CWA ICWA Inter Stage II : Quantitative Methods - December 2008

CWA ICWA Inter - Stage II • Quantitative Methods • December 2008

This Paper has 57 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.
SECTION I (Mathematical Techniques — 40)
Answer Question No. 1 (Compulsory — 10 marks) and two other questions (15x2=30 marks) from this section.
Marks

The figures in the margin on the right side indicate full marks.

SECTION I (Mathematical Techniques — 40)

Answer Question No. 1 (Compulsory — 10 marks) and two other questions
(15x2=30 marks) from this section.

Marks

Attempt any five questions:

2x5=10

(a)

Evaluate (using property):

2
–4
3

1
–2
0

–3
6
2

(0)

(b)

If the position vector a of the point (3, n) is such that |a| = 5 then find the value of n.

(0)

(c)

If A =

(

1
3

1
b

)

, B =

(

1
0

0
1

)

and C=

(

2
a

1
3

)

find the value of a and b for which A + B = BC.

(0)

(d)

Find the domain of f(x) =

x² – 9

(0)

(e)

A function is defined as follows:
f(x) = x², x ≠ 2
= 2, x = 2
Is f(x) continuous at x = 2?

(0)

(f)

Evaluate

lim
x→∝

(x + 1) (2x + 3)

(x + 2) (2x + 5)

(0)

(g)

Find the gradient of the curve log xy = x² + y².

(0)

(h)

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

(0)

(i)

If u = x²y + y²x find the value of x

∂u

∂x

+ y

∂u

∂y

when x = y = 1.

(0)

(j)

Find the value of ƒ log x dx.

(0)

(a)

Express 16i – 10j – 15k as a linear combination of vectors 4i + 2j – k and – i + 4j + 3k.

(0)

(b)

Solve by matrix method the following system of equations:

2x – y + 3z = 13
x + 3y – z = –2
3x + 2y + z = 9

(0)

(c)

Show that

yz – x²
zx – y²
xy – z²

zx – y²
xy – z²
yz – x²

xy – z²
yz – x²
zx – y²

x
y
z

y
z
x

z
x
y

(0)

(a)

If x√1 + y + y √1 + x = 0 the show that

(1 + x)²

+ 1 = 0. Consider x ≠ y.

(0)

(b)

A wire of length 4 cm is to form a rectangle. Find the dimentions of the rectangle so that it has maximum area.

(0)

(c)

Examine for maximum and minimum values of the function

ƒ(x, y) = x³ + y³ = 3xy.

(0)

(a)

Evaluate

2
∫
1

√x + x

(0)

(b)

Find the area bounded by the curve x = 3 (y – 1) (5 – y) and y–axis.

(0)

(c)

Solve the game whose pay–off matrix is

		Player B →
		B₁	B₂	B₃	B₄	B₅
Player A →	A₁	–2	0	0	5	2
	A₂	3	2	1	2	2
	A₃	–4	–3	0	–2	6
	A₄	5	3	–4	3	–6

(0)

(a)

Obtain an initial BFS to the given TP by VAM:

	D₁	D₂	D₃	D₄	Supply
O₁	50	60	100	50	20
O₂	100	40	70	50	38
O₃	90	70	30	50	16
Demand	10	18	22	24

(0)

(b)

Using simplex method, solve the following LPP;
Maximise z = 3x₁ + 2x₂
Subject to x₁ + x₂ ≤ 15 and 2x₁ – x₂ ≤ 5 (x₁, x₂≥ 0.

(0)

(c)

In a service department manned by one server, on an average one customer arrives every 10 minutes. It has been found out that each customer requires 6 minutes to be served. Then determine

(i)
(ii)
(iii)

Average queue length;
Average time spent in the system; and
The probability that there would be 2 customers in the queue.

(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.

2x5=10

(a)

If P(A) = 0.5, P(B) = 0.4, P(A∪B) = 0.70 then P (A/B) is

(i)

0.3

(ii)

0.4

(iii)

0.5

(iv)

0.6

(0)

(b)

A problem in Statistics is given to three students whose chances of solving it are

respectively. The chance that the problem will be solved is

(i)

(ii)

(iii)

(iv)

(0)

(c)

If ƒ(x) = k, x – 1, 2, 3, 4, 5, 6
= 0, elsewhere
is a probability mass function, then the value of k is

(i)

(ii)

(iii)

(iv)

(0)

(d)

The variance of the probability distribution

x	0	1	2
P	¼	½	¼

(i)

0.25

(ii)

0.50

(iii)

0.75

(iv)

(0)

(e)

If the probability of a male birth is 0.5, then the probability that in a family of 4 children there will be at least 1 boy is

(i)

(ii)

(iii)

(iv)

None of these.

(0)

(f)

If X is a Poisson variate with parameter 1, then Prob. (2 < X < 4) is

(i)

0.0613

(ii)

0.1603

(iii)

0.0062

(iv)

0.6221 (given e^–1 = 0.3679.

