CWA ICWA Inter Stage II : Quantitative Methods - December 2005

CWA ICWA Inter - Stage II • Quantitative Methods • December 2005

This Paper has 58 answerable questions with 0 answered.

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.

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 =

(

2
2

1
2

)

, B =

(

4
5

3
6

)

show that |AB| = |A| ×|B|.

(0)

(b)

If i + 2j + 3k and 2i + 4j + 4k be the position vectors of points A and B, find the unit vector along

(0)

(c)

If f(x) = e^{ax + b} then show that

f(l) f(m) f(n)

f(l + m + n)

= e^2b

(0)

(d)

Draw the graph of f(x) =[x] for −1 < x < 1.

(0)

(e)

Evaluate

lim
x→1

x² − 3x + 2

x² − 5x + 4

(0)

(f)

Differentiate x⁵ w.r.t x².

(0)

(g)

If y = 3^x log x, prove that x

(

− y log3

)

=3^x

(0)

(h)

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

(0)

(i)

Evaluate

∫xe^x

(0)

(j)

If f(x, y) = e^{x² + y²} find f_xx

(0)

(a)

If (5, 2, 3), (−1, −1, 5) and (2, 4, −3) be the position vectors of three points A, B and C respectively with respect to origin O (0, 0, 0) then show that they are vertices of a right angled isosceles triangle.

(0)

(b)

If A =

(

1
2

2
0

)

and B =

(

3
2

2
4

)

then show that AB ≠ BA but (A + B) (A − B) =(A² − B² −(AB −BA)

(0)

(c)

Solve by Cramer’s rule: x + y + z = 24, 2x + 3Y + z = 2 and 8y − 3z = 2.

(0)

(a)

Verify that the function u = 6x² − 2y² − 12x − 16y has a saddle point.

(0)

(b)

A function f(x) is defined as follows:

f(x) = 2x − 1 if x < 3,

= K if x = 3,

= 8 − x if x > 3.

For what value of k f(x) is continuous at x = 3? With this value of k draw the graph.

(0)

(c)

If y = ( x +

√

x² + 4

)

ⁿ

, show that (x² + 4)^3/2

d²y

dx²

+ n (x – n

√

x² + 4

) y = 0.

(0)

(a)

For a certain establishment cost and revenue functions are c= x³ − 12x² + 48x + 11 and R = 83x − 4x² − 21 both in rupees respectively, obtain the maximum profit, Here x = output.

(0)

(b)

Evaluate

∫

x⁴ − 1

x³

e^{x + 1/x} dx

(0)

(c)

Find the area of the region bounded by the curves y = x² and y = √x

(0)

(a)

Maximize z = 5x₁ + 7x₂ subject to 2x₁ + 3x₂< 13, 3x₁ + 2x₂< 12, x₁> 0, x₂> 0 by the simplex method.

(0)

(b)

Obtain the initial basic feasible solution to the following transportation problem by least cost method and determine the cost associated with this solution :

To	→	D₁	D₂	D₃	Supply
From	↓
	O₁	12	19	16	14
	O₂	19	13	22	16
	O₃	8	28	14	12
	Demand	17	15	10	42

(0)

(c)

In a bank cheques are cashed at a single “teller“ counter. Customers arrive at the counter in a Poisson manner at an average rate of 30 customers per hour. The teller takes on the average a minute and a half to cash a cheque. The service time has been shown to be exponentially distributed, Calculate the percentage of time the letter is busy and average time a customer is expected to wait in the system.

(0)

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

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)

Probability of obtaining an even number in a single throw of an unbiased die is

(i)

(ii)

(iii)

(iv)

(0)

(b)

If P(A) =

, P(B) =

and P(A/B) =

, then the value of P(B/A) is

(i)

(ii)

(iii)

(iv)

(0)

(c)

If for two independent events A and B, P(A) = 0.4, P(A U B) = 0.7, the value of P(B) is

(i)

0.25

(ii)

0.3

(iii)

0.2

(iv)

0.5

(0)

(d)

If a random variable X assumes only two values + 1 and − 1 such that P(X = 1) = P(X = −1), then V(X) is

(i)

(ii)

(iii)

(iv)

(0)

(e)

If for a binomial distribution, P (success in a trial) =

, P (no failure) =

then the probability of at least one success is

(i)

(ii)

(iii)

(iv)

(0)

(f)

For a Poisson distribution with mean 4, the coefficient of variation is

(i)

200%

(ii)

100%

(iii)

50%

(iv)

25%

(0)

(g)

For a standard normal variable Z, P(0 <Z <1) = 0.34. Then P(Z >− 1) is equal to

(i)

0.32

(ii)

0.84

(iii)

0.68

(iv)

0.16

(0)

(h)

If for two variables X and Y, correlation coefficient = −0.6, V(X) = 9, V(Y) =16, then the regression coefficient of X and Y is

(i)

0.45

(ii)

0.8

(iii)

−0.8

(iv)

−0.45

(0)

(i)

In order to test H₀ :p =

against H₁ :p =

, a coin is tossed 4 times

Where p is the probability of a head in tossing a coin once. H₀ is rejected if and only if the number of heads is 0 or 4. Then power of the test is

(i)

(ii)

(iii)

(iv)

(0)

(j)

If a random sample of size 4 with mean 50 is drawn from a normal population with mean u and

variance 25, then the 95% confidence interval for u (where

∞
∫
1.96

(2π)^½ e^−t/2 dt = 0. 025) is

(i)

(45.1, 54.9)

(ii)

(−4.9, 4.9)

(iii)

(47.55, 52.45)

(iv)

(0, 54.9)

(0)

(a)

Box 1 contains 2 white and 2 black balls, Box 2 contains 2 white and 1 black balls. One of the boxes is selected at random and one ball is drawn from it. Find the probabilities that (i) it turns out to be White, (ii) it is selected from Box 1 if the ball drawn is white.

