No.

of Printed Pages : 7 MEC-003

MASTER OF ARTS (ECONOMICS)

Term-End Examination

December, 2014

MEC-003 : QUANTITATIVE TECHNIQUES

Time : 3 hours Maximum Marks : 100

Note : Answer the questions from each sections as directed.

SECTION - A
Answer any two questions from this section. 2x20=40
1. Given the demand and supply for cobweb model
as :
Qdt = 18 - 3 t and Qst - 3 + 4Pt-i
(a) Find the inter temporal equilibrium price
and determine whether the equilibrium is
stable.
(b) Establish the stability condition of
Samuelson's multiplier accelerator model.
2. (a) Write down the distribution functions of the
Binomial distribution and Poisson
distribution. When is a Poisson distribution
an approximation of Binomial
distribution ? Obtain the mean and
variance of the Binomial and Poisson
distribution.
(b) Write down the properties of a normal
distribution. For a standard normal
distribution, write the density function.

MEC-003 1 P.T.O.
3. (a) What are the difference between open and
closed input-output models ?
(b) An economy has 2 sectors agriculture and
industry. The input-output coefficients of
these sectors are given as :
Input sector
Output sector
Agriculture Industry
Agriculture 0.10 0.50
Industry 0.20 0.25
(i) If the final demand of these sectors are
300 and 100 respectively, determine
the gross output of the two sectors.
(ii) If the input coefficients for labour of
the two sectors are 0.50 and 0.60
respectively, determine the total labour
that would be required.
4. A revenue maximising monopolist requires a
profit of at least 1500. His cost and demand
functions are C = 500 + 4q + 8q2 and P = 304 — 2q.
(a) determine his output level and price.
(b) contrast these values with those that would
be achieved under profit maximisation.
SECTION - B
Answer any five questions from this section. 5x12=60
5. A bag contains 8 blue balls and 5 black balls.
2 successive draws of 3 balls are made without
replacement. Find the probability that the first
drawing will give 3 black balls and the second
3 blue balls.
6. (a) Solve graphically Min C = 0.6x1 + .x2
Sub to 10x1+ 4x2 > 20
5x1 + 5x2 20
2x1 + 6x2 > 12
Xi, X2 > 0
(b) Why does the solution occur at the corner
point only ? Give reasons.

MEC-003 2
7. Explain the method of maximum likelihood for
estimating the value of a population parameter.

4 1 —1
8. Find inverse of 0 3 2
3 0 7

9. (a) What is sampling distribution ?

(b) State control limit theorem.
(c) State properties of point estimators.

10. When we roll a die and are told that the number
is even ? What is the probability that it was 4 ?

11. Find the extreme values of :

2 2 2
z = 2X1 - X1 X2 4x2 x1X3 + x3 + 2 . Using
the Hession matrix check whether the extreme
value(s) is/are maximum or minimum.

12. Solve the following Linear Programming Model

in x1 and x2.
Maximise z = 45x1 + 55x2
Sub to 6x1 +4x2<-120
3x1 +10x2-L:180
x2 ?..-

MEC-003 3 P.T.O.
 No. of Printed Pages : 8 I MEC-003 I
MASTER OF ARTS (ECONOMICS)
Term-End Examination

02290 December, 2015

MEC-003 : QUANTITATIVE METHODS

Time : 3 hours . Maximum Marks : 100

Note : Answer the questions from each section as directed.

SECTION A
Answer any two questions from this section. 2x20=40

1. (a) Discuss the Hawkins-Simon condition in

the context of input-output analysis.

(b) From the following technology matrix find

equilibrium prices if the wage rate is 100
per day :

Steel Coal Final Demand

Steel 0.4 0.1 50
Coal 0.7 0.6 100
Labour 5 2

MEC-003 P.T.O.
2. (a) Given demand and supply for cobweb
model as Qdt = 18 - Pt and
Qst 3 + 4 Pt - 1
Find the intertemporal equilibrium price
and determine whether the equilibrium is
stable.
(b) Establish the stability condition of
Samuelson's multiplier-accelerator
interaction model.

