CWA/ICWA Inter :: Quantitative Methods: December 2004

*I-12(QNM)* *Revised Syllabus*
Quantitative Methods
Time Allowed : 3 Hours	Full Marks : 100
The figures in the margin on the right side indicate full marks.
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)

Evaluate the determinant

1
2
3

2
3
1

3
1
2

(b)

If A =

(

1
− 1

− 1
1

)

, B = (

− 1
− 1

2
2

), find AB.

(c)

For the vectors

and

where

= 2i + 3j + k,

=3i − j − k, find the value of

+ 3

(d)

If f(x) =

x + 1

x − 1

find f(f(x)) for x ≠ 1.

(e)

Find

lim
x→0

a^x − b^x

(f)

Draw the graph of f(x) =

|x|

(g)

Differentiate x^logx w.r.t. x.

(h)

Find

d²y

dx²

when x = at² and y = 2at.

(i)

Evaluate

∫

x + √x

(j)

Evaluate

∫ logx dx

(a)

Find a vector

which is perpendicular to both 4i + 5j + k and i − 4j − and which satisfies the relation

= − 1where

= 3i + j − k.

(b)

Prove that

a² + 1
ab
ac

ab
b² + 1
bC

ac
bc
c² + 1

= 1 + a² + b² + c².

(c)

Using matrices solve the following equations:

2x + 4y + z = 5
x + y + z = 12
2x + 3y + z = 16

Please turn over

( 2 )

I-12(QNM)
Revised syllabus

Marks

(a)

If y = xlog(

a + x

), show that x³

d²y

dx²

= ( x

− y)

(b)

If f(x, y) = x² — 3xy + y², then prove that

δf

δx

−

δf

δy

= (x − y)

(

δ²f

δx²

&minus

δ²f

δx δy

)

(c)

Evaluate

2
∫
1

(

x⁴ − 1

x³

)

e^{x − 1/x}

(a)

Find the area bounded by the curve y² = 8x and the line y = x.

(b)

Solve the following problem by simplex method:

Maximize z = 4x + 2y subject to x +2y < 15, 2x — y < 5, x, y > 0.

(a)

For a certain establishment the total cost function C and the total revenue function R is given by C = x³ — 12x² + 48x + 11 and R = 83x — 4x² — 21 for output x. Obtain the output for which the profit maximum. What is maximum profit?

(b)

Using graph maximize f = x + 3y subject to 2x + y < 20, x + 2y < 20, x, y > o.

(c)

Two firms are competing for business under the conditions so that one firm's gain is another firm's loss. Firm A's pay-off matrix is given below :

		Firm B
		no advertising	medium advertising	heavy advertising
	no advertising	10	5	—2
Firm A	medium advertising	13	12	15
	heavy advertising	16	14	10

SECTION II(Statistical Techniques — 30 marks)

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

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

2x5=10

(a)

Two dice are thrown simultaneously and the points on the face on the dice are multiplied together. The probability tht the product is 4, is

(i)

(ii)

(iii)

(iv)

(b)

A card is drawn from a well-shuffled pack of 52 cards. What is the probability of the card being black or an ace?

(i)

(ii)

(iii)

(iv)

none of these

(c)

The mathematical expectation of number of heads, when a balanced coin is tossed twice, is

(i)

(ii)

(iii)

(iv)

(d)

For a binomial distribution mean=2 and p (success) =

, P (at least one) is

(i)

(ii)

(iii)

(iv)

Please turn over

( 3 )

I-12(QNM)
Revised Syllabus

(e)

If x is a Poisson variate with mean λ then P(x + 1) is

(i)

x + 1

p(x)

(ii)

p(x)

(iii)

x + 1

p(x)

(iv)

p(x)

(f)

If Z is a normal deviate and

1
∫
- ∞

e^−t²/z

√2π

dt = 0.84 then P(|Z| < 1) is

(i)

0.16

(ii)

0.68

(iii)

0.34

(iv)

0.32

(g)

If for two random variables X and Y, V(x) = V(y) = V(x — y) = 2, the; correlation coefficient between x and y is

(i)

(ii)

(iii)

—1

(iv)

—1

(h)

If two regression lines are x + 2y = 4 and 2x + y = 5 for two variables x and y, then mean (x, Y) is

(i) (1, 2) (ii) (2, 1) (iii) (1, 1) (iv) (2, 2)

(i)

To test H₀ : u = 0 against H₁ : u ≠ 0, independent sample observations

are drawn from a normal population with mean and variance (unknown) and the sample mean and sample variance are calculated as 6 and 4 respectively, then the value of the test statistic will be

(i) 6 (ii) 4 (iii) 10√3 (iv) 9

(j)

A simple random sample of size 100 has mean 15, the population variance being 25. Then the 95% confidence interval for mean is

(i) (13.71, 16.29) (ii) (14.02, 15.98) (iii) (13.71, 14.12) (iv) (14.14, 16.31)

(a)

There are 4 clerks and 3 officers in a Bank. A committee of 3 is to be formed at random. Find the probability that at least on clerk and at least one officer are included in the committee.

(b)

Urn 1 contains 4 red and 6 black balls and urn 2 contains 6 red and 4 black balls. One Urn is chosen at random and a ball is drawn from it. The colour of the ball drawn is black. What is the probability that it has been drawn from urn I?

