CWA/ICWA Inter :: Quantitative Methods: December 2003

*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 (15x2 = 30 marks) from this section.

Attempt any five of the following:

2x5=10

(a)

If A − 2B =

(

− 7
4

7
− 8

)

and A − 3B =

(

− 11
4

7
− 13

)

find the matrices A and B.

(b)

For the vectors

= i + 2j − k,

= 2i — j + k, find

−

(c)

Solve by Cramer's rule: x + y — 3 = 0, 4x — 5y + 6 = 0.

(d)

If y = f(x) =

ax + b

cx − a

prove that f(y) = x

(e)

Evaluate

lim
x→0

e^x² − 1

(f)

If y = √x +

√x

show that 2x

+ y = 2 √x

(g)

If u = f

(

)

show that x

δu

δx

+ y

δu

δy

= 0

(h)

If x = at, y =

find

d^{2^y}

dx²

(i)

Evaluate:

1
∫
0

xe^x dx

(j)

Find the domain of f(x) =

x² − 1

(a)

If 5i + 2j + 3k, —i — j + 4k and 2i + 4j — 3k be the position vectors of the points A, B and C respectively, then show that A, B, C form a right angled isosceles triangle. Give diagram (i, j, K have their usual significances)

(b)

Evaluate:

Lim
x→− 4

(

1 + x

x² − 4

)

(c)

If y = { log (x +

√

1 + x²

)

}

then show that (1 + x²) y₂ + xy₁ = 2

Please turn over

( 2 )

I-12(QNM)
Revised syllabus

Marks

(a)

If u = x²y + y²x, find the value of x

δu

δx

+ y

δu

δy

when x = y = 1.

(b)

Evaluate:

logz
∫
0

e^x

e^x + 1

(c)

A radio manufacturing company produces two types of radios :

Type	Mfg. cost
A B	Rs. 280 Rs. 310

During a week, the company produced 128 radios, total cost of these radios being Rs. 37.280. Using the matrix algebra, find the number of A-type and B-type radios produced during the week

(a)

By using the properties of determinant.

Prove that

−a²
ab
ac

ab
− b²
bc

ac
bc
−c²

is a perfect square

(b)

Mr. X quite often files from town A to town B. He can use the airport bus that costs him Rs. 13 but if he takes it, there is a 0.08 chance that he will miss the flight. A hotel limousine costs Rs. 27 with a 0.96 chance of being on time for the flight. For Rs. 50, he can use a taxi which will make sure 99 of 100 flights. If Mr. X catches the plane on time, he will conclude a business deal that will earn him a profit of Rs. 1,000, otherwise he will lose it. Which mode of transport should Mr. X use? Give your answer on the basis of EMV criterion.

(c)

Find the area cut off from the parabola y² = 12x by its latus rectum. Give diagram.

(a)

A firm produces x units output at a total cost of Rs.

300x − 10x² +

find

(i)	Output, at which marginal costs is minimum
(ii)	Output, at which average cost is minimum and
(iii)	Output, at which average cost is equal to marginal cost.

(b)

Using simplex method solve the following

Maximixe : Z = 3x₁ + 5x₂

Subject to: x₁ + x₂ < 10; 2x₁ — 3x₂ < 12; x₁, x₂ > 0.

(c)

Find the extreme values of the function

f(x, y) = x² — y² + xy + 5x subject to x + y + 3 = 0

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)

If A and B are independent events such that P(A) = 0.3 and P(A U B) = 0.5 then P(B)

(i)

(ii)

(iii)

(iv)

(b)

If 5 persons are arranged in a line, then probability that e particular persons will be always together is:

(i)

(ii)

(iii)

(iv)

(c)

If A and B are two events with P(A + B) = ¾ and P(AB) = ¼ then P(B)

(i)

(ii)

(iii)

(iv)

(d)

A card is drawn from a full pack of cards. The probability that it is either an ace or red card is:

(i)

(ii)

(iii)

(iv)

Please turn over

( 3 )

I-12(QNM)
Revised Syllabus

(e)

A random variable x takes values 1, 2 and 4 with respective probabilities 0.25, 0.50 and 0.25 then E(x²) is:

(i)

2.52

(ii)

5.25

(iii)

6.25

(iv)

None of these

(f)

Consider a binomial distribution with n = 18, q = 1/3 then s.d is

(i)

—1

(ii)

(iii)

—2

(iv)

(g)

If x is a Poisson variate with parameter 1, then prob(2 < x < 4) is [given e^—1 = 0.3679]

(i)

0.0613

(ii)

0.1603

(iii)

0.0062

(iv)

0.6212

(h)

Pearson's coefficient of correlation of two variates x and y is 0.46 then covariance is 3.45. If the variance of y is 9, then s.d of x is

(i)

2.0

(ii)

2.5

(iii)

5.2

(iv)

3.0

(i)

If V(x) = V(y) = 1/4 and v(x — y) = 1/3, then correlation coefficient of x and y is:

(i)

(ii)

—1

(iii)

(iv)

—2

(j)

From a population of size 41, a sample of size 5 is drawn with SRSWOR if population S.Q is 5, then SE is

(i)

√2

(ii)

2√2

(iii)

3√2

(iv)

none of these

(a)

Two persons X and Y appear in an interview for two vacancies in the same post. The probability of X's selection is 2/11 and that of y's selection is 1/7. What is the probability that (i) both of them will be selected (ii) only one of them will be selected (iii) none of them will be selected?