3. (a) Consider the aggregate production function

Q= L1 a, where Q, K and L are all
functions of time. Depict and solve for the
time path of capital output ratio.
(b) Solve :

(t + 2y) dy + (y + 3t 2) dt = 0

4. (a) If i is the sample mean, prove that the

expected value of i , E(i) is equal to g (the
population mean).
(b) Describe the process of testing hypotheses
about population proportion of a given
attribute.

MEC-003
 SECTION B
Answer any five questions from this section. 5x12=60

5. A monopolists demand curve is P = 100 — 2Q.

(a) Find her MR function.
(b) What is the relation between slopes of AR
and MR curves ?
(c) At what price is MR = 0 ?
[AR = Average Revenue; MR = Marginal
Revenue]

6. Suppose we roll a die and are told that the

number is even. What is the probability that it
was 2?

7. Solve the following linear programming model in

x1 and x2
Maximize z = 45x1 + 55x2

subject to 6x 1 + 4x2 120, xi 0

3x1 + 10x2 180, x2 ?.. 0

8. The SD of output per acre from a sample of

34 representative firms producing paddy is 83 kg.
Is the hypothesis that SD of output per acre for all
firms producing paddy is 107 kg rejected at 5%
level of significance ? (Use large sample test)
[SD = Standard Deviation].

MEC-003 3 P.T.O.
9. What is a binomial distribution ? Find the mean
and standard deviation of it with n and p as the
two parameters.

10. Write short notes on the following :

(a) Eigenvalue and Eigenvector
(b) Rank of a matrix
(c) Taylor's expansion

4 1 -1

11. Find the inverse of 0 3 2

3 0 7

12. (a) What is a test statistic ?

(b) Distinguish between one-tailed and
two-tailed tests.
(c) What is p-value ?

MEC-003 4
 No. of Printed Pages : 8 MEC-003

MASTER OF ARTS (ECONOMICS)

Term-End Examination
06722 June, 2014

MEC-003 : QUANTITATIVE METHODS

Time : 3 hours Maximum Marks : 100

Note : Answer the questions from each section as directed.

SECTION A

Answer any two questions from this section. 2x20=40

1. A revenue maximising monopolist requires a

profit of at least 1500. His demand and cost
functions are
P = 304 — 2q and C = 500 + 4q + 8q2.
(a) Determine his output level and price.
(b) Contrast these values with those that would
be achieved under profit maximization.

2. (a) If 5E. is the sample mean, prove that the

expected value of 51 , E( ) equals the
population mean IA.
(b) Describe the process of testing hypothesis
about population proportion of a given
attribute.
MEC-003 1 P.T.O.
3. Consider the Cobb-Douglas production function
Q = A La K1 1', a > O.

Prove that
(i) it is homogeneous of degree 1.
(ii) the marginal and average productivities of L
and K, the two inputs, depend on the ratio of
the two inputs.
(iii) elasticity of substitution is unity.
4. When do you need the help of a differential
equation ? Discuss the role of initial condition in
solving differential equation. If your objective is
to examine the stability of equilibrium, show
with the help of an example, how a second order
differential equation helps address your concern.

MEC-003 2
 SECTION B

Answer any five questions from this section. 5x12=60

5. Consider the matrices

—1 —2 3 1 1 —1

P= 6 12 6 Q= 2 —3 4

5 10 5 3 —2 3

Find the rank of P, Q, P + Q, PQ, QP.

6. Suppose x has the following probability

distribution :

x 0 1 2 3 4

P(x) 0.2 0.2 0.1 0.3 0.2

Find the mean and variance of the distribution.

7. Find the inverse of the matrix

4 1 —1

0 3 2

[3 0 7_

8. A sub-committee has 6 members. It has to be

formed out of a group of 7 men and 4 ladies.
Calculate the probability that the sub-committee
will consist of
(a) exactly 2 ladies and
(b) at least 2 ladies.

MEC-003 3 P.T.O.
9. (a) What is a test statistic ?
(b) Distinguish between one-tailed and
two-tailed tests.
(c) What is p-value ?

10. (a) Find "zi ±when

cbc
(i) y = log (ex + 3)
1
(ii) y=
V x2 +a2
x,
(b) If y = , find the total differential.
xi + x2

11. What is a Binomial distribution ? Find the Mean

and Standard Deviation of the binomial
distribution with parameters n and p.
12. Write short notes on :
(a) Taylor's expansion
(b) Kuhn-Tucker condition

MEC-003
No. of Printed Pages : 7 I MEC-003 I
MASTER OF ARTS (ECONOMICS)
05
Term-End Examination
June, 2016
C;$
MEC-003 : QUANTITATIVE TECHNIQUES

Time : 3 hours Maximum Marks : 100

Note : Answer questions from each section as directed.

