CWA/ICWA Inter :: Quantitative Methods : June 2007

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).
1. Attempt any five of the following: 2x5=10
(a) Find the value of t for which the vectors 3i + j + tk and 2i − 2j + 4k are perpendicular to each other.
(b)
Evaluate:   2
— 4
3		 1
— 2
0		 — 3
0
2		  
(c)
If f(x) =
x − 1
x + 1
 , show that
f(x) − f(y)
1 + f(x) f(y)
 =
x − y
1 + xy
(d)
If y = log√x + √x2 + a2, find
dy
dx
(e)
If y = Aemx + Be–mx find the value of k such that
d2y
dx2
 + ky = 0.
(f)
Evaluate:     1
∫
  0
x2 – x – 6
x + 22 – 3x – 6
dx.
(g) The MR function of output x is MR = 3 (x + 1)2 + 3. Find the total revenue function.
(h)
If A = ( 1
2		 3
4		 ) , find AA⁄.
(i)
Evaluate:lim-----------------------------not completed
x → 0
(j) A function is defined as follows:
f(x) =
=		 x2, x # 2
2,   x = 2
Is f(x) continuous at x = 2?
2. (a) Find the area of the triangle OAB (O being the origin). A = (1,2,3) and B = (–3, –2, 1). 5
(b) Show that
yz – x2zx – y2xy – z2   =xyz2
zx – y2xy – z2yz – x2yzx
xy – z2yz – x2zx – y2zxy
 5
(c) A radio manufacturing company produces two types of items:

TypeManufacturing Cost (in Rs.)
A
B		280
310

During a week the company produces 128 items, the total cost of these items being Rs. 37,280. Using matrix algebra, find the number of A type and B type items which have been produced during the week.

 5
3 (a)
Evaluate:lim-----------------------------not completed
h → 0
 5
(b)
If y = x1/x + (
1
x
 )x, find
dy
dx
5
 
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I-12(QNM)
Revised syllabus		 Marks

(c)
Evaluate: ∫
xex
(1 + x)2
 dx.
 5
4. (a) Determine the points of inflexion of the function f(x) = x4 − 3x3 + ex2 + 5x + 11. 5
(b)
If u =
x3 + y3
x + y
 prove that x
∂u
∂x
 + y
∂u
∂y
 = 2u.
(c) Solve graphically the following LPP:

Minimize Z = 4x + 6y

subject to the constraints 2x + 3y ≥ 6, x + y ≤ x, y ≥ 1 and (x, y) ≥ 0.

 5
5. (a) Solve the game by using dominance property

Player B
IIIIII
Player A 1
2
3		 2
7
7		 8
3
2		 3
8
7

5
(b) Determine the initial basic feasible solution to the following Trasportation Problem by Vogel's Approximation Method.

To
D1D2D3Capacity
From F1 10 7 8 45
F2 15 12 9 15
F3 7 9 12 40
Demand 25 55 20

5
(c) In a tool crib manned by a single assistant, operators arrive at the tool crib at the rate of 10 per hour. Each operator needs 3 minutes on an average to be served. Find out the loss of production due to waiting of an operator in a shift of 8 hours if the rate of production is 100 units per shift. 5
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.
 
6. Attempt any five of the following

Choose the correct alternative stating proper reason:

 2x5=10
(a) If A and B be two independent events such that P(A) = 0.4, P(A ∪ B) = 0.7 then the value of P(B) is
(i)
1
2
 (ii)
1
3
 (iii)
1
4
 (iii)
1
6
 
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I-12(QNM)
Revised Syllabus

(b) If A, B and C be mutually exclusive and exhaustive events such that 1/3 P(C) = ½ P(A) = P(B) then the value of P(B) is
(i)
1
6
 (ii)
1
3
 (iii)
1
2
 (iii)
1
5
(c)
If P(A) =
1
3
 , P(B) =
3
4
 and P(A ∪ B) =
11
12
 , then P(A/B) is
(i)
1
9
 (ii)
2
9
 (iii)
1
3
 (iv)None of these
(d) If 6 fair coins are tossed simultaneously, then the probability of getting at least 4 heads is
(i)
7
32
 (ii)
9
32
 (iii)
11
32
 (iv)None of these
(e) For the following probability distribution
x :123
p(x) :½1/31/6
the variance is
(i)
5
3
 (ii)–
5
6
 (iii)
5
9
 (iii)
5
12
(f) If the distribution of a random variable X is normal with mean 0 and variace 1 with P(0 ≤ X ≤ 1) = 0.3413, then P(| X | > 1) is

(i)0.6587, (ii)0.3413, (iii)0.3174, (iv)0.6826.

(g) If the correlation of a coefficient between x and y is 0.4, then the correlation coefficient between -3x and +5y is

(i)0.4, (ii)−0.4, (iii)0.3, (iv)0.5.

(h) If a random sample of size 5 is drawn without replacement from a finite population of 41 units with σ = 10, then the SE of sample mean is

(i)√2 (ii)2√2 (iii)3√2 (iv)None of these.

(i) A random sample of size 100 has mean 15, the population variance being 25, then the 95% confidence interval for the population mean is

(i)(13.71, 16.29), (ii)(14.02, 15.98), (iii)(13.71, 14.12), (iv)(14.16, 16.31).

