CWA ICWA Inter Stage II : Quantitative Methods - December 2006

CWA ICWA Inter - Stage II • Quantitative Methods • December 2006

This Paper has 55 answerable questions with 0 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 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.

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 (15x2 = 30 marks) from this section.

Attempt any five of the following:

2x5=10

(a)

If A =

(

9
1

1
3

)

B =

(

1
7

5
12

)

find the matrix X such that 5A +3B + 2X = 0

(0)

(b)

Evaluate (using properties):

1
2
3

2
3
4

3
4
5

(0)

(c)

If a = 2i + j − k, b = i + 2j + k, c = 3i − 2j + 2k, find

a − 2b + c

(0)

(d)

Find the value of | f(c) − f(−c)| when f(x) =

|x|

and c (≠ 0) is a real number.

(0)

(e)

If y = 5^x find

d²y

dx²

(0)

(f)

For f(x) =\|x\|, find	lim	f(x)
	x → 0

(0)

(g)

If y = (log x)², prove that x²

d²y

dx²

+ x

= 2.

(0)

(h)

If f′(x) =

e^x

(1 + x log x), x > 0, find f(x).

(0)

(i)

f (x, y) = x³ + y³ + 3xy, show f_xy = f_yx.

(0)

(j)

Evaluate:

1/3
∫
0

√1 −3x dx

(0)

(a)

Using properties of determinant, show that

a²
b²
2ab

2ab
a²
b²

b²
2ab
a²

is a perfect square.

(0)

(b)

If a = 3i + j + 2k and b = 2i − 2k + 4k, find: (i) a x b, (ii) | a x b |, (iii) unit vector perpendicular to
both a and b.

(0)

(c)

If y = x log(

a + bx

),show that x

d²y

dx²

= (

−

)²

(0)

(a)

If A =

(

2
3

3
5

)

The show that A² = 7A − I, where I is the unit matrix of order 2. Hence find A^-1. Obtain

two integers p and q such that one can express A³ in the form pA + qI.

(0)

(b)

Evaluate:

4
∫
3

3x² − 3x − 6

(0)

(c)

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

		Firm B
	Strategy	No advertising	Medium advertising	Heavy advertising
Firm A:	No advertising Medium advertising Heavy advertising	10 13 16	5 12 14	-2 15 10

Suggest optimal strategies for the two firms and the net outcome there of.

(0)

(a)

Verify that a saddle point exists for the function
z = 18x² − 6y² − 36x − 48y.

(0)

(b)

Solve the following L.P.P. by simplex method.

Maximise	z = 3x₁ + 5x₂
subject to	x₁ + x₂ ≤ 10; 2x₁ − 3x₂ ≤ 12; x₁, x₂ ≥ 0.

(0)

(c)

A T.V. repair–man finds that the time spent on his job has an exponential distribution with mean 30 minutes. If he repairs sets on "first–come first–served" basis and if the arrival of sets is approximately Poisson with an average 10 per 8 hours a day, then what is the repair–man’s expected idle time each day? Also obtain average number of units in the system.

(0)

(a)

The supply function of a product is y = 3x² − 4, Find the producer’s surplus when 4 units are supplied.

(0)

(b)

Obtain an initial B.F.S. to the transportation problem by north–west corner method.

To →	W₁	W₂	W₃	W₄	Capacity
From ↓
F₁	21	16	25	13	11	(a₁)
F₂	17	18	14	23	13	(a₂)
F₃	32	27	18	41	19	(a₃)
Demand	6	10	12	15	43
	(b₁)	(b₂)	(b₃)	(b₄)

(0)

(c)

The total cost function of a company is c = 1/3 x³ − 5x² + 28x + 10, where c is the total cost and x is output. A tax at the rate of Rs. 2 per unit of output is imposed and the producer adds it to his cost. If the market demand function is given by p = 2530 − 5x, where Rs. p is the price per unit of output, find the maximum profit.