SECTION - A
Answer any two questions from this section : 2x20=40
1. A revenue maximising monopolist requires a
profit of at least 1500. His demand and cost
functions are :
D =304 —2Q and C =500 +4Q + 8Q 2
Determine his price and level of output. Contrast
these values with those that would be achieved
under profit maximisation.
2. (a) Write a linear first-order differential
equation and work out its general solution.
(b) How will you solve Harrod-Domar
formulation of steady growth through
differential equations ?
3. A production function is given by y = x1Y3 x2Y3 ,
where y is the output and xi and x2 are the two
inputs. If price of output P y =15 and prices of
inputs Pxi =5, Px2 = 3 then
(a) Derive profit maximising inputs ; and
(b) Verify that these inputs are _profit
maximising.

MEC-003 1 P.T.O.
4. If x1 , x2 and x3 are a random sample of size
3 from a population with mean 11, and variance
cr2 and T i, T2, T3 are the estimators used to
estimate the mean value IL where T 1 = x1 + x2 — x3;
1
T2 = 2x1 — 4x2 3x3 and T3 = —3 (ax1 + x2 + x3)
(a) Are Ti and T2 unbiased estimators of ?
(b) For what value of a, T3 will be unbiased
estimator of ?
(c) For what value of a, will T3 be a consistent
estimator ?
(d) Which of the 3 is the best estimator ?

SECTION - B
Answer any five questions from this section :
5x12=60
1 1 —1 —1 —2 3
5. A= 2 —3 4 B= 6 12 6
3 —2 3 5 10 5
Find ranks of AB, BA and A + B.

6. A subcommittee of 6 is to be formed out of a group

of 7 men and 4 ladies. Calculate the probability
that the subcommittee will have :
(a) exactly 2 ladies (b) at least 2 ladies

7. (a) Find --dd x when

I
1
(i) y = log(ex + 3 ) (ii) y—
NiX 2 +a2
(b) Find the total differential given
x1
y xi +x2

MEC-003 2
 0.2 0.3 0.2
8. Suppose A = 0.4 0.1 0.2 be the technology
0.1 0.3 0.2_

10
matrix. Let D= 5 be the final demand vector.
6
Find the level of production of the three goods.

9. Suppose a die is rolled. We are told that the

number is even. What is the probability that it
was '2' ?

10. The standard deviation of the distribution of

income of a sample of 100 household was 6970.
Test the hypothesis that the standard deviation
of the distribution of income for all households is
4700. (Use large sample test).

11. Solve the following Linear Programming Model

Max z = 45x1 + 55x2
Sub. to 6x1 + 4x2 5 120
3x1 + 10x2 5 180
0, x2 0

12. What is Poisson Distribution ? Find its mean and

variance.

MEC-003 3 P.T.O.
 No. of Printed Pages : 12 MEC-003

MASTER OF ARTS (ECONOMICS)

Term-End Examination
June, 2011

MEC-003 : QUANTITATIVE TECHNIQUES

Time : 3 hours Maximum Marks : 100

SECTION - A
Attempt any two questions from this section. 2x20=40
1. A two product firm faces the following demand
and cost functions :

Q i =40-2Pi —P2
demand function
Q2 =35-P1 -P2
c=Q12+2 rm,22
,2 +10 = cost function
(a) Find the output levels that satisfy the first
order conditions for maximum profit.
(b) Check the second order for sufficient
condition. Can you conclude that this
problem possesses a unique absolute
maximum ?
:.D (c) What is the maximal profit ?

2. (a) Given the demand and supply for cobweb

model as :
Qdt = 18 — 3Pt
Qst = — 3 + 4Pt _ 1
Find the intertemporal equilibrium price and
determine whether the equilibrium is stable.

MEC-003 1 P.T.O.
 (b) Establish the stability condition of
Samuelson's multiplier - accelerator
interaction model.

