CWA ICWA Inter - Stage II : Quantitative Methods - June 2010

CWA ICWA Inter - Stage II • Quantitative Methods • June 2010

This Paper has 62 answerable questions with 2 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 marks)
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 marks)

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.

2 x 5

(a)

(

7	6	3
1	4	5

)

+ 3

(

0	2	3
2	1	4

)

(A)

(

14	8	15
18	11	23

)

(B)

(

8	11	22
14	18	15

)

(C)

(

14	18	15
8	11	22

)

(D)

none of these

(1)

(b)

The value of

1	2	3	²
3	1	2
2	3	1

(A)

(B)

324

(C)

(D)

none of these

(0)

(c)

If p = 2i − j + k and q = i + j + k, then the magnitude of p x q is

(A)

(B)

√14

(C)

√41

(D)

none of these

(0)

(d)

If f(x−1) = x² − 4x+3 then f(x+1) is

(A)

x²

(B)

x² – 1

(C)

x² + 1

(D)

none of these

(0)

(e)

The limit of

√4 + x – √4 - x

as x → 0 is

(A)

(B)

(C)

(D)

none of these

(0)

(f)

If y = (a √x + b) (b √x + a) then the value of

for a = b = 1 is

(A)

√x

(B)

1 +

√x

(C)

1 –

√x

(D)

none of these

(0)

(g)

The value of

1
∫₀

e^x

e^x + 1

dx is

(A)

log

e+1

(B)

log(e+1)

(C)

(D)

none of these

(0)

(h)

If the cost function c(x) =

x³ – 3x² + 25x + 5

then at what output level, marginal cost is minimum?

(A)

(B)

(C)

(D)

none of these

(0)

(i)

The extreme values of the function f(x) = x³ – 6x²+9x – 8 are at

(A)

(1,2)

(B)

(1,3)

(C)

(–1, –3)

(D)

none of these

(0)

(j)

If f(x) = x³ + 2xy + y³ then the values of f_xy and f_yx are

(A)

(2,2)

(B)

(3,3)

(C)

(2, 3)

(D)

none of these

(0)

(a)

If a = 2i + 5j + 3k, b = 3i + 3j + 6k, c = 2i + 7j + 4k, find (i) (a – b) x (c – a), (ii) | (a – b) x (c – a).

(0)

(b)

If A =

1	–1	1
2	–1	0
1	0	0

, then find A² and show that A² = A^–1.

(0)

(c)

Solve by Cramer's rule showing the condition of consistency of solution:
x+y+z = 6, x+2y+z =8, x+y+2z = 9

(0)

(a)

For what value of a,
f(x) = x +1 when x ≤ 1
3a – x when > 1
is continuous at x = 1? Hence determine f(2) − f(0)

(0)

(b)

if y = x + √1 – x² prove that (y₁ – 1)√1 – x² + x = 0 and y₂ (1 – x²) – xy₁ + y = 0 where y₁ =

and y₂ =

d²y

dx²

(0)

(c)

The production function of a commodity is given by q = 20x+2x² –

x²

where q is the total output for x

units of input. Verify that when average product is maximum, it is equal to marginal product.

(0)

(a)

Evaluate ∫

2x³ - 2x - 12

2x² - 2x - 4

dx for x ≠ 2.

(0)

(b)

Using graph paper minimize the objective function z = 2x − y subject to the constraints x + y ≤ 5, x+2y ≤ 8, 4x+3y ≥ 12, x ≥ 0, y ≥ 0.

(0)

(c)

Is there any saddle point in the pay off matrix of the game between two players A and B?

Player A

Player B

2	7	-5
9	10	12
11	13	7

Determine the optimum strategies of the players and the value of the game.

(0)

(a)

In an Auto Hub, automobile cars arrive at a rate of 30 cars per day. Assuming that the inter-arrival time follows an exponential distribution and the service time distribution is also exponential with average 36 minutes, calculate the following.