(0)

SECTION II(Statistical Techniques — 30 marks)
Answer Question No. 6 (compulsory — 10 marks) and two other questions (10x2 = 20 marks) from this section.

SECTION II(Statistical Techniques — 30 marks)

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

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

2x5=10

(a)

Probability of getting 2 heads in a single throw of 4 perfect coins is

(i)

(ii)

(iii)

(iv)

None of these.

(0)

(b)

P {4 < x < 6} of a Poisson variate with mean parameter 1 (given e⁻¹ = 0.3678) is

(i)

0.001,

(ii)

0.002,

(iii)

0.0031,

(iv)

0.0189.

(0)

(c)

Probability of getting 8 points by throwing 2 unbiased dice is

(i)

(ii)

(iii)

(iv)

(0)

(d)

Given P {0 ≤ z ≤ 0.45} = 0.1736 and P {0 ≤ z ≤ 1.92} = 0.4726 then P {−0.45 ≤ z ≤ 1.92} is

(i)

0.6462,

(ii)

0.2990,

(iii)

0.9452,

(iv)

0.3472.

(0)

(e)

For the two regression lines 9x + 3y = 46 and 3x + 12y = 19, the correlation coefficient is

(i)

2√3

(ii)

−

2√3

(iii)

(iv)

None of these.

(0)

(f)

In order to test whether a coin is fair or not it is tossed 5 times. The null hypothesis of fairness is rejected if and only if the number of heads is 0 or 5. The probability of type −I error of the test is

(i)

(

)³

(ii)

(

)⁴

(iii)

(

)⁵

(iv)

None of these

(0)

(g)

Probability of getting a zero value of a binomial distribution with mean 4 and variance 3 is

(i)

(

)¹⁶

(ii)

(

)¹⁶

(iii)

(

)¹⁶

(iv)

None of these

(0)

(h)

The p.d.f. of a continuous random variate X is given by f(x)	=kx² (1 − x),	0 < x < 1 , elsewhere
	= 0

. Then the constant k is

(i)

(ii)

10,

(iii)

11,

(iv)

None of the these.

(0)

(i)

A random sample of size 25 has been drawn from a normal population with variance 9. If the sample mean is 5 the 95% confidence interval for the population mean is

(i)

(3.284, 6.716),

(ii)

(3.824, 6.176),

(iii)

(3.482, 6.617),

(iv)

None of these

(0)

(a)

If P(A) = ¼ P(B) = 2/5 and P(A ∪ B) ½ find (i) P(A ∩ B) and (ii) P(A ∪ B), where A, B are mutually exclusive.

(0)

(b)

A class consists of 50 students out of which the number of girl students is 10, In the class 2 girls and 5 boys are rank holders in the previous examination. If a student is selected at random from the class and is found to be a rank holder, what is the probability that the student selected is a girl.

(0)

(a)

Fit a Poisson distribution to the following data:

No. of mistakes per page
No. of pages

:
:

0
109

1
65

2
22

3
3

4
1

(0)

(b)

Find the mean and s.d. of a normal distribution where 8% of the items are over 64 and 31% are under 45. Given : P{0 < z < 0.495 = 0.19} and P{0 < z < 1.405} = 0.42, where z is a N(0, 1) variate.

(0)

(a)

A group of 5 patients treated with medicine A, weight 42, 39, 48, 60, 41 kg. while a second group of 7 patients from the same hospital treated with medicine B, weight 38, 42, 56, 64, 68, 69, 62 kg.
Do you agree with the claim that medicines increase the weight equally?
[Value of t at 5% level of significance of 10 d.f. is 1.812].

(0)

(b)

A die was thrown 90 times with the following results:

Face
Frequency

:
:

1
10

2
12

3
16

4
14

5
18

6
20

Total
90

Are these data consistent with the hypothesis that the die is unbiased?

[Given

X²
0.05

= 11.07 at 5 d.f.; = 12.59 at 6 d.f.]

(0)

10.