3. (a) Consider the Cobb Douglas production

function q = AL" k1 ' ; A, a > 0
Prove that :
(i) It is homogeneous of degree 1
(ii) the marginal and average
productivities of the 2 inputs L and k
depend on the ratio of the 2 inputs
(iii) elasticity of substitution is unity.
(b) Determine whether the following function
is homogeneous. If so, to what degree ?

f (x, y) = ( x2 _y2

4. (a) What is point estimation and how is it

different from interval estimation. What are
the characteristics of a good estimator.
(b) If x1, x2, ..., xn is a random sample from an
infinite 'n' size population with variance o-
2 and is the sample mean. Show

( - 22
k Xi —X)
2.1 is a biased estimator of o- 2, but
i=i
the bias becomes negligible for large n.

MEC-003 2
 (c) If x1, x, and x3 is a random sample of size 3
from a population with meanµ and variance
2 and T i, T2 and T3 are the estimators used
to estimate the mean value p. where
T, +x-2 —.113, T2 = 2X1 4r2 3x3 and

T3 = 3 [a xi +x2 + xl]

(i) Are Ti and T2 unbiased estimator

of
(ii) For what value of a will T3 be unbiased
estimator of R.
(iii) With this value of a will T3 be a
consistent estimator.
(iv) Which of the 3 is the best estimator.

MEC-003 3 P.T.O.
 SECTION - B
Answer any 5 questions from this section. 5x12=60
5. A subcommittee of 6 members is to be formed out
of a group consisting of 7 men and 4 ladies.
Calculate the probability that the sub-committee
will consist of (a) exactly 2 ladies and (b) atleast
2 ladies.

6. Consider the matrices

11 —1 —1 —2 3
A = 2 —3 4 and B= 6 12 6
3 —2 3 5 10 5

Find Rank of matrices A, B, [A + B], [A B] and

[B A]

7. (a) What is Karl Pearson's Correlation

Coefficient ? Prove that it lies between +1
and —1.
(b) In a certain examination 10 students
obtained the following marks in Maths and
Economics.

Roll No. 1 2 3 4 5 6 7 8 9 10
Maths 90 30 82 45 32 65 40 88 73 66
Economics 85 42 75 68 45 63 60 90 62 58

Find Spear man's rank correlation coefficient.

MEC-003 4
8. Marks obtained by 12 students in college test (x)
and the university test (y) are as follows.

x : 45 41 50 68 47 77 90 100 80 100 40 43
y 63 60 60 48 85 56 53 91 '74 98 65 43

What is your estimate of the marks a student

would have got in the university test if he got 60
in the college test but was ill at the time of
university test ?

9. What is a Binomial Distribution ? Find the mean

and standard deviation of a binomial distribution
with parameters n and P.

10. Solve the following problem.

Max 10x1 +10x2 + 20x3 + 20x4
Subject to 12x1 + 8x2 + 6x,, + 4x4 5_210
3x1 + 6x2 +12x3 +24x4 5..210
x 1 '• 2' •Y , •1:4 0

11. Given the input - output matrix and final demand

vector as :

0.05 0.25 0.34 1800

A= 0.33 0.10 0.12 and d 200
0.19 0.38 0.0 901)

Find the solution the output levels of three

industries.

MEC-003 5 P.T.O.
 -3
L
12. (a) Let q = — + 2L2 +12 L is the production
3
function with L = labour employed. Find
the maximum L beyond which the average
return from labour starts diminishing.

(b) Find d'Y when

dx
(i) it = log (ex + 3)
1
(ii) Y NI X2 a2

MEC-003 6.
No. of Printed Pages : 6 MEC-003

MASTER OF ARTS (ECONOMICS)

Term-End Examination
December, 2018 02341
MEC-003 : QUANTITATIVE TECHNIQUES

Time : 3 hours Maximum Marks : 100

Note : Answer questions from each section as per instructions.

SECTION - A
Answer any two questions from this section. 2x20=40

1. (a) Given the supply and demand for the

Cobweb model as :
Qst = 6 Pt-51 and Qdt =19 - 6Pt
Find the intertemporal equilibrium price and
determine whether the equilibrium is stable.
(b) Establish the stability condition of
Samuelson's multiplier accelerator
interaction model.