(b)

There are two identical urns containing 3 red and 5 white and 7 red and 3 white balls. An urn is chosen at random and a ball is drawn from it. If the ball is white. What is the probability that it is drawn from the first urn?

3+2

(a)

A normal variable has mean 165 and s.d 2. If the probability that the variable exceeds a particular value is one in one-thousand, find the value

(Given that

∫φ(t)dt = 0.998, where φ(t)

is the standard normal probability

density function.]

(b)

A computer while calculating the correlation coefficient between the variables X and Y obtained the following results :

N = 30, Σx = 120, Σx² = 600, Σy = 90, Σy² = 250, Σxy = 356

It was, however, later discovered at the time of checking that it had copied down two pairs of observations :

x
8
12

y
10
7

while the correct values are

x
8
10

y
12
7

Find the correct value of correlation coefficient between X and Y

(a)

A machine put out 20 imperfect items in a sample of 500. After the machine was overhauled it put out 5 imperfect items in a batch of 150. Has the machine being improved after overhauling?

(b)

A sample of 600 screws is taken from a large consignment and 75 are found to be defective.

(i)	Estimate the percentage of defectives in the consignment, and
(ii)	Find the 99% confidence limits of the population proportion.

Please turn over

( 4 )

I-12(QNM)
Revised syllabus

Marks

10.

(a)

Find the regression equation of y on x from the following data:

Σx = &Sigmay = 56; &Sigmax² = Σy² = 476; Σxy = 469 and n = 7.

(b)

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

the hypothesis that skilled fathers have intelligent boys? Use χ²

for 1 d.f.

at 5% level is 3.84

11.

(a)

A company is trying to manufacture a new type of toy. The company is attempting to decide whether to bring out a full, partial or minimum product line. The company has three levels of product acceptance and has estimated their probabilities of occurrence. Management will make the decision on the basis of maximising the expected profit from the first year of production. The relevant data are shown in the following table where first-year profits are given in thousand rupees

(d)

In complete information obtained from a partly destroyed record on cost of living analysis is given below:

State of nature
probability
Product Line
Full
Partial
Minimum

Good
0.2

80
70
50

Fair
0.4

50
45
40

Poor
0.4

—25
—10
0

(i)	What is the optimum product line and its expected profit?
(ii)	Develop an Opportunity Loss Table and calculate the EOL value. What the optimum value of EOL and the optimum course of action?

(b)

From a population of 5 members Viz., 3, 6, 9, 12, 15 draw all possible SRS of size of 3 without replacement.. Verify that the mean of the sample means is exactly equal to the population mean. Obtain the standard error of the sample mean.

SECTION III(Economic Techniques — 30 marks)

12.

Attempt any five of the following:

2x5=10

(a)

If 7½% fall in the price causes 1.5% rise in demand then find price elasticity of demand and its nature.

(b)

If the demand law is p =

1 + x

find the elasticity of demand in terms of x

(c)

"I_19x5.1983 = 97". What does it mean ?

(d)

During a certain period the cost of living index goes up from 110 to 200 and the salary of a worker is also raised from Rs. 330 to Rs. 500. Does he really gain?

(e)

Find Laspeyre's quantity index number from the following table :

(f)

With which component of a time series would you associate each of the following?

(i)	Decrease of death rate due to advancement of medical science.
(ii)	Increase of sale of clothes and garments during the months of festivals.

(g)

Calculate the value of R_1.12 when r₁₂ = r₂₃ = r₃₁ = r.

(h)

Given total cost = 400 + 30x —p 12x² + x³; find at what level of x diminishing marginal return begins.

Please turn over

( 5 )

I-12(QNM)
Revised Syllabus

13.

Answer any four of the following

5x4=20

(a)

If the demand function is given by = 48 — 4x², then for what value of x the elasticity of demand is unity? For p = 12, determine whether demand is elastic or inelastic, where P is the price for demand of quantity x.'

(b)

In a study of sales, a company obtained the following least squares trend equation: y = 16 + 2x. Origin 1985. x units = 1 year, y= total number of units (sold per year). The company has physical facilities to produce only 28 units a year and it believes that at least for the next decade the trend will continue as before.

(i) What is the average annual increase in the number of units sold?

(ii) By what year the company's expected sales have equaled its present capacity?.

(iii) Estimate the annual sales for the year 1998.

(c)

The seasonal indices of the sales of garments of a particular type in a certain shop are given below :

Quarter	Seasonal index
Jan — March April — June July — Sept Oct — Dec	97 85 83 135

If the total sales in the first quarter of a year be worth Rs. 20,000 and sales are expected to rise by 55 in each quarter, determine how much worth of garments of this type be kept in stock by the shop-owner to meet the demand for each of three quarters of the year?

(d)

In complete information obtained from a partly destroyed record on cost of living analysis is given below:

Group
Food
Clothing
Housing
Fuel & light
Miscellaneous Group index
134
140
105
120
130 % of total expenditure
60
Not available
20
5
Not available

The cost of living index with % of total expenditure as weight was found to be 127.9. Estimate the weights used for clothing and miscellaneous.

(e)

From a trivariate distribution, the following correlation coefficients are obtained:

Prove that 1 &minus R²_1.23 = (1 − R²₁₂) (1 − R²_13.2) where R₁₂₃ and r₁₃₂

correlation coefficient and partial correlation coefficient respectively.

(f)

Are the multiple Write short note on any one of the following:

(i)	Input-output analysis
(ii)	Moving average method.
(iii)	Least squares theory.

__________

CWA/ICWA Inter :: Quantitative Methods : December 2003