CWA/ICWA Inter :: Quantitative Methods: June 2005

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

Attempt any five of the following:

2x5=10

(a)

compete A³ if A =

(

0
− i

− i
1

)

(b)

Prove that x = —9 is a root of

x + 1
2
2

3
x + 2
3

5
5
x + 4

(c)

Find the direction cosines of the vector 2i — 3j + 5k.

(d)

If f(x) = x + |x| show that f(3) ≠ f(− 3)

(e)

Draw the graph of the function y =x |x| where x is real

(f)

Show that f(x) = |x| is continuous at x = 0

(g)

If y = √ x +

√x

show that x

y = √ x

(h)

If y= 2x find

d²y

dx²

(i)

Evaluate : ∫

3x + 4

6x + 7

(j)

If	∫(x, y) = x³ + y³ + 3xy then prove that ∫xy = ∫yx = 23

(a)

Express — 6i + 7j — 8k, as a linear combination of the vectors 3i + 2j — K and 4i — j + 2k.

(b)

A factory finds that it makes a profit of Rs. 20 on each article if the number of articles produced, is less or equal to 800 daily. The profit per article decreases by 2 paise per article produced in excess of 800. How many articles should be produced per day to yield maximum profit?

(c)

A company is to employ 40 labourers from either A or B comprising of persons in different age groups as under:

category
party A
party B I(20-25 yrs.)
25
10 I(26-30 yrs.)
10
20 I(31-40 yrs.)
5
10

Rate of labour applicable to categories I, II and III are Rs. 1500, Rs. 1200 and Rs. 1000 respectively. Using matrices find which party is economically preferable over the other.

(a)

If xy = ax² +

show that x²y₂ + 2(xy₁ — y) = 0

(b)

Show that

Lim
x→0

a^x − b^x

= log

(c)

Evaluate:

∫

logx

(1 + logx)²

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I-12(QNM)
Revised syllabus

Marks

(a)

Minimize z = 4x + 6y subject to the constraints: 2x + 3y > 6, x + y < 4, y > and x, y > 0.

(b)

The Parker Flower shop promises its customers to deliver within 4 hours on all flower orders. All flowers are purchased on the previous day and delivered to Parker by 8 a.m. in the next morning. Parker's daily demand for roses is as follows:

Dozens of roses
Probability 7
0.1 8
0.2 9
0.4 10
0.3

Parker purchases roses for Rs. 10 per dozen and sells them at Rs. 30 All unsold roses are donated to a local hospital. How many dozens of roses should parker order each evening to maximise its profits? What is the optimum expected profit?

(c)

The demand function for a commodity is p = 20 — 3q and the supply function on the market is p = 2q. Find the consumer's surplus under pure competition.

(a)

Obtain an initial basic feasible solution to the given transportation problem by North-West Corner Rule:

Demand D₁ D₂ D₃ D₄ Supply

Origin O₁ 4 3 2 5 6

O₂ 6 1 4 3 9

O₃ 7 2 4 6 7

Demand 4 6 6 6

(b)

Customers arrive at a sales counter manned by a single per according to a Poisson process with a mean rate of 10 per hour. The time required to serve a customer has an exponential distribution with a mean of 200 seconds. Find the

(c)

Solve the following game using the dominance property:

			Player B
			I	II	III
Player A	1 2 3	[	1 6 6	7 2 1	2 7 6	]

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)

If P(A) =

P(B) =

P(A/B) =

then P(A^c/B) is

(i)

(ii)

(iii)

(iv)

(b)

If the events A and B are independent and P(A) = ½, P(B) = 1/3, then the probability of occurrence of almost one of A and B is

(i)
5
6
(ii)
1
6
(iii)
2
3
(iv)
1
2

(c)

An unbiased coin is tossed thrice. If first two tosses show tail the probability that other toss shows a head is

(i)
1
8
(ii)
1
2
(iii)
1
4
(iv) 1

(d)

When two unbiased coins are tossed once, the expected number of heads is

(i)

(ii)

(iii)

(iv)

(e)

A random variable X is binomially distributed with mean 4 and standard deviation √2. Then the set of parameters (n, p) is

(i)

(4, ½)

(ii)

(8, ½)

(iii)

(16, ¼)

(iv)

(8, ¼)

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I-12(QNM)
Revised Syllabus

(f)

If a random variable X follows a Poisson distribution with parameter 1 then P(0 < X < 3) is

(i)

(ii)

(iii)

(iv)

(g)

If a random variable X follows normal distribution with mean 0 and variance 1 and P(x < 1.96) = 0.95, then P(X > 1.96) is

(i) 0.975 (ii) 0.05 (iii) 0.025 (iv) 0.475

(h)

The results of a bivariate distribution are
Σx = 30, Σ y = 60, Σ y² = 760, Σ xy = 360, n = 10, Σ x² = 190.
Then regression coefficient of x on y is

(i)

(ii)

0.45

(iii)

0.025

(iv)

0.5

(i)

For a simple random sample of size 25, drawn without replacement from a population of size 50 with variance 16, the standard error of the sample mean is

(i)
4
5
(ii)
5
25
(iii)
4
7
(iv)
1
7

(j)

In order to test whether a coin is fair or not it is tossed 4 times. The null hypothesis of fairness is rejected if and only if the number of heads is 0 or 4. The probability of type 1 error of the test is

(i)
1
16
(ii)
5
16
(iii)
7
8
(iv)
1
8

(a)

There are two identical boxes containing 4 white and 3 red balls, and 3 white and 7 red balls. A box is chosen at random and a ball is drawn from it. Find the probability that the ball is red

(b)

In pen factory machines M_1', M_2', M_3' manufacture respectively 30, 40 and 45 percent of the total output. Of their output 4, 3 and 3 percent respectively are defective pens. One pen is drawn at random from the product and is found to be defective. What is the probability that it is manufactured in the machine M₃?