2. (a) Given the marginal propensity to import

M'(y) = 0.1 and also M = 20 when y = O.
Find the import function M(y).
(b) Given c'(y) = 0.8 + 0.1y -1/2 is the marginal
propensity to consume and c = y when
y=10. Find c(y) the consumption function.

MEC-003 1 P.T.O.

••••
3. (a) What is point estimation and how is it
different from interval estiMation ? What
are the characteristics of a good estimator ?
(b) If xi, x2„ xn is a random sample from an
infinite 'N' size population with variance
o-2 and y is the sample mean. Show
7)2

L, is a biased estimator of cr2, but

i =1
as 'n' becomes large, the bias becomes
negligible.

4. The input coefficient matrix P for an economy is

r0.0 0.3 0.3

given by P = 0.3 0.1 0.1 and the final
0.2 0.4 0.0_

180
demand vector D = 20 Find the output levels.
80

• SECTION - B
Answer any five questions from this section :
5x12=60
5. Solve Max z = 50yi + 30y2
sub to + y2 — 9 0
0 12 — 2yi — y2
Y2 °

6. Write notes on (i) Eigen value and Eigen vector

(ii) Rank of a matrix

1
7. Find the solution to Yt+i — yt = 5 for yo= 2.
4

MEC-003 2
8. A monopolist's demand curve is given by
P =100 — 2q.
(a) Find his marginal revenue function.
(b) At what price is marginal revenue zero ?

9. What is a Binomial distribution ? Find its mean

and variance given that the parameters of the
distribution are n and p.

10. Assume that on an average one telephone number

out of 15 is busy. Find the probability that if 6
randomly selected telephone numbers are picked
up,:
(a) not more than .5 are busy
(b) at least 3 of them are busy

11.. Find the inverse of the following matrix.

4 1 —1-
A= 0 3 2
3 0 7

12. Firid the rank of matrices P, Q, P + Q, PQ and QP

given.
_
-1 —2 3- 1 1 —1
P = 6 12 6 Q = 2 —3 4
5 10 5 3 —2 3

MEC-003 3
 No. of Printed Pages : 8 MEC-003

MASTER OF ARTS (ECONOMICS)

Term-End Examination
wzr December, 2012
Cr)
N-
Lt) MEC-003 : QUANTITATIVE TECHNIQUES
—Time : 3 hours Maximum Marks : 100

Note : Answer the questions from each section as directed.

SECTION-A
Answer all the questions from this section. 2x20=40
1. Suppose an economy has two consumers A and
B and two commodities 1 and 2. The endowments
of the two agents are respectively

WA = (WA, \A/3. ) and WB = . The

utility functions are :
uA = (xi )a (x2A )1 — a

b (4)1- b
and UB = (X0

with 0 < a, b < 1,

(a) Solve the consumer demand functions.
(b) Use the feasibility conditions to solve for the
relative prices.
OR
(a) Write a linear first - order differential
equation and work out its general solution.
(b) How will you solve Harrod Domar
formulation of steady growth through
differential equations ?

MEC-003 1 P.T.O.
2. (a) Write down the distribution functions of the
binomial distribution and poisson
distribution. When is a poisson distribution
an approximation of binomial
distribution ? Obtain the mean and
variance of the binomial and poisson
distribution.
(b) Write down the properties of the normal
distribution. For a standard normal
distribution, write the density function.
OR
(a) If -
x is the sample mean, prove that the

expected value of -
x , E (-
x ) equals the
population mean (11).
(b) Describe the process of testing hypothesis
about population proportion of a given
attribute.

MEC-003 2
 SECTION-B
Answer any five questions from this section.
5x12=60
3. Find the inverse of the matrix
3 2
3 10 6 •
A= [1
2 5 5

4. (a) What is sampling distribution ?

(b) State the Control Limit Theorem.
(c) State the properties of point estimators.

5. Suppose the technology matrix is

0.2 0.3 0.2
A. 0.4 0.1 0.2
0.1 0.3 0.2

10

Let the final demand sector be D= 5

6
Find the level of production of the three goods.

6. From the following data, obtain the two regression

equations Y on X and X on Y.
X 2 4 6 8 10
Y 5 7 9 8 11

7. A monopolist's demand curve is given by

P=100-2q.
(a) Find his marginal revenue function.
(b) At what price is marginal revenue zero ?

