CWA/ICWA Inter :: Quantitative Methods: June 2003

*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)

= 2i + 3j + K,

= i + 4j + 2K, find

(b)

Find the value of the determinant

3
1
1

2
1
− 3

− 1
1
− 1

(c)

If A =

(

3
3

2
3

)

and B =

(

1
− 1

− 2
1

)

find AB

(d)

If A =

(

1
1

1
− 1

)

find out A^{− 1}

(e)

Evaluate :

Lim
x→∞

6 − 5x²

4x + 15x²

(f)

A function f(x) is defined as follows:

f(X)

=
=
=

2x — 1,
k,
8 — x,

x < 3
x = 3
x > 3

For what value of K, f(x) is continuous at x = 3?

(g)

If y = x², find

(h)

The average cost function (AC) for a certain commodity is given by

AC = 2x − 1 +

in terms of output x. Find marginal cost.

(i)

Find the function whose derivative is

x²

x + 1

(j)

Examine f(x) = x³ — 6x² + 9x — 8 for maximum and minimum values

(a)

If f(x) =

x − 1

x + 1

then show that f{f(

(x)

)} =

f{f(

f(x)

)} for x ≠ 0, 1, 0, 1

(b)

Find a vector α which is perpendicular to both 4

+ 5

−

and

− 4

− 5

which satisfies the relation

α.β = 21 where β = 3

−

(c)

Solve the equations by inverse matrix 2x + y + Z = 1; x — 2y — 3z = 1 and 3x + 2y + 4z = 5.

(a)

If a ≠ b ≠ c and

a
a
c

a²
b²
c²

1 + a²
1 + b²
1 + c²

show that abc + 1 = 0

(b)

If y = log(x +

√

a² + x²

) then prove that (a² + x²)

d²y

dx²

+ x

(c)

Evaluate :

(a² + x²)

d²y

dx²

+ x

Please turn over

( 2 )

I-12(QNM)
Revised syllabus

Marks

(a)

If f(x, Z) = x² − 5xy + y², then show that

∫

t(t³ + 1)

(b)

Find the area bounded by the parabola y = 8(x — 1)(4 — 1) and the x-axis

(c)

Using graph, maximise f = x + 3y subject to
2x + y = 20, x + 2y < 20, x> 0, y> 0

(a)

Solve the following LPP by simplex method
Maximize z = 7x + 5y subject t x + 2y < 6, 4x + 3y < 12 (x,y > 0)

(b)

Obtain initial B.F.S. to the transportation problem (T.P.) by NOrth-West corner method :

To	→	W₁	W₂	W₃	W₄	Capacity
From	↓
	F₁	21	16	25	13	11
	F₂	17	18	14	13	16
	F₃	32	27	18	41	19
	Demand	6	10	12	15	43

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. Choose the correct alternative, stating proper reason.

2x5=10

(a)

If P(A) = 0.4, P(B) = 0.7, P( at least one of A and B) = 0.8 then p (only one of A and B) is

(i)

0.2

(ii)

0.5

(iii)

0.7

(iv)

(b)

Two unbiased dice are thrown. The probability that the sum of the faces is not less than 10 is

(i)

(ii)

(iii)

(iv)

(c)

6 unbiased coins are tossed simultaneously. The probability of getting at least 4 heads is

(i)

(ii)

(iii)

(iv)

(d)

If P(A) =

P(B) =

and P(A ∪ B) =

then P(A/B) is

(i)

(ii)

(iii)

(iv)

none of these

(e)

Number of observations of a binomial distribution with mean = 4 and variance = 8/3 is

(i)

(ii)

(iii)

(iv)

(f)

If a random variable x follows a Poisson distribution such that prob.(X = 1) = prob.(x = 2) then prob.(x = 0) is

(i) e^—4 (ii) e^—3 (iii) e^—2 (iv) e^—1

Please turn over

( 3 )

I-12(QNM)
Revised Syllabus

(g)

Given Σx = 21, Σ y = 20, Σ x² = 91, Σ xy = 75\4, n = 0 from these data, the regression coefficient which you think to be fit is

(i)

0.481

(ii)

0.489

(iii)

0.948

(iv)

0.498

(h)

If a random sample of size 5 is drawn without replacement from a finite population of size 41 units with standard deviation as 10, then the standard error of the sample mean is

(i) √2 (ii) 2√2 (iii) 3√2 (iv) none of these

(i)

A simple random sample of size 100 has mean 10, the population variance being 16. Then the 95% confidence interval for the mean is

(i) (9.22, 10.78) (ii) (9.22, 11.78) (iii) (10.22, 10.78) (iv) (10.22, 11.78)

(j)

If v(x) = v(y) = 1/4 and v(x — y) = 1/3 then the correlation coefficient between random variables x and y is

(i)
1
3
(ii)
-1
3
(iii)
2
3
(iv)
-2
3

(a)

A class consists of 60 students out of which the number of girl students is 10. In the class 2 girls and 6 boys are rank holders in the previous examination. If a student is selected at random from the class and is found to be rank holder. What is the probability that the student selected is a girl?

