CWA/ICWA Inter :: Quantitative Methods: June 2006

*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. (Notations and symbols have their usual meanings.)
SECTION I(Mathematical Techniques — 40 marks)
Answer Question No. 1 (compulsory — 10 marks) and two other questions from this section (15x2 = 30 marks).

Attempt any five of the following:

2x5=10

(a)

If A =

(

1
a

1
2

)

B =

(

1
0

0
1

)

and C =

(

b
3

1
3

)

find the values of a and b

for which A + B = BC.

(b)

Evaluate:

1
1
1

a
b
c

b + c
c + a
a + b

(c)

For the vectors

=i + 2j − k,

= 2i − j + 2k find

+ 2

(d)

If y= f(x) =

1 − x

1 + x

find f(y).

(e)

Determine	lim x→3	f(x) where f(x)	= 2x + 3, x > 3
			= 3x + 1, x ≤ 3.

(f)

Find the gradient of the curve log(xy) = x² + y² at (1, 2).

(g)

If u = x²y + y²x, find the value of x

δu

δx

+ y

δu

δy

where x = y = 1.

(h)

Evaluate:

∫

1+ e^2x

e^x + e^{− x}

(a)

Show that the vectors 2i — j + k, i — 3j — 5k and 3i — 4j — 4k are coplanar.

(b)

Obtain the inverse of the matrix

[

2
3
1

4
1
3

− 1
2
− 3

]

and hence solve the following system of

equations : 2x + 4y — z = 9, 3x + y + 2z = 7, x + 3y — 3z = 4.

(c)

Evaluate:

lim
x→0

3x + |x|

7x − 5|x|

(a)

A firm has found from past experience that its profit in terms of number of units produced is given by p(x) = x³/3 + 729x — 2500, 0 < x < 35
Calculate (i) value of x that maximises the profit, and (ii) per unit profit of the product when this maximum level is achieved.

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I-12(QNM)
Revised syllabus

Marks

(b)

If y = log(x +

√

1+ x²

) then show that (1 + x²)y₂ + xy₁ = 0.

(c)

If u =

x³ + y³

x + y

show that x

δu

δx

+ y

δu

δy

= 2u

(a)

Evaluate

∫

3x² − x + 1

(x + 1)²

(b)

Solve graphically the following LPP:
Maximise z =2x + y subject constraints 5x + 10y < 50, x + y < 1, y <4, (x, y) > 0.

(c)

In a service station manned by one server, on an average one customer arrives every 10 minutes. If has been observed that eeach customer requires 6 minutes to be served. Determine (i) average queue length, (ii) average time spent in the system.

(a)

Find the area of the region bounded by the curve y² = 12x, x-axis and the semi-latus rectum.

(b)

A company with factories at F₁, F₂ and F₃ supplies to warehouses at W₁, W₂ and W₃. Factories capacities are 200, 160 and 90 units respectively while weekly warehouses requirements are 180, 120 and 150 units respectively.

Warehouses W₁ W₂ W₃ Supply

Factories

F₁ 16 20 12 200

F₂ 14 8 18 160

F₃ 26 24 16 90

Demand 180 120 150 450

To minimise shipping costs, determine an initial BFS to above Transportation problem using North-West Corner Method.

(c)

What is the optimal strategy in the game described by the matrix?

(

− 5
5
− 4

3
5
− 2

1
4
0

20
6
− 5

)

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I-12(QNM)
Revised Syllabus

SECTION II(Statistical Techniques — 30 marks)

Answer Question No. 6 (compulsory — 10 marks) and two
other questions (10 x 2 = 20 marks) from this section.

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

2x5=10

(a)

Two dice are thrown simultaneously and the point on the dice are multiplied together. Then the probability that the product is 4, is

(i)
2
3
(ii)
1
6
(iii)
1
4
(iv)
1
12

(b)

Let the events A and B be independent with P(A) = 0.5 and P(B) = 0.8. Then the probability that neither of the events occurs is

(i) 0.1 (ii) 0.2 (iii) 0.3 (iv) 0.4

(c)

Let A and B are two events such that P(A) = 0.4, P(A U B) = 0.7 and P(B) = p. For what choice of p are A and B independent?

(i)

(ii)

(iii)

(iv)

(d)

The mathematical expectation of the number of points if a balanced die is thrown is

(i)

(ii)

(iii)

(iv)

(e)

If Z is N(0, 1) variate then P(—0.75 < Z< 2.04) is [given

z
∫
0

√2π

e^{-1/z t²}

= 0.2734 and 0.4793 for z = 0.75 and 2.04 respectively]

(i)

0.2509

(ii)

0.2059

(iii)

0.7527

(iv)

0.72257

(f)

For a Poisson distribution p(x = 1) = p(x = 2), then p(x = 1 or 2) is

(i)

2e^-2

(ii)

3e^-2

(iii)

4e^-2

(iv)

5e^-2

(g)

The p.d.f. of a normal distribution is f(x) = −

√π

e^-4x², − ∞ < x < ∞.

Then its

deviation is

(i)

√2

(ii)

2√2

(iii)

4√2

(iv)

5√2

(h)

For a simple random sample without replacement of size 16 drawn from a population of size 65 and variance 4 the standard error of sample mean is

(i)

(ii)

(iii)

(iv)

(i)

If the regression lines are perpendicular to each other, then the correlation coefficient between the variables.