(a)

Fit a binomial distribution to the following data:

x :	0	1	2	3	4
f :	28	62	46	10	4

(b)

An automatic machine makes paper clips from coils of wire and on the average one in 400 clips is defective. If 100 paper clips are packed in each box, what are the probabilities that any given box of clips will contain (i) one defective, (ii) at least one defective? [Given that e^0.25 = 1.284]

(a)

The height of a student in a college follows normal distribution with mean 5.5 ft and standard deviation 2 ft. Find the probability that a student of that college has height (i) below 5 ft, (ii) between 5 ft and 6 ft.

[Given that

0.25
∫
0

e^−t²/z

√2π

dt = 0.0987 ]

(b)

For a certain bivariate data, following results are obtained:

n = 25, Σx = 125, Σy = 100, Σx² = 125, Σy² = 125, Σxy = 125,

Determine the regression coefficients and the correlation coefficient.

Please turn over

( 4 )

I-12(QNM)
Revised syllabus

Marks

10.

(a)

A sample of 900 members has a mean 3.4 cm and s.d. 2.16 cm. Can the sample be regarded as drawn from a population with mean 3.25 cm? Find the 95% confidence limits for the population mean.

(b)

In order to test p = P (head) = 1/2 against p = 1/4 in a coin it is tossed 6 times. Reject the null hypothesis and only if no head or 6 heads occur, Otherwise accept it. What are then probabilities of type I error and type II error of the test?

11.

(a)

A die was thrown 60 times with the following results :

Face
Frequency

1
6

2
10

3
8

4
13

5
11

6
12

Total
60

Are the data consistent with the hypothesis that the die is unbiased? [Givenx²_.05 = 11.07 for 5 degrees of freedom]

(b)

A magazine distributor assigns probabilities to the demand for a magazine as follows :

Copies demanded
probability

3
0.4

4
0.3

5
0.15

6
0.15

A copy of magazine which he sells at Rs. 8 costs Rs. 6. How many should he stock to get the maximum possible expected profit if the distributor can return back unsold copies for Rs. 4 each?

SECTION III(Economic Techniques — 30 marks)

12.

Attempt any five of the following:

2x5=10

(a)

If 12½% fall in price causes only 12½% rise in demand, then find price elasticity of demand and its nature.

(b)

The total daily cost (in Rs.) for producing 'x' toys is TC = 2.5x + 1200. If each toy sells for Rs. 5, what is the break-even point?

(c)

Show that the elasticity of demand with respect to price for the demand function x = 2p^—2 is constant, p and x being the price and quantity in demand respectively.

(d)

If r₁₂ = 0.70, r₁₃ = 0.50, r₂₃ = —0.20, then show that these are consistent.

(e)

"I_{1998, 1999} = 92" , what dies it indicate?

(f)

During a certain period the cost of living index goes up from 200 to 250 and the salary of a worker is alos raised from Rs. 800 to Rs. m960. Does he really gain?

(g)

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)	Hot wave for 3 or 4 days in the month January at Delhi in 1951.

(h)

In a study of sales, a company obtained by 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 32 units a year and it believes that at least for the next decade the trend will continue as before.

Find by what year the company's expected sales have equalled to its present capacity.

Please turn over

( 5 )

I-12(QNM)
Revised Syllabus

13.

Answer any four of the following

5x4=20

(a)

The demand function for a particular brand of pocket calculators is p = 75 — 0.3Q — 0.05Q². Find the consumer's surplus at the quantity of 15 calculators.

(b)

In a budget enquiry for working class in towns A and B, it was found that an average working class family's expenditure on food and other items are as follows:

Item group
Food
Other items

Town A
64%
36%

Town B
50%
50%

In 2001 the consumer price index stood at 279 for town A and 265 for town B with base year 1991. It was found that the rise in the prices of all the articles consumed by the working class was same for A and B. What were the price indices of food and other items in 2001?

(c)

Find the 5 year weighted moving average with weights 1, 2, 2, 2, 1 respectively for necessary trend of the following time series:

year	:	1974	75	76	77	78	79	80	81	82	83
Sales (in lakh Rs.)	:	22	26	21	25	23	27	22	26	24	26

(d)

Fit a trend line to the following data of sales of a commodity in shop using least square method :

year
Volume of sales
(in '000 tons)

:
:

1995
2.1

1996
2.4

1997
2.6

1998
3.2

1999
3.4

2000
3.8

(e)

In a certain trivariate distribution s₁ = 2, s₂ = 4, s₃ = 2, X₁ = 50, X₂ = 60, X₃ = 40 r₁₂ = r₃₁ = .5 where x₁, s₁ are mean and s.d. of variable x_i i = 1, 2, 3 and r_ij = correlation coefficient between x_i and x_j i is not equal to j, i, j = 1, 2, 3.

Find the multiple regression line of x₁ on x₂ and x₃ and estimate x₁ for x₂ = 62, x₃ = 45.

(f)

The coefficient matrix of input is

[

.5
.7

.2
.6

]

and the vector of final demand is

[

80
120

]

Find the

gross production

__________

CWA/ICWA Inter :: Quantitative Methods : December 2004