(0)

(b)

The probability that A speaks the truth is 0.4 and that B speaks the truth is 0.7. What is the probability that they will contradict each other?

(0)

(a)

What is the probability of guessing correctly at least 6 of 10 answers in a TRUE–FALSE objective test?

(0)

(b)

If the random variable X follows a normal distribution with mean 18 and variance 625, find the value of (i) P(X > —31) and (ii) P(X < 67 / X > —31)

where it is given that

8
?
1.96

(2π)^½ e^–t/2 dt = 0. 025) is

(0)

(a)

For the variables x and y the regression equations are 4x − y + 8 = 0 and 7x − 3y + 39 = 0. Identify the regression equation of x and y and that of y on x. Find the means of x and y correlation coefficient between x and y.

(0)

(b)

From a population 3,5,5,7,9,10 of 6 units, find the sampling distribution of sample mean of simple random samples without replacement of size two. Then show that mean of sample means is exactly equal to the population mean.

(0)

10.

(a)

Pay−offs of acts X,Y and Z and the states of nature L, M and N are as follows :

Act State of nature	X	Y	Z
L M N	25 400 650	−10 440 740	−125 400 750

The probabilities of the states of nature are respectively 0.1, 0.7 and o.2. Calculate EMV and conclude which of the acts can be chosen as the best.

(0)

(b)

The following results were obtained from the record of age (x) and blood pressure (y) of a group of 10 women:

	x	y
Mean	53	142
variance	130	165

Σ(x –

) (y –

) = 1220

Find the regression equation of y on x and use it to estimate the blood pressure of a woman of age 45.

(0)

11.

(a)

I.Q. test was administered to 5 persons before and after they are trained. The results are given below :

Candidates
I.Q. before training:
I.Q. after training:

A
70
80

B
80
78

C
83
85

D
92
96

E
85
81

Test whether there is any improvement in I.Q. after the training. Given that t_0.05,4 = 2.13.

(0)

(b)

A die is thrown 120 times of which 1 comes 20 times, 2 or 3 comes 45 times, 4 or 5 comes 40 times and 6 comes 15 times. Test whether the die is perfect or not. Given upper 5% point of the Chisquare distribution at 3 and 5 d. f are 7.81 and 11.07 respectively.

(0)

SECTION III(Economic Techniques — 30 marks)

12.

Attempt any five of the following:

2x5=10

(a)

If (x + 3)³p = 4, compute the elasticity of demand in terms of x.

(0)

(b)

Find the average cost function when the marginal cost function is 3x − 4x + 6 Rs. for output of x units and fixed cost of 30 Rs.

(0)

(c)

If the quarterly trend equation of production of sugar in a company is y = 5.27 + 0.04 t with origin = 3rd quarter of 1983 and unit of t = 1 quarter, find the trend equation based on annual sales with origin 1983 and unit of t = 1 year.

(0)

(d)

Show that Paasche’s price index number can be written as weighted harmonic mean of price relatives.

(0)

(e)

Σ
i

p_ni q_ni = 979 and Laspeyre’s price index number is 110%,

Σ
i

p_oi q_oi find the sum

(0)

(f)

Deseasonalise the data using multiplicative model of tea production:

Quarters
Production (’000 tons)
Seasonal index

:
:
:

I
26.4
110

II
27
90

III
24
120

IV
32
80

(0)

(g)

Find r₂₃ if r_23.1 = 0.6 and r ₁₃ = 0.5.

(0)

(h)

If the multiple regression equation of X₁ on X₂ and X₃ is X₁ = 0.6 X₂ + 0.4₃ − 14, r_13.2 = 0.8 and V(X₃) = 16, find V(X₁).

(0)

13.

Answer any four of the following.

5x4=20

(a)

A demand law is given by x = 80 − 4p where x is the quantity demanded and p is the price of a commodity. If the price elasticity of demand at p = 8 is increased by 50%, obtain the percentage decrease in demand.

(0)

(b)

Find the trend values from the following time series data by method of moving average with suitable period to be obtained by you from the data :–

Year	:	1990	1991	1992	1993	1994	1995	1996	1997	1998	1999	2000	2001
Profit(in lakh Rs.)	:	463	495	479	481	513	494	493	528	518	514	540	500

(0)

(c)

Calculate the price index number by Fisher’s method:

year →	2000 (base)		2004 (current)
Commodity	Price (Rs.)	Total value (Rs.)	Price (Rs.)	Total value (Rs.)
↓
A	10	100	12	120
B	12	144	14	196
C	14	140	16	192
D	16	192	18	216

(0)

(d)

In a trivariate distribution,

Mean
s.d
Correlation coefficient

:
:
:

x₁ = 60,
x₂ = 70,
x₃ = 100,

s₁ = 3,
s₂ = 4,
s₃ = 5,

r₁₂ = 0.7,
r₁₃ = 0.6,
r₂₃ = 0.4,

Find the multiple regression line of x₁ on x₂ x₃. Estimate x₁ for x₂ = 80, x₃ = 120.

(0)

(e)

Given the following input–output table:

production	Consumer		Final demand
	Steel	Coal
Steel (in ton)	0.3	0.1	40
Coal (in ton)	0.6	0.4	80
Labour (mandays)	6	3

Find the gross production of steel and coal and also the total labour required.

(0)

(f)

Writer short notes on any one of the following:

(i)

Elasticity of demand.

(0)

(ii)

Partial correlation coefficient.

(0)

(iii)

Least square theory

(0)

Disclaimer ♣ Privacy Policy ♣ Who We Are ♣ Terms of Service