MEC-003 3 P.T.O.
 (c) What is the relationship between the slopes
of the average and marginal revenue
curves ?

8. Suppose x has the following probability

distribution.
x 0 1 2 3 4
P(n) 0.2 0.2 0.1 0.3 0.2
Find the mean and variance of the distribution.

9. (a) The supply function of a certain commodity

is :
Q = a + bP2 + (a < 0, b > 0) where R is
rainfall. Find the price elasticity of supply.
(b) Find the total differential, given :
x1
x1 + X2 •

MEC-003 4
 No. of Printed Pages : 7 MEC-003

MASTER OF ARTS (ECONOMICS)

Term-End Examination
,r)
December, 2013

MEC-003 : QUANTITATIVE TECHNIQUES

Time : 3 hours Maximum Marks : 100

Note : Answer the questions from each sections as directed.

SECTION - A
Answer any two questions from this section : 2x20=40
1. A production function is given by y = x 20 , x2
y= output and xl, x, are two inputs. If the price
of output is P =15 and prices of inputs are
Px i =5 and Px 2 =3. Then
(a) Derive profit maximising inputs
(b) With the help of Hession matrix verify that
these inputs are profit maximising.

2. Give examples of the problems where you can

use :
(a) Poisson distribution. Does it have a
probability density function ? Why or why
not ? Discuss your answer in the context of
mean and variance of the Poisson
distribution.
(b) In a sequence of 4 trials with probability of
1
success p= 3 , what is the probability that
there will be exactly two successes ?

MEC-003 1 P.T.O.
3. (a) Explain the process of drawing inferences
and testing hypothesis about the difference
between two population means, when the
population variances are known.
(b) How would you draw inferences about the
variance of a population ?

4. The input coefficient matrix P for an economy is

0.0 0.3 0.3
0.3 0.1 0.1
given by P = and the final
0.2 0.4 0.0

180-
demand vector D= 20 Find the output levels.
90 j

SECTION - B
Answer any five questions from this section : 5x12=60
5. Let the production function Q=1 (L, K) be
homogenous of degree 2. If Q= output,
K = capital and L = labour. Find
(a) The MPPk function
(b) Is MPPk function homogenous in K and L ?
If so, of what degree ?

6. Solve the following :

Max z = 50x1 + 30x2
Sub to xi +x? 9
2 xi + x2 12
if xi 0
x2 > 0

MEC-003 2
 1 1 —1- —1 —2 3-
6
7. A1 = 2 —3 4 A2 = 6 12
3 —2 3 5 10 5
Find the rank of A1, A2, Al + A2, A1A2

8. A committee of 6 is to be formed out of a group of

7 men and 4 women. Find the probability that
the committee will have
(a) exactly 2 women
(b) atleast 2 women

9. Estimate the regression equation of x on y

from :
x 5 3 8 5 10
y 8 11 6 9 8

10. Write notes on :

(a) Rank of a matrix
(b) Eigen value and eigen vector

11. (a) Supply function of a commodity is

Q = a + bp2 + (a < 0, b > 0), R = rainfall.
Find price elasticity of supply.
x1
(b) If y x1+x2 , find the total differential.

12. (a) What is a test statistic ?

(b) Distinguish between one-tailed and
two tailed tests.
(c) What is p-value ?

MEC-003 3 P.T.O.
 No. of Printed Pages : 7 I MEC-003

MASTER OF ARTS (ECONOMICS)

c--1 Term-End Examination
v—i December, 2016
*4-
MEC-003 : QUANTITATIVE TECHNIQUES

Time : 3 hours Maximum Marks : 100

Note : Answer questions from each section as directed.

SECTION - A
Answer any two questions from this section. 2x20=40
1. (a) Given the supply and demand for the
Cobweb model as :

Qst = 6Pt_1 and Qdt =19 — 6Pr

Find the inter temporal equilibrium price
and determine whether the equilibrium is
stable.
(b) Establish the stability condition of
Samuelson's multiplier accelerator model.
2. The input coefficient matrix for an economy is
given by :
0.0 0.3 0.3
A = 0.3 0.1 0.1 and the final demand
0.2 0.4 0.0

180-
D = 20
L 90
Find the level of output.

