CWA/ICWA Inter :: Quantitative Methods: December 2002

*I-12(QNM)* *Revised Syllabus*
Quantitative Methods
Time Allowed : 3 Hours	Full Marks : 100
SECTION I(Mathematical Techniques — 40 marks)
Arithmetic (15 marks)
Answer Question No. 1 (compulsory — 10 marks) and two other questions (15x2 = 30 marks) from this section.

Attempt any five of the following:

2x5=10

(a)

If a = 2i — 3j +5k b = —2i + 2j +2k. find a + b and 2a − b

(b)

Solve

a − b

b − c

c − a

b − c

c − a

a − b

c − a

a − b

b − c

(c)

If x

(

− 1
2

3
1

)

find matrix X.

(d)

If A is a square matrix, show that A + A' is a symmetric matrix where A' is transpose matrix of A.

(e)

Evaluate

Lim
x→1

x² + 1

x − 1

(f)

If x = at², y = 2at, find

(g)

Evaluate:

∫

(1 + log)²

(h)

If f(x, y) = 3x³ — 5x²y + 2y³, then show that: x

δf

δx

+ y

δf

δy

= 3 f(x, y)

(i)

What is the function whose derivative is:

x^1/3

(j)

Find the domain of f(x) =

x² − 4

(a)

Find the area of a parallelogram by the vectors i + 2j + 3k and 3i — 2j + k. Also find the unit vector perpendicular to each of them.

3+2

(b)

Solve by Cramer's rule

x — 2y + z = 0
3x — y = 2
y — z = —1

3+2

For what type/degree of equation and for what value of determinate. Cramer's rule is applicable?

(c)

Define orthogonal matrix. Show that the matrix

1+4

[

− 1
− 2
2

2
1
2

− 2
2
1

]

is orthogonal.

Please turn over

( 2 )

I-12(QNM)
Revised syllabus

Marks

(a)

Evaluate:

∫

√	x² − 7x + 12

(b)

If x^y = e^{x — Y}, show that

logx

(1 + logx)²

(c)

Discuss the continuity of the function f(x) at x = —1, where

f(x)

{

x +

x + 1

|x + 1|

}

, at x ≠ − 1

− 1

, at x = − 1

(a)

Solve the following L.P.P. graphically:
Maximise A = 7x + 3y subject to 4x + 5y < 40, x > 3, Y < 4; x, y > o.

(b)

A steel plant produces x tons of steel per week at a total cost of

Rs.(

x³ − 9x² + 79x + 175 ) find the output level at which the marginal cost attain its minimum.

cost attains its minimum.

(c)

Find the area of the region above the x-axis bounded by 2x — 3y — 6 = 0, x = 4, x = 6 (rough sketch is necessary)

(a)

If u = log

√

x² + y²

find the value of x²

δ²u

δx²

− y²

δ²u

δy²

at x = 1, y = 1

(b)

Given the pay-off matrix for player A, obtain the optimum strategies for both the players and determine the value of the game.

player B

player A

[

6
− 3

− 3
0

7
4

]

(c)

A radio repairman finds the time spent on his jobs has an exponential distribution with mean 50 minutes. If he repairs sets in the order in which they come in, and it the arrival of sets is approximately Poission with average of 10 per 10 hour day.

(i) What is the utilisation factor?

(ii) What is the repairman's expected idle time each day?

(iii) How many jobs are ahead of the average set just brought in?