(i)	Mean queue size(line length);
(ii)	Probability that the queue size exceeds 10

(0)

(b)

Find the area bounded by the curves x² = y and y² = x by drawing the graphs

(0)

(c)

write short note on any one of the following

(i)

Slack and Surplus variable used in LPP

(0)

(ii)

Consumer’s Surplus and Producer’s Surplus

(0)

SECTION II (Statistical Techniques — 30 marks)

Answer Question No. 6 (Compulsory — 10 marks) and any 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)

For two independent events A and B, P(A) =

, P(B) =

then P (none of the events A and B occur) is

(i)

(ii)

(iii)

(iv)

none of these

(1)

(b)

If a random variable x follows Poisson distribution such that P(x = 1) = P(x = 0) = P(x = atmost 1) is

(i)

2e^-1

(ii)

e^-1

(iii)

e^-1

(iv)

none of these

(0)

(c)

If var (X) = var (Y) =

and var (X + Y) =

the correlation coefficient between random variables X and Y is

(i)

(ii)

(iii)

(iv)

none of these

(0)

(d)

6 fair coins are tosses simultaneously. Probability of getting at least 4 heads is

(i)

(ii)

(iii)

(iv)

none of these

(0)

(e)

If x be a normal variate with mean 100 and variance 9 then the modal ordinate is

(i)

100√2Π

(ii)

9√2Π

(iii)

3√2Π

(iv)

none of these

(0)

(f)

If mean and variance of a binomial distribution are 4 and

respectively, the values of n and p are

(i)

(12,

)

(ii)

(12,

)

(iii)

(6,

)

(iv)

none of these

(0)

(g)

The correlation coefficient of bivariate X and Y is 0.60. Variance of X is 2.25 and regression coefficient of Y on X is b_yx = 0.8. The value of s.d. of Y is

(i)

(ii)

(iii)

(iv)

none of these

(0)

(h)

A lot of 100 items contains 20 defectives. If a sample random sample of size 10 is drawn without replacement (SRSWOR), the standard error(S.E.) of the sample proportion of defective items is

(i)

2√11

(ii)

5√10

(iii)

5√11

(iv)

none of these

(0)

(i)

If the samples of size n are drawn at random without replacement from the set {1, 2, 3, ......., N}, then var (x) is

(i)

(N+1)(N -n)

12n

(ii)

N+1

12n(N - n)

(iii)

n(N+1)

12(N - n)

(iv)

none of these

(0)

(j)

To test unbiasedness of a coin it is tossed 3 times and null hypothesis is accepted when number of heads obtained is atmost 2. Then probability of type I error is

(i)

(ii)

(iii)

(iv)

none of these

(0)

(a)

In a class of 10 students, any student can choose one of the two subjects X and Y. 6 students take X and the rest take Y. 3 students are selected at random (without replacement). Find the probability that (i) they all take X (ii) they all take Y (iii)at least one takes Y

(0)

(b)

A box contains 10 screws of which 5 are defectives. Obtain the probability distribution of the number of defective screws(x) in a sample of 4 screws chosen at random. Also find var (x).

(0)

(a)

In an office 40% of employees have scooters, 20% have cars and 80% of the employees who have scooters do not have cars. What is the probability that an employee has a car given that he does not have a scooter?

(0)

(b)

Find the Spearman's rank correlation coefficient from data of marks in Statistics and Economics of 9 students.

Marks in Statistics:	64	57	66	47	58	69	40	36	35
Marks in Statistics:	56	57	65	48	66	50	35	34	40

(0)

(a)

A company is trying to manufacture a new type of item. The relevant data are shown in the following table where first year profits in thousand rupees are given:

Product line\State of nature		Good	Fair	Poor
Product line\State of nature	Probability:	0.2	0.4	0.4
Full		70	40	-15
Partial		60	45	-10
Minimal		40	30	0

(i)	Find the optimum product line and its expected profit
(ii)	Find the optimum value of expected opportunity hours and then optimum course of action

(0)

(b)

A confectioner sells confectionery items. Past data of demand per week (in hundred kilograms) with frequency is given below:

Demand
per week	:	0	5	10	15	20	25	30
Frequency	:	2	9	6	18	9	4	2

Using following sequence of random numbers generate the demand for next 15 weeks and also find the average demand per week.

Random Numbers
60	66	67	56	94	83	35	57
34	73	23	13	90	52	35	19

(0)

10.