(i)

(ii)

0.5

(iii)

(iv)

none of these

(j)

If random variable X is uniformly distributed with probability density function f(x) = 1, 0 < x < 1, then V(X) is

(i)

(ii)

(iii)

(iv)

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I-12(QNM)
Revised syllabus

Marks

(a)

The odds in favour of one student passing an examination are 3 : 7. The odds against another student passing an examination are 3 : 5. What are the probabilities that (i) both will pass; (ii) both will fail?

(b)

The Wage distribution of workers in a factory is normal with mean Rs. 200 and standard deviation Rs. 25. If the wages of 40 workers be less than Rs. 175, find the number of workers whose income is less than Rs. 225.

[Given

1
∫
0

√2π

e^{-1/z t²}

dt = 0.34]

(a)

For a binomial distribution, mean is 4 and variance is 2. Find the probability of getting (i) at least 2 successes and (ii) atmost 2 successes.

(b)

A sample of 100 arrivals of customers in a departmental store is according to the following distribution:

Time between arrivals (in minutes)		Frequency
0.5 1.0 1.5 2.0 2.5 3.0		0.5 21 36 19 7 5

Use the following random numbers to simulate for the next 10 arrivals. [Given: Random numbers: 25, 39, 65, 76, 12, 05, 73, 89, 19, 49]

(a)

(a)	Find the sample size such that the probability of the sample mean differing from the population mean by not more than 1/10th of standard deviation is 0.95. [given P(Z > 1.96) = 0.025 where Z is N90, 1) variate]
(b)	Generate 2 random numbers from the 'seed' 7534.

(b)

Two variables have the regression lines 3x + 2y = 26 and 6x + y = 31. Find the mean values, the correlation coefficient between x and y and the ratio of variances of the variables.

10.

(a)

Marketing staff of an industrial unit has submitted the following pay-off table, giving profits in million rupees, concerning a certain proposal depending upon the rate of technological advance:

Technological advance Decision

Accept Reject

Much
Little
None 2
5
–1 3
2
4

The probabilities are 0.2, 0.5 and 0.3 for much, little and none technological advance, respectively. What dicision should be taken?

(b)

Consider the following table:

	Poor eye-sight	Good eye-sight
No. of Males :	200	350
No. of Females :	200	250

Can we conclude at 5% level of significance that sex has no bearing on the quality of eye-sight?[Given: values of at 5% level of significance are are 3.84, 5.99, 7.81, 9.49 for d.f. = 1, 2, 3 and 4 respectively?]

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I-12(QNM)
Revised Syllabus

11.

(a)

A random sample of 100 articles taken from a batch of 2,696 articles contains 5 defective articles. Find95% confidence interval for the proportion of defective articles in whole batch.

(b)

There are two brands of car tyres A and B in the market. A sample of 100 tyres of brand A has an average life of 37,500 km with a standard deviation of 2,500 km. Another sample of 75 tyres of brand B has an average life of 39,000 km with a standard deviation of 3000 km. Can we conclude that brand B is better than brand A ?

SECTION III(Economic Techniques — 30 marks)

12.

Attempt any five of the following:

2x5=10

(a)

With the base year 2002, the CLI in 2006 stands 146. A person who was getting a monthly salary of Rs. 2,500 in the year 2002 gets Rs. 3,700 in the year 2006. Is he getting more or less and to what extent?

(b)

If 12 ½% fall in price causes a 25% rise in demand, then find price elasticity of demand and its nature.

(c)

The total daily cost (in Rs.) for producing 'x' plastic chairs is TC = 15x + 3000. If each chair sells for Rs. 65, find the break-even point

(d)

For the demand law p =

(x + 2)²

find the elasticity of demand in term of x.

(e)

Write down the normal equations in fitting a second degree equations in fitting a second degree equation y = a bt + ct² by the method of least squares.

(f)

If the seasonal index is 110 and production is 25.4, then what is the deseasonalised value?

(g)

If total cost = 250 + 10x — 12x² + x³, find at what level of x diminishing marginal return begins.

(h)

Compute Fisher's Index Number from the following data:

ΣP₀Q₀ = 280, ΣP₀Q₁ = 384, ΣP₁Q₀ = 344, ΣP₁Q₁ = 472

13.

Answer any four of the following

5x4=20

(a)

The demand functions of two commodities are
x₁ = 1 — 2p₁ + p₂ and x₂ = 5 — 2p₁ — 3p₂, with usual symbols. Examine the nature of commodities x₁ and x₂.

(b)

Show that r₁₂ = 0.80, r₁₃ = 0.60 and r₂₃ = —0.20 are inconsistent in multivariate distribution.

(c)

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

	Group	Group index	Percentage of total expenditure
(i) (ii) (iii) (iv) (v)	Food Clothing Housing Fuel and electricity Miscellaneous	134 140 105 120 130	60 Not available 20 5 Not available

The CLI with percentage of total expenditure as weight was found to be 127.9. Estimate the weights used for 'clothing' and 'miscellaneous'

(d)

Apply the semi-averages method and obtain the trend values for following 8 years and also for next 2 years from the following data:

Year	:	1998	1999	2000	2001	2002	2003	2004	2005
Value	:	45	43	47	49	46	45	42	43

(e)

From the following transaction matrix, find gross output to meet demand of 300 units of agriculture and 900 units of industry:

Purchasing sector	Agriculture	Industry	Final demand
Producing sector
Agriculture	200	500	100
Industry	300	900	300

(f)

A production function Q =

√

is given where K and L are the two variable

inputs and prices per unit of K and L are Rs. 2 and Rs. 3 respectively with total cost at Rs. 60. Determine the maximum output subject to the cost constraint.

__________

CWA/ICWA Inter :: Quantitative Methods : June 2006