(0)

(g)

The standard deviation of the normal distribution whose pdf is given by ƒ(x)=

√π

e^–25x²– ∞ <x< ∞, is

(i)

5√2

(ii)

5√3

(iii)

2√5

(iv)

3√5

(0)

(h)

If u + 3x = 5, 2y – v = 7 and r_xy = 0.12, then r_uv is

(i)

– 0.12

(ii)

0.12

(iii)

– 0.21

(iv)

0.21

(0)

(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)

(ii)

(iii)

(iv)

(0)

(j)

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

(i)

(13.71, 16.29)

(ii)

(14.02, 15.98)

(iii)

(13.71, 14.12)

(iv)

(14.14, 16.31)

(0)

(a)

The chance that doctor A will diagnose disease B correctly is 60%. The chance that a patient will die by his treatment after correct diagnosis is 40% and the chance of death by wrong diagnosis is 70%. A patient of doctor A who had disease B died. What is the chance that his disease was correctly diagnosed?

(0)

(b)

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

(0)

(a)

Fit a Poisson distribution to the following data and calculate the theoretical frequencies, given e^–0.5 = 0.61.

Number (x):
Number (y):

0
122

1
60

2
15

3
2

4
1

(0)

(b)

A normal variable has mean 165 and sd 7. If the probability that the variable exceeds a particular value is one in one–thousand, find the particular value.

(Given

3.1
∫
–3.1

Φ (t) dt = 0.998 where Φ (t) is standard normal p.d.f)

(0)

(a)

Calculate Pearson’s coefficient of correlation from the following data taking 75 and 125 as the assumed average of X and Y respectively.

X :
Y :

78
125

89
137

97
156

69
112

59
107

79
136

61
123

61
108

(0)

(b)

The lines of regression of y on x and x on y are respectively y = x + 5 and 16x = 9y – 94. Find the variance of x if the variance of y is 16. Also find trhe covariance of x and y.

(0)

10.

(a)

For a random sample of size 10 from a normal population the mean is 12.1 and the s.d. is 3.2. Is it reasonable to suppose that the population mean is 14.5? Test at 5% significance level (clearly state the null and alternative hypotheses and assumptions).

[Given t_0.025 at 9 df. = 2.262, t_0.05 at 9 df. = 1.833]

(0)

(b)

A coin is tossed 400 times, which turns up head 219 times. Do the data justify the hypothesis of an unbiased coin?

(0)

11.

(a)

A population consists of four numbers 0, 2, 4, 6. Draw all possible samples with replacement of size from the population and hence find the sampling distribution of sample mean.

(0)

(b)

A retailer has to decide as to the optimum number of units to be stocked of a certain item under the following conditions:

(i)

Cost price in season : Rs. 12;

(ii)

Selling price in season : Rs. 18;

(iii)

Bargain price after the season : Rs. 9;

(iv)

Cost of holding an item beyond the season : Re. 1;

(v)

The probability distribution of demand based on past data is as follows:

Units demanded
Probability

:
:

7
0.20

8
0.20

9
0.25

10
0.15

11
0.20

Determine the optimum stock level based on expected monetary value criterion.

(0)

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.

(0)

(b)

If the demand law is P =

x + 1

, find the elasticity of demand in terms of x

(0)

(c)

For the functions x₁

P₁

P₂

and x₂

P₁²

P₂

, examine the nature of the commodities.

(0)

(d)

If the seasonal index is 90 and production is 28, what is the deseasonalised value?

(0)

(e)

The daily cost of production C for x units of an assembly is given by C(x) = Rs. 12.5x + 6400 and selling price of each unit is Rs. 25. If now the selling price is reduced by Rs. 2.5 per unit, what will be the break–even point?

(0)

(f)

During a certain period the cost of living index goes up from 110 to 200 and the salary of a worker is also raised from Rs. 330 to Rs. 500. Does he really gain?

(0)

(g)

For the Cobb–Douglus production function Q = 4√KL, find the share of K in Q and that of L in Q.

(0)

(h)

If r₁₂, r₂₃, r₁₃ are all equal, then what is the value of r_13.2?

(0)

13.

Answer any four of the following:

4x5=20

(a)

The demand function for a particular brand of pocket calculator is given by = = 75 – 0.3Q – 0.05Q². Find the consumer’s surplus at a quantity of 15 calculators.

(0)

(b)

Determine the trend line by method of least squares from the data of sales of a company given below:

Year Sales	: :	1986 70	1987 74	1988 80	1989 86	1990 90
(in Rs. lakh)

Estimate the sales of that company in 1991. What is the slope of the trend line?

(0)

(c)

In a certain trivariate distribution S₁ = 3, S₂ = S₃ = 5, r₁₂ = 0.7, r₂₃ = r₁₃ = 0.6.
Find (i) the partial correlation coefficient r_12.3’ and
(ii) Regression coefficient b_12.3

(0)

(d)

The following table gives the CLI for different groups for the year 2008 (base year : 2000), (Figures are hypothetical)

Food
Clothing
Fuel & light
House rent
Miscellaneous

:
:
:
:
:

500
450
400
350
300

If the weights of above commodities in order are in the ratio of 10 : 1 : 2 : 3 : 4, the obtain the overall CLI. If the person was earning Rs. 5000 in the year 2000 then what should be his salary in 2008 to maintain the same standard of living as in 2000?

(0)

(e)

The coefficient matrix of input is

[

0.4
0.7

0.1
0.6

]

and the vector of final demand is

[

50
100

]

Find the gross production.

(0)

(f)

Write short notes on any one of the following:

(i)

Scatter diagram,

(0)

(ii)

Components of Time series.

(0)

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