MEC-003 1 P.T.O.
 3. Consider the Cobb-Douglas production function,
Q = ALa K1 ; A, a > O. Prove that,
(a) It is homogeneous of degree 1.
(b) The marginal and average productivities of
L and K the two inputs depend on the ratio
of the 2 inputs.
(c) Elasticity of substitution is unity.

4. (a) Define standard error of a statistic. Explain

how is it useful in testing of hypothesis and
decision making.
(b) If xi; i =1, 2, . . n are n sample values from
a normal distribution with mean p, and s.d.
cr, derive the pdf of the sample mean x .

SECTION - B
Answer any five questions from this section.
5x12=60
5. Find the expected value and variance of the
following data :
Books sold
0 1 2 3 4 5 6
per day, x;
P (x ,) 0.02 0.10 0.21 0.32 0.20 0.09 0.06

6. Write short notes on :

(a) Mean Value Theorem
(b) Taylor's expansion

7. Solve :
Max : 10x1 + 10x2 + 20x3 + 20x4
Sub to : 12x
1 + 8x2 + 6x3 + 4x4 5_210
3x1 + 6x2 + 12x3 + 24x4 5_210
xi, x2, x3, x4 0

MEC-003 2
8. What is a Binomial distribution ? Find its mean
and variance given that the parameters of the
distribution are n and p.

9. Assume that on an average one telephone number

out of 15 is busy. Find the probability that if 6
randomly selected telephone numbers are picked
up :
(a) not more than 3 are busy
(b) at least 3 of them are busy

10. Find the Inverse of the following matrix :

4 1 —1
0 3 2
3 0 7

11. (a) What is a test statistic ?

(b) Distinguish between one tailed and two
tailed tests.
(c) What is p-value ?

12. (a) If y - , find the total differential.

1 + X2
(b) Write notes on Eigen value and Eigen vector.

MEC-003 3 P.T.O.
 No. of Printed Pages : 8 I MEC-003
MASTER OF ARTS (ECONOMICS)

CV Term-End Examination
CD June, 2012
LO
C:) MEC-003 : QUANTITATIVE TECHNIQUES
Time : 3 hours Maximum Marks : 100
Note : Answer the questions from each section as directed.

SECTION-A
Answer all the questions from this section. 2x20=40
1. (a) Discuss the Hawkins-Simon condition in
the context of input-output analysis.
(b) You are given the following technology
matrix. Find the equilibrium prices if the
wage rate is Rs 100 per day.

Steel Coal Final Demand

Steel 0.4 0.1 50
Coal 0.7 0.6 100
Labour 5 2

OR
(a) Explain the importance of duality of linear
programming in economic analysis.
(b) Consider the linear programming problem:
Maximise Z = 5x1 +10x2
Subject to x1 +3x2__
< 50
4x1 + 2x2... 60
xi—<5
xi,x2?-

MEC-003 P.T.O.
 (i) State the dual of the above linear
programming problem.
(ii) Given that (5,15) is an optimal solution
to linear programming problem above,
find the optimal solution to the dual.

2. (a) Explain the process of drawing inferences

and testing hypothesis about the difference
between two population means, when the
population variances are known.
(b) How would you draw inference about the
variance of a population ?
OR

(a) What is the normal probability distribution

function ? State its properties.

(b) The concentration of impurities in a

semiconductor used in the production of
microprocessors for computers is a normally
distributed random variable with mean 127
parts per million and standard deviation 22.
A semiconductor is acceptable only if its
concentration of impurities is below 150 parts
per million. What proportion of the
semiconductor are acceptable for use ?

(The area under the standard normal curve

for the value of Z =1.04 is 0.3508)

MEC-003 2
 SECTION-B
Answer any five questions from this section. 5x12=60

3. Find the extreme value (s) of

Z=2x12 —x1x2 +4x22 fx,x3 +4 + 2 , and using the

Herbon matrix. Check whether the extreme value
(s) is (are) maximum or minimum

4. Explain the method of maximum likelihood for

estimating the value of a population parameter.

5. Find the inverse of the matrix :

4 1 —1
0 3 2
3 0 7

6. (a) Given the values of x and y

x 1 2 3 4 5
y 3 7 5 11 14
Regress x on y
(b) Given the values of x and y
x 25 25 30 30 1.6
y 2 3 5 1 8
Regress y on x.