(a)

An unbiased coin is tossed 4 times. Let X be the number of heads. Find the probability distribution, mean and variance of X

(b)

Fit a Poisson distribution to the following data and calculate the expected frequencies [given e^{—0.5 = 0.607]:

No. of deaths
Frequency
0
122
1
60
2
15
3
2
4
1}

(a)

Ten competitors in a voice contest are ranked by two judges in the following order:

1st Judge
2nd Judge

1
3

6
5

5
8

10
4

3
7

2
10

4
2

9
1

7
6

8
9

Find the rank correlation coefficient.

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I-12(QNM)
Revised syllabus

Marks

(b)

The heights of 10 students of a college are 68, 70, 62, 67, 61, 68, 70, 64, 66, 64 inches. Is it reasonable to9 believe that the average height is greater than 64 inches? Test at 5% level of significance considering population is normal. [Given: upper 5% points of t distribution with 9 and 10 degrees of freedoms are 1.83 and 1.81 respectively.]

10.

(a)

Test whether inoculation is effective in preventing tuberculosis from the following data:

	Tuberculosis
	Attacked	Not attacked
Inoculated	135	465
Not inoculated	265	1135

[Given: upper 5% points of chisquare distribution with 1 and 4 degrees of freedoms are 3.84 and 9.49 respectively.]

(b)

A sample size of 600 persons selected at random from a large city shows that the percentage of male in the sample is 53%. It is believed that male to total population ratio in the city is ½. Test whether this belief is confirmed by the observation.

11.

(a)

Suppose that a decision maker faced with 3 decision alternatives and 2 states of nature. Apply (i) maximin and (ii) minimax regret approach to the following pay-off table to recommend the decisions:

Acts States of nature

s₁ s₂

A₁ 12 17

A₂ 22 14

A₃ 32 13

(b)

From a population of 5 numbers 11, 13, 15, 17 and 21 draw all possible samples of size 3 with SRSWOR. Obtain the sampling distribution of sample mean and its standard error

SECTION III(Economic Techniques — 30 marks)

12.

Attempt any five of the following:

2x5=10

(a)

If the demand function is p = 48 — x², for what value of x be elasticity of demand will be unity where p and x represent price and quantity respectively?

(b)

Show that the elasticity of demand =

AR − MR

where AR and MR

are average and marginal revenue respectively at any output.

(c)

If trend equation of annual sales of a company is y = 80 + 5.2t, origin 1988, t unit = 1 year, find the quarterly trend equation.

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I-12(QNM)
Revised Syllabus

(d)

Show that Laspeyre's price index number can be written as weighted arithmetic mean of price relatives.

(e)

If the cost of living index number in 1998 with 1995 as base year is 118% and an worker draws Rs. 2360 in 1998, find the amount, the worker draws in 1995 if he maintains same standard of living in 1995 and 1998.

(f)

The total cost function of a firm is c = Rs. 14.3x + 6000 where C is the total cost and x is the output. A tax at the rate of Rs. 2.80 per unit of output is imposed and the producer adds if to his cost. If the selling price of each unit is Rs. 24.60 what will be the break even point?

(g)

For a certain trivariate data the following results are obtained:

x₁

= 40,

x₂

= 50,

x₃

= 30, b_12.3 = 0.41, b_13.2 = 0.23

Estimate the value of x₁ when x₂ = 52 and x₃ = 36.

(h)

Given Σ p₀q₀ = 1360, Σ p_nq₀ = 1900, Σ p₀q_n = 1344, Σp_nq_n = 1880

show that Fisher's ideal index number satisfies Factor Reversal Test.

13.

Answer any four of the following

5x4=20

(a)

If the demand curve is of the form p = ad^—mx, where p is the price and x is

the demand, prove that the elasticity of demand is

Hence deduce the elasticity of demand for the

curve p = 5e^—x/3

(b)

Below are given the figures of production (in thousand tons) of a sugar factory:

Year Production
('000 tons)

:
:

1989
77

1990
88

1991
94

1992
85

1993
91

1994
98

1995
90

(i)	Fit straight line the method of least squares and show the trend values.
(ii)	What is the monthly increase in production?

(c)

An enquiry into the budgets of the middle class families in a certain city are as follows:

Expenditure

Price (in Rs.) 1975
Price (in Rs.) 1976

food
35%
150
145

fuel
10%
25
23

garments
20%
75
65

rent
15%
30
30

mise
20%
40
45

Find the C.L.I. of 1976 considering 1975 as base period.

(d)

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 800 and 1600 units respectively.

Industry Input to Final demand

1
2 1
160
80 2
200
400
40
320

(e)

From a trivariate distribution the following correlation coefficients are obtained r₁₂ = 0.7, r₁₃ = 0.6 and r₂₃ = 0.4

Prove that 1 − R₁₂₃² = (1 − r₁₂²) (1 − r₁₃₂²), where R_1.23 and r_13.2 are the

multiple correlation coefficient and partial correlation coefficient respectively.

(f)

On the basis of quarterly sales (in Rs. lakhs) of a certain commodity to the year 1971-75, the following calculations were made: Trend : y = 250 + 0.6 t with origin at 1st quarter of 1971, t = time unit (one quarter), y = quarterly sales (Rs. in lakhs)

Seasonal variation
Quarters
Seasonal Indices

:
:
:

I
90

II
95

III
110

IV
105

Estimate the quarterly sales for the year 1972 (Use multiplicative model).

__________

CWA/ICWA Inter :: Quantitative Methods : June 2005