(b)

The probability that Arun can solve a problem of Geometry is 3/7, Barun can solve it is 2/3 and Mithun can solve it is 4/5. If all of them try independently. find the probability that the problem will be solved

(a)

A box contains 10 screws of which 5 are defectives. Obtain the probability distribution of the number of defective screws (x) in sample of 4 screws chosen at random and variance(x)

(b)

If the distribution of marks received in an examination is normal. 44% of candidates got marks below 61.4% of candidates got marks above 80, then find the percentage of candidates having got marks above 65 marks.

[Given

√2π

π
∫
0

e^{-1/z t²} dt = 0.06, 0.10, 0.46 according as x = 0.15, 0.25]

(a)

Following are the marks obtained by 8 students in Mathematics and Statistics papers in an examination. Calcualte the spearman's rank correlation coefficient

Marks in Maths
Marks in Statistics 25
50 30
40 38
60 22
45 50
30 70
20 35
48 90
70

Please turn over

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I-12(QNM)
Revised syllabus

Marks

(b)

The equation of two regression lines between two variables are expressed as 2x - 3y = 0 and 4y - 5x - 8 = 0.

(i) Identify which of two can be called regression of y on x on y?

(ii) Find the means of the variates.

(iii) Determine r_xy and justify its sign (+/-).

10.

(a)

A die is thrown 120 times with following results

Face
frequency

1
16

2
30

3
22

4
18

5
14

6
20

[Given χ²_0.05,5 = 11.07 and χ²_0.05,6 = 12.59] Test the hypothesis that the die is unbiased.

(b)

For a random sample of size 10 from a normal population, the mean is 10.4 and the s.d. is 2.8. Is it reasonable to suppose that the mean is 12.3? Test at 5% significance level (clearly state the null and alternative hypothesis and assumptions). [Given t_0.025 at 9 d.f. = 2.262, t_0.05 at 9 d.f. = 1.833]

11.

(a)

Apply (i) Maximin (ii) Maximax (iii) Minimax regret to the following pay-off matrix:

States of Nature	→	S₁	S₂
Acts	↓
	A₁	—5	3
	A₂	4	2
	A₃	8	—3

(b)

For a population comprising the values 1, 3, 5, 7, 9 list all possible sample of size 2 without replacement and show that mean is equal to the population mean

SECTION III(Economic Techniques — 30 marks)

12.

Attempt any five of the following:

2x5=10

(a)

If 12½% fall in price causes a 25% rise in demand, then find price elasticity of demand and its nature.

(b)

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.

(c)

Determine price index for the year 2000 w.r.t. 1990 from following data

Price/Unit (Rs.)
commodity	weight	1990	2000
A B C	20 20 10	10 40 5	10 5 10

(d)

Taking 1960 as base year, CLI of the year 2001 was 250. A person drawing monthly salary of Rs. 750 in 1960 drew Rs. 2000 in 2001 and approached management for increased salary to maintain his standard of living. Should his demand be met and to what extent?

(e)

Given r₁₂ = 0.5, r₂₃ = 0.7 find r₂₃₁

Please turn over

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I-12(QNM)
Revised Syllabus

(f)

If r₁₂ = 0.7, r₂₃ = 0.4 and r₃₁ = 0.6, then calculateR₁₂₃

(g)

With which component of a time series would you associate each of the following?

(i)	Due to natural calamity institutions, officers are closed for a day or two.
(ii)	Decline of ice-cream sales during winter season.

(h)

The trend equation is y = 15 + 0.9141 t with the origin at the midpoint of 1989 and 1990 and t unit = ½ year. Find the slope and trend value of 1993.

13.

Answer any four of the following

5x4=20

(a)

The demand function for a particular brand of pocket calculator is p = 75 — 0.3Q — 0.05Q². Find the consumer's surplus as a quantity of 15 calculators.

(b)

From the following series of observations series of observations, calculate the three-yearly moving averages with weights 1, 2, 3 respectively:

Year
Value 1
2 2
4 3
5 4
7 5
8 6
10 7
13

(c)

Calculate the price index number of 1996 with 1986 as base year by using (i) Laspeyre's (ii) Paasche's and (iii) Fisher's formulae:

Commodities Unit 1986 1996

Price
(Rs.) Value
(Rs.) Qty.
(kg.) Value
(Rs.)

A Kg 10 1500 160 1760

B Kg 12 1080 100 1300

C Metre 150 900 60 960

D Packets 9 450 40 480

(d)

Fit a trend line of the following data of sales of a commodity in a shop using least square theory and estimate the volume of sales in 1997:

Year : 1990 1991 1992 1993 1994

Vol of sales (thou. Tonnes) : 24 2.8 3.1 3.6 4.2

(e)

Given the input matrix and the final demand vector:

[

0.07
0.32
0.18

0.28
0.12
0.36

0.35
0.74
0.00

]

, D =

[

1600
250
850

]

(i)	Explain the economic meaning of the elements 0.32, 0.00, 250
(ii)	Explain the economic meaning, if any of the second column sum.

(f)

Write short notes on any one of the following:

(i)	Stratified sampling.
(ii)	Cost of living index.
(iii)	Parameter and statistic.

__________

CWA/ICWA Inter :: Quantitative Methods : June 2003