SECTION II(Statistical Techniques — 30 marks)

Answer Question No. 6 (compulsory — 10 marks) and two
other questions (10x2 = 20 marks)

Attempt any five of the following
choose the correct alternative, stating proper reason:

2x5=10

(a)

From a pack of 52 cards, the probability of drawing an ace of a diamond is:

(i)

(ii)

(iii)

(iv)

(b)

Two identical dice are tossed. The probability that the same number will appear on each of them is:

(i)

(ii)

(iii)

(iv)

(c)

If P(A) = 0.4, P(A U B) = 0.7 then for two independent events A and B, P(B) is

(i)

0.5

(ii)

0.75

(iii)

0.3

(iv)

0.2

(d)

The mean of a binomial distribution is 12 and its variance is 9. Then the parameters (n, p) are given by

(i)

(

36,

)

(ii)

(

36,

)

(iii)

(

48,

)

(iv)

None of these

Please turn over

( 3 )

I-12(QNM)
Revised Syllabus

(e)

In a Poisson distribution if x is a variate with parameter 1, then the probability (3 < x < 5) is given by for e = 2.7182)

(i)

0.0153

(ii)

0.153

(iii)

0.1055

(iv)

None of these

(f)

For a normal distribution if mean is m, median is m₀ and mode is m_o' then we get

(i) m > m₀ > m_o' (ii) m < m₀ < m_o' (iii) m = m₀ = m_o' (iv) none of these

(g)

For a random variable x, then p.m.f is: f(x) = ax_i x = 1, 2, 3, ......, n or x = 0 otherwise. then the expectation of x is

(i)

n(n +1)

(ii)

(2n + 1)

(iii)

n(n +1)(2n +1)

(iv)

none of these

(h)

While studying a set of pairs of reated variates, the following results are obtained:

N = 100. Σx_i = 57, Σy_i = 43, &Sigma(x_i −

)² = 169, Σ (y −

)² = 64,

Σ (x_i −

) (y −

) = 101 Then correlation coefficient is equal to

(i)

0.79

(ii)

0.97

(iii)

1.00

(iv)

—0.563

(i)

In a sample of 400 parts manufactured by a factory, the number of defective parts was found to be 28. the company, however, claimed that only 4% of their products is defective. The S.E. is given by:

(i)

0.0198

(ii)

0.0098

(iii)

0.098

(iv)

0.98

(j)

Describe a method of drawing sample without replacement of size 3 from a population of size 11.

(a)

A bag contains 5 red and 3 black balls and the second one 4 red and 5 black balls. One of these is selected at random and draw of two balls is made from it. What is the probability that one of them is red and other is black?

(b)

In an examination 50% and 40% of the students, failed in Mathematics and Accountancy respectively while 20% failed in both the subjects. A student is selected at random. Find the probabilities that:

(i) The students failed in Accountancy if known that he failed in Mathematics;

(i) The student failed in exactly one subject.

3+2

(a)

If the distribution of marks received in an examination is normal, 44% of the candidates got marks below 61.4% of candidates got marks above 80, find the percentage of candidates received above 65 marks.

(Given that

√2π

∫ e^{− 1/z t²}

dt = 0.06, 0.10, 0.46 according as x = 0.15, 0.25, 1.75)

(b)

An oil exploration firm funds that 4% of the test wells it drills yield a deposit of natural gas. If the firm drills 5 wells. What is the probability that: (i) exactly 2 wells, (ii) at least one well yield gas?

3+2

(a)

The average number of articles produced by two machines per day are 200 and 250 with s.d. 20 and 25 respectively on the basis of records of 25 days production. Can you regard both the machines equally eddicient at 1% level of significance?

(b)

In a survey of 200 boys of which 75 were intelligent, 40 had skilled fathers, while 85 of the unintelligent boys had unskilled fathers. Do these figures support the hypothesis that skilled fathers have intelligent boys?

Use χ²test. Value of χ² for 1 d.f. at 5% level is 3.84.

Please turn over

( 4 )

I-12(QNM)
Revised syllabus

Marks

10.

(a)

the regression equation of production (x) on capacity utilisation (y) of a certain firm is 3x — 2.76y + 102.30 = 0. The average capacity utilisation of the firm was 70% and the variance of capacity utilisation is (9/16)^th of the variance of production. Find the average production and coefficient of correlation between production and capacity utilisation.

