price Index from the following table:

Commodity	p₀	p₁	q₀	q₁
A B C D	2 3 4 7	4 5 6 9	3 4 5 8	5 6 7 1

(0)

(g)

Calculate the overall cost of living index number for the year 2008 with base year = 2000, for the following data:

Food
Clothing
Light and fuel
House rent
Miscellaneous

150
70
60
50
100

40
20
20
15
5

(0)

(h)

If Laspeyre’s price index number is written as weighted arithmetic mean of price relatives, what will be the weights?

(0)

13.

Answer any four of the following:

5x4=20

(a)

Prove that the elasticity of demand =

AR – MR

where AR and MR are average and

marginal revenue respectively at any output. Verify the result for the demand law : p = 3 + 4x where p is the price per unit, and x is the output in demand. Describe the nature of demand when x = 2 units.

(0)

(b)

Find the trend values for the following series using a four–year moving average with weights 1,2,1,1.

Year
Values

1
3

2
4

3
7

4
8

5
9

6
11

7
14

(0)

(c)

Determine the price index number and quantity index number of the year 2001 with 1991 as base year using Fisher’s method from the following data

Commodity	Unit	Quantity		Price (Rs.)
Commodity	Unit	1991	2001	1991	2001
A B C	Kg Dozen Watt	5 5 2	7 10 3	2 3 5	4 5 9

(0)

(d)

For a multivariate distribution of 3 variables x, y, z the following results were found:

x =50	y = 60	z = 70
s_x = 2	s_y = 3	s_z = 4
r₁₂ = 0.5	r₁₃ = 0.4	r₂₃ = 0.3

Where x, y, z are means of variables x, y, z; s_x, s_y, s_z are standard deviations of x, y, z; r₁₂, r₁₃, r₂₃ are the correlation coefficients between x and y, between x and z and between y and z respectively. Find the multiple regression line of x on y and z and estimate the value of x for y = 65 and z = 75.

(0)

(e)

The input – output table of 2–product economy is given below. Derive the gross output level of 2 commodities and also total labour.

Producer	Consumer		Final demand
	Product 1	Product 2
Product 1: Product 2: Labour per unit product in mandays:	0.6 0.4 6	0.2 0.3 5	120 140

(0)

(f)

Write short notes on any one of the following:

(i)

Components of a time series

(0)

(ii)

Production function.

(0)

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