CWA/ICWA Inter :: Quantitative Methods : December 2005

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 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.
1. 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|.
(b)
If i + 2j + 3k and 2i + 4j + 4k be the position vectors of points A and B, find the unit vector alongAB
(c)
If f(x) = eax + b then show that
f(l) f(m) f(n)
f(l + m + n)
 = e2b
(d) Draw the graph of f(x) =[x] for —1 < x < 1.
(e)
Evaluate lim
x→1
x2 − 3x + 2
x2 − 5x + 4
(f) Differentiate x5 w.r.t x2.
(g)
If y = 3x log x, prove that x (
dy
dx
 − y log3 ) =3x
(h) Find the slope of the curve log xy = x2 + y2 at the point (1,1).
(i)
Evaluate ∫xex
(j)
If f(x, y) = ex2 + y2 find fxx
2. (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.

 5
(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)
5
(c) Solve by Cramer's rule: x + y +z = 24, 2x + 3Y + z =2 and 8y — 3z = 2. 5
3. (a)Verify that the function u = 6x2 — 2y2 — 12x — 16y has a saddle point. 5
(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. 5
(c)
If y = ( x + √ x2 + 4 )  n , show that (x2 + 4)3/2
d2y
dx2
 + n (x − n √ x2 + 4 ) y = 0.
5
 
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I-12(QNM)
Revised syllabus		 Marks
4. (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.

5
(b)
Evaluate ∫
x4 − 1
x3
 ex + 1/x dx
5
(c)
Find the area of the region bounded by the curves y = x2 and y = √x
 5
5. (a) Maximize z = 5x1 + 7x2 subject to 2x1 + 3x2 < 13, 3x1 + 2x2 < 12, x1 > 0, x2 > 0 by the simplex method. 5
(b)

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

 5
To→D1D2D3Supply
From↓
O112191614
O219132216
O38281412
Demand17151042
(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.

 5
 
SECTION II(Statistical Techniques — 30 marks)
Answer Question No. 6 (compulsory — 10 marks) and two
other questions (10x2 = 20 marks) from this section.
 
6. 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)
1
4
 (ii)1 (iii)
1
2
 (iv)
1
3
(b)
If P(A) =
2
3
, P(B) =
1
2
and P(A/B) =
1
6
, then the value of P(B/A) is
(i)
1
2
 (ii)
1
3
 (iii)
1
4
 (iv)
1
6
(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
(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)
1
2
 (ii)1 (iii)
1
4
 (iv)0
(e)
If for a binomial distribution, P (success in a trial) =
1
3
, P (no failure) =
1
81
then the probability of atleast one success is
(i)
65
81
 (ii)
16
81
 (iii)
32
81
 (iv)
80
81
 
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I-12(QNM)
Revised Syllabus

(f)For a Poisson distribution with mean 4, the coefficient of variation is
(i)200% (ii)100% (iii)50% (iv)25%
(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
(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
(i)
In order to test H0 :p =
1
2
against H1 :p =
1
3
, 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)
17
81
 (ii)
16
81
 (iii)
1
81
 (iv)
20
81
(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		 (2&pi)&frac 12; 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)
7. (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.

 5
(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?

 5
8. (a)

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

 5
(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)

 5
where it is given that ∞
∫
1.96		 (2&pi)&frac 12; e-t/2 dt = 0. 025) is
9. (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.

 5
(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.

 5
 
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I-12(QNM)
Revised syllabus		 Marks
10. (a) Pay-offs of acts X,Y and Z and the states of nature L, M and N are as follows : 5
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.
(b)

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

 5
xy
Mean53142
variance130165
&Sigma(x − x ) (y − y ) = 1220
Find the regression equation of y on x and use it to estimate the blood pressure of a woman of age 45.
11. (a)

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

 5
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 t0.05,4 = 2.13.
(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.

 5
 
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.
(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.
(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.
 
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( 5 )

I-12(QNM)
Revised Syllabus

(d)

Show that Paasche's price index number can be written as weighted harmonic mean of price relatives.
(e)
If Σ
i		 pni qni = 979 and Laspeyre's price index number is 110%, Σ
i		 poi qoi find the sum
(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
(g) Find r23 if r23.1 = 0.6 and r 13 = 0.5.
(h)

If the multiple regression equation of X1 on X2 and X3 is X1 = 0.6 X2 + 0.43 — 14, r13.2 = 0.8 and V(X3) = 16, find V(X1).
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.
(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:199019911992199319941995199619971998199920002001
Profit(in lakh Rs.):463495479481513494493528518514540500
(c) Calculate the price index number by Fisher's method:
year   →2000 (base)2004 (current)
CommodityPrice (Rs.)Total value (Rs.)Price (Rs.)Total value (Rs.)
↓
A1010012120
B1214414196
C1414016192
D1619218216
(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.
(e) Given the following input-output table:
productionConsumerFinal demand
SteelCoal
Steel (in ton)0.30.140
Coal (in ton)0.60.480
Labour (mandays)63
Find the gross production of steel and coal and also the total labour required.
(f) Writer short notes on any one of the following:
(i)Elasticity of demand.
(ii)Partial correlation coefficient.
(iii)Least square theory.

__________

 

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