(b)

The Alpha flower shop promises its customers delivery within four hours on all flower orders. All flowers are purchased on the prior day and delivered to Alpha by 8 A. M. next day. Alpha's daily demand of roses is as follows:

Dozen of Roses	:	7	8	9	10
Probability	:	0.1	0.2	0.4	0.3

Alpha purchases roses for Rs. 10.00 per doze and sells them for Rs. 30.00. All unsold roses are donated to a local hospital. How many dozen of roses should Alpha order each evening to maximise its profits? What is the optimum expected profit?

11.

(a)

A population consists of the numbers 1, 5, 3, 7, 9. consider all possible samples of size two which can be drawn without replacement from this population. Find the mean of the sampling distribution of mean.

(b)

Suppose a departmental store manager is studying the inventory situation and is interested in generating possible sales for 10 days. Assuming that the number of sales per day is Poison with mean 5,generate 10 days of sales by Monte Carlo method using the following random numbers:

1048
1792 0150
2368 1101
4657 5360
3255 2011
9585 8164
3933 7916
0995 4669

SECTION III(Economic Techniques — 30 marks)

12.

Attempt any five of the following:

2x5=10

(a)

For the demand law p =

(2x + 1)²

find the elasticity demand in terms of x

(b)

Show that the elasticity of demand e_p =

AR − MR

where AT and MR

are respectively average and marginal revenue at any output.

(c)

Write down the normal equations in fitting a third degree equation y = a + bx + cx² + dx³ by the method of least squares.

(d)

The demand function for two commodities are x₁ = p₁^-1.1. P₂^0.3 and x₂ = p₁^0.2. P₂^-0.6. Where commodities x₁, x₂ are sold at prices p₁ and p₂ respectively. Determine whether the commodities are complementary or competitive.

(e)

If r₁₂ = 0.65, r₁₃ = 0.60 and r₂₃ = 0.90 then calculate r_12.3 (by usual notations).

(f)

Construct C.L.I. from the following data:

Group	Weight	Index No (Basic:1952 — 53 = 100)
Food Clothing Fuel and light Rent Miscellaneous	50 2 3 16 29	241 221 204 256 179

(g)

Complete the sentence
"A Laspeyre's index uses .............as weights while a Paasche's Index uses .............. as weights."

(h)

Define a production function.

Please turn over

( 5 )

I-12(QNM)
Revised Syllabus

13.

Answer any four of the following

5x4=20

(a)

From the following time series obtain 4-quarter moving averages and also deviations from trend.

year	Quarterly output ('000 tons) of coal for 4 years
	I	II	III	IV
1978 1979 1980 1981	65 58 70 60	56 63 56 51	58 63 59 55	61 67 52 58

(b)

Fit by the method of lease squares a parabolic curve to the following data:

year	:	1991	1992	1993	1994	1995	1996	1997
Production (in tons)	:	23	20	18	18	14	13	13

(c)

From the given data, show that Fisher's ideal index number satisfies the Time Reversal Test:

Commodity	1990 (base year)		1996 (current year)
	Price	Quantity	Price	Quantity
A B c	8 10 7	6 5 8	12 11 8	5 6 5

(d)

In a working class consumer price index no. of a particular town, the weights corresponding to different groups of items were as follows:

Food — 55, fuel — 15, clothing — 10, rent — 8 and miscellaneous — 12. In Oct. 1992, D.A. was fixed by a mill of that town at 182 percent of the workers which fully compensated for the rise in prices of food and rent, but did not compensate for anything else. Another mill of the same town paid D.A. of 46.5 percent which compensated for the rise in fuel and miscellaneous groups. It is known that rise in food is double that of fuel and rise in miscellaneous group is double that of rent.
Find the rise of food, rent, fuel and miscellaneous groups.

(e)

For the transaction matrix given below, find the matrix of technological coefficients and hence find out the total output for each industry if the new final demands are 1000 and 2000 units respectively:

Industry	Input		Final
	1	2	demand
1 2	200 100	100 500	50 400

(f)

Write short note on any one of the following :

(i)	Components of time series;
(ii)	Multiple correlation;
(iii)	Long run cost curve.

__________

CWA/ICWA Inter :: Quantitative Methods : December 2002