(j) To test the null hypothesis H0: μ = 0 against H1; μ # 0, 10 independent sample observations are drawn from a normal population with mean μ and unknown variance σ2. If the sample mean and variance are 6 and 4 respectively, the value of the appropriate test statistic is

(i)6, (ii)4, (iii)10√3 (iv)9.

7. (a) A purse contains 4 nickel coins and 9 copper coins, while another purse contains 6 nickel and 7 copper coins. A purse is chosen at random and a coin is drawn at random from it. What is the probability that it is a nickel coin? 5
(b) A problem is given to 5 students and their chances of solving it are ½, 1/3, ¼, 1/5 and 1/6. What is the probability that the problem will be solved? 5
8. (a) A random variable X is binomially distributed with mean 4 and s.d. = &radic2.4. Find the probability that more than half the trials are successes. 5
(b) A book contains 100 misprints distributed at random throughout its 100 pages. What is the probability that a page observed at random contains at least three misprints? [Given e = 2.718] 5
9. (a) Calculate the coefficient of correlation for the following data;
x
y		:
:		1
9		2
8		3
10		4
12		5
11		6
13		7
14		8
16		9
15
 5
(b) The regression lines of two correlated variables x and y are 5x — 6y + 90 = 0 and 15x — 8y — 130 = 0. Identify the regression lines x on y and y on x. Find the means and the correlation coefficient of the variables. 5
 
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I-12(QNM)
Revised syllabus		 Marks
10. (a) The coefficient of rank correlation of the marks obtained 10 students in two particular subjects was found to be 0.8. It was then detected that the difference in ranks in the two subjects obtained by one of the students was wrongly taken to be 7 in place of 9. What should be the correct rank correlation coefficient? 5
(b) In 120 throws of a single die the following distribution of faces was observed:
Face
Frequency		:
:		1
30		2
25		3
28		4
12		5
11		6
13		7
14		8
16		9
15
 5
11. (a) Suppose a decision maker faced with three decision alternatives and two states of nature. Apply (1) maximin and (ii) minimax regret approach to the following pay-off table to recommend the decision.
State of nature
ActS1S2
A11217
A22214
A33213
 5
(b) In one sample of 8 observations, the sum of squares of the deviation of the sample values from the sample mean was 84.4 and in the other sample of 10 observations, it was 102.6. Test whether these differences are significant at 5% level, given that F0.05;(7,9) = 3.29 5
 
SECTION III(Economic Techniques — 30 marks)
 
12. Attempt any five of the following: 2x5=10
(a) The demand and supply function under perfect competition respectively are y = 16 - x2 and
y = 2(x2 + 2). Find the market price.
(b) If the demand law is p = 1 + x, find the elasticity of demand in terms of x.
(c) The total daily cost (in Rs.) for producing x plastic chairs is TC = 25x + 4500. If each chair sells for Rs. 40, find the break even point.
(d) The price and quantity of video recorder in the year 2004 are Rs. 438 and 37 respectively. The corresponding values for the year 2005 are Rs. 462 and 18. Find the price relative and quantity relative for 2005.
(e) Convert fixed base index (FBI) number of the following three years into corresponding chain base index numbers.
Year
FBI		:
:		2001
80		2002
88		2003
105.6
(f) For the Cobb - Douglus production function Q = 3K1/5 L4/5, write down the relative share of K and that of L.
(g) If r12, r23 and r13 are all equal, then what is the value of r13.2?
(h) Determine the normal equations for fitting the exponential curve y = abx (where a, b are constants) to a set of observed data by the method of least squares.
 
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I-12(QNM)
Revised Syllabus

13. Answer any four of the following: 5x4=20
(a)
Show that the elasticity of demand =
AR
AR – MR
 where AR and MR are respectively
average and marginal revenue at any out put. Verify this for linear demand law p = a + bx.
(b) Below are given the figure of production (in '000 tonees) of a sugar factory.
Year
Production ('000 tonnes)		:
:		1991
77		1992
88		1993
94		1994
85		1995
91		1996
98		1997
90

 (i) Fit a straight line by the method of "least squares" and find the trend values.

(ii) What is the months increase in production.
(c) Calculate the index number of price for 2005 on the basis of 2000 from the given data;
CommodityWeightPrice per unit
2000
(Rs.)		Price per unit
2005
(Rs.)
A
B
C
D		40
25
20
15		16
40
12
 2		20
50
15
 3

If the weights of commodities A, B, C, D are increased in the ratio 1 : 2: 3: 4 what will be increase in index number?
(d) Given r12 = 0.8, r13 = 0.6; find the maximum possible value of r23.
(e) Find the 5 year weighted moving average with weights 1, 2, 2, 2, 1 respectively for necessary trend of the following time series:
Year
Sales (in lakh Rs.)		:
:		1984
22		1985
26		1986
21		1987
25		1988
23		1989
27		1990
22		1991
26		1992
24		1993
26
(f) 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 1000 and 2000 units respectively.
IndustryInputFinal Demand
12
1
2		200
100		250
500		50
400

__________

 

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