(a)

A confectioner sells confectionery items. Past data of demand per week (in ‘00 kg) with frequency is given below:

Demand per week
Frequency

:
:

0
2

5
11

10
8

15
21

20
5

25
3

Using the following sequence of random numbers, simulate the demand for the next 10 weeks. Also find the average demand per week:

35
35

52
83

90
94

13
56

23
67

73
66

34
60

(0)

(b)

The following are the marks obtained by 7 students in two subjects. Compute the rank correlation coefficient.

Marks in Mathematics
Marks in Statistics

:
:

21
59

62
89

39
28

48
68

60
80

78
92

25
81

(0)

11.

(a)

A bag contains defective articles the exact number of which is unknown. A sample of 100 from the bag gives 8 defective articles. Find the possible limits for the proportion of defective articles in that bag.

(0)

(b)

An educational entrepreneur in order to run one out of two software courses CUR and FUT, has obtained following estimates from the software experts:

(i)	Probability of getting a return of Rs. 5 crores is 0.6 for FUT and 0.2 for CUR;
(ii)	Probability of getting a return of Rs. 40 lakhs is 0.4 for FUT and 0.8 for CUR;
(iii)	FUT will cost the entrepreneur Rs. 1 crore and CUR only Rs. 30 lakhs. Using EMV, obtain the optimum strategy.

(0)

SECTION III(Economic Techniques — 30 marks)

12.

Attempt any five of the following:

2x5=10

(a)

If the demand law is p =

x + 1

find the elasticity of demand in terms of x.

(0)

(b)

If Laspeyre’s index number is 114.4 and Paasche’s index number is 114.6, find out the Fisher’s ideal index number.

(0)

(c)

The total daily cost (in Rs.) for producing x dot–pens is TC = 2.5x + 3000. If each dot–pen sells for Rs. 5, what is the break–event point?

(0)

(d)

Write down the normal equations in fitting a parabolic equation y = a + bx + cx² by the method of least square.

(0)

(e)

If 12½% fall in price causes a 25% rise in demand, then find price elasticity of demand and its nature.

(0)

(f)

If r₁₂ = 0.7, r₂₃ = 0.4 and r₃₁ = 0.6, calculate R_1.23.

(0)

(g)

With the base year 1960 as the base, the C.L.I in 1972 stood at 250. X was getting a monthly salary of Rs. 500 in 1960 and Rs. 750 in 1972. How much should X have received as extra allowance in 1972 to maintain his standard of living in 1960?

(0)

(h)

Given total cost = 250 − 15x − 12x² + x³, find at what level of x diminishing marginal return begins.

(0)

13.

Answer any four of the following:

5x4=20

(a)

The total revenue from the sale of q units is given by R = 40q − 5q². Calculate the price elasticity of demand, when the marginal revenue is 10.

(0)

(b)

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

Commodity	1996 (base year)		2000 (current year)
	Price	Quantity	Price	Quantity
A B C	8 10 7	6 5 8	12 11 8	5 6 5

(0)

(c)

The table below gives the number of production of a commodity:

Year (t)
Production (y) (in ‘000 tons)

:
:

1980
28

1981
38

1982
46

1983
40

1984
56

Using the method of least squares fit a straight line. The company has physical facilities to produce only 70.6 units (in ‘000 tons) a year and it believes that at least for the next decade the trend will continue as before.
By what year the company‘s expected production has equalled its present capacity?

(0)

(d)

If r₁₂ = 0.8, r₁₃ = 0.6, r₂₃ = −0.2 then examine whether these are inconsistent.

(0)

(e)

Suppose the demand curve is given by q = 10 − p. What is the total consumer’s surplus from consuming 6 units of the item? If the price changes from Rs. 4 to Rs. 6, what is the change in consumer’s surplus?

(0)

(f)

Quarterly sales in (Rs. ’000) of a company are given below:

Quarter	Year
	1996	1997	1998
I II III IV	7.2 5.0 7.8 9.2	7.4 6.8 7.4 9.0	8.4 6.0 6.2 7.6

Calculate the seasonal indices.

(0)

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