(a)

Find the means of two variables x and y and the correlation coefficient between x and y from two regression lines x+6y = 6 and 2x+3y = 9

(0)

(b)

From a population of 5 members 10, 14, 14, 16, 18 draw all possible samples of size 2 with SRSWOR. Obtain the sampling distribution of sample mean and hence find its standard error.

(0)

11.

(a)

Fit a Poisson distribution to the following data. Calculate the theoretical frequencies. [Given e^-0.5 = 0.61]

x	:	0	1	2	3
f	:	121	61	15	3

(0)

(b)

Write short note on any one of the following:

(i)

Bayes theorem with one example;

(0)

(ii)

Properties of normal distribution.

(0)

SECTION II (Economic Techniques — 30 marks)

12.

Answer any five of the following:

2x5

(a)

If 15

% fall in price causes 6% rise in demand then demand is

(i)

elastic

(ii)

inelastic

(iii)

none of these

(0)

(b)

If the demand function is p = 12-x² then for what value of x is the elasticity of demand unity?

(i)

(ii)

(iii)

(iv)

none of these

(0)

(c)

For the demand law p =

(x+2)²

,the elasticity of demand in terms of x is

(i)

x+2

(ii)

x-2

(iii)

x+2

(iv)

x-2

(0)

(d)

Given Ʃp₀q₀ = 200, Ʃp₀q₁ = 300, Ʃp₁q₀ = 250, Ʃp₁q₁ = 400 the Fisher's Ideal Index Number is

(i)

157

(ii)

167

(iii)

177

(iv)

none of these

(0)

(e)

An index is at 100 in the year 2000. It rises 5% in 2001, falls 6% in 2002, further falls 7% in 2003 and then rises 10% next year. The net rise/fall w.r.to the base year is

(i)

no rise

(ii)

no fall

(iii)

1% rise

(iv)

1% fall

(0)

(f)

If the cost C of output q is C =

q³ - 5q² then the marginal cost is minimum at output level

(i)

(ii)

(iii)

(iv)

none of these

(0)

(g)

when r₁₂ = 0.6, r₂₃ = 0.8, r₃₁ = 0.5, then value of R_1.23 is

(i)

0.5

(ii)

0.6

(iii)

0.7

(iv)

none of these

(0)

(h)

Taking 2000 as the base year, CLI in the year 2010 stands at 250. An employee drawing a monthly salary of Rs.800 in the year 2000, gets Rs. 1800 in 2010. How much is he getting more or less?

(i)

Rs.200 more

(ii)

Rs.200 less

(iii)

Rs.400 more

(iv)

Rs.400 less

(0)

(i)

The second degree polynomial passing through the points (1, 3), (0, 4), and (-1, 9) is

(i)

2x² -3x+4

(ii)

3x² -2x+4

(iii)

4x² -2x+3

(iv)

none of these

(0)

13.

Answer any four of the following:

5x4

(a)

The demand curve for a commodity is given by p = 20 - √x where p and x are the price and amount of the commodity respectively. Find the marginal quantity demand and also average quantity demand. Find the elasticity of demand at a price p = 2 and state whether the demand is elastic or inelastic.

(0)

(b)

The demand function for a particular brand of pocket radio is stated below:

p = 100 - 0.4 Q - 0.006 Q²

Find the consumer's surplus at a quantity of 20 radios.

(0)

(c)

Fit a straight line trend by the method of least squares from the following data:

Year	:	2004	2005	2006	2007	2008	2009
Value	:	15	18	16	18	20	19

and estimate the value for 2010.

(0)

(d)

Given the following data:

Commodity	Base year		Current Year
Commodity	Price (p₀)	Quantity (q₀)	Price (p₀)	Quantity (q₀)
A B	1 1	10 5	2 x	5 2

Find x, if the ratio between Laspeyre’s and Paasche’s index numbers is 28 : 27.

(0)

(e)

If r₁₂ = 0.5, r₁₃ = 0.6, r₂₃ = 0.7 determine r_1.23 and r_12.3.

(0)

(f)

The coefficient matrix of input is

0.5	0.2
0.7	0.6

and the vector of final demand is

100

200

Find the gross production.

(0)

(g)

write short note on any one of the following;

(i)

Moving average method;

(0)

(ii)

Components of a time series;

(0)

(iii)

Multiple regression.

(0)

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