MEC-003 3 P.T.O.
7. Explain the relevant considerations of making a
choice between one-tailed and two-tailed tests.
How would you determine the level of significance
in the above tests.

8. What economic interpretation would you attribute

to the method of finding the 'time-path' using a
difference equation ? Explain your answer with
the help of the Cobweb model.

9. Given the rate of investment is I(t) = 12 tY3 , where

't' is time. Suppose the initial capital stock, Ko is
25.
(a) Find the time path of capital stock
(b) Find- the amount of capital accumulation
during the time intervals (0,1] and [1,3].

MEC-003 4
 No. of Printed Pages : 8 MEC-003

MASTER OF ARTS (ECONOMICS)

In Term-End Examination

June, 2013

MEC-003 : QUANTITATIVE TECHNIQUES

Time : 3 hours Maximum Marks : 100

Note : Answer the questions from each section as directed.

SECTION-A 2x20=40

Attempt any two questions from this section.

1. Consider a monopolist producing 2 commodities.

Her demand function is Q1 = 40 — 2P1 + P2 and
Q2 = P1 —P2 +15 where Q1, Q2 are quantities of 2
goods with P1 and P2 their prices respectively.

Let C = Q 22 + Qi Q2 ± Q12 be the cost function.

Find the profit maximising output levels and
prices. Also find the Hessian Matrix.

2. What is Baye's theorem ? Explain how you would

make use of the results of this theorem to derive
the law of total probability.

MEC-003 1 P.T.O.
3. The input coefficient matrix X, for an economy is

0.3 0.0 0.3

given by X = 0.1 0.3 0.1
0.4 0.2 0.0_

180
If the final demand vector is Y = 20 find the
90

output levels for all the sectors.

4. Let u =f x2,..., xn) be the utility function where

xi's are the quantities of n goods consumed. Let
P : be the price of xi, i = 1,2,..,n. Let M be the
income of the consumer. Show that the
Lagrangian multiplier of the utility maximisation
problem equals Marginal utility of income.

MEC-003 2
 SECTION - B 5x12=60
Answer any five questions from this section.

4 —1 1-
5. Find the inverse of the matrix 0 2 3
3 7 0

6. Solve the following linear programming model in

x1 and x2.
Max y = 45x1 + 55x2
sub. to 6x1 + 4x25 120
180 10x2 + 3x1

x2"

7. What is Binomial distribution ? Find the mean

and standard derivation of a binomial distribution
with parameters n and p.

8. Write short notes on :

(a) Taylor's Expansion
(b) Kuhn- Tucker condition

9. Suppose a jar contains 6 blue balls, 8 red balls

and 6 yellow balls, 2 balls are selected at random
without replacement.
(a) What is the probability that the first ball will
be red and second blue ?
(b) What is the probability that neither balls will
be red ?

MEC-003 3 P.T.O.
 10. For 150 beams of a particular variety, the mean
and standard deviations of breadth were found
to be 8.5 mm and 0.5 mm respectively. Test if the
observed mean differs significantly from 8 mm.

11. Find the solution to the equation

1
Yt + ± 4Yt= 5 for yo = 2

12. Assume a normal distribution with a mean of 90

and SD of 7. What limits would include the middle
65% of the cases ?

MEC-003 4
No. of Printed Pages : 8 MEC-003

MASTER OF ARTS (ECONOMICS)

Term-End Examination
501.
June, 2017

MEC 003 : QUANTITATIVE TECHNIQUES

-

Time : 3 hours Maximum Marks : 100

Note : Answer the questions from each section as directed.

SECTION A
Answer any two questions from this section. 2x20=40

1. A monopolist produces two commodities A and B.

His demand function is Q 1 = 40 + P2 — 2P1 and
Q2 = 15 - P2 + P1. P1 and P2 are prices and Q 1 and
Q2 are quantities of A and B. Let the cost function
of the monopolist be C = Q 2 + Q 1Q2 + Q 21 . Find
his profit maximising output and prices. Also, find
the Hessian matrix.

2. (a) The demand and supply for Cobweb model

are : Qdt = 18 - 3Pt and Qst = 4Pt-1 - 3 .
Find the inter-temporal equilibrium price ,

and determine whether the equilibrium is

stable.

MEC-003 1 P.T.O.
 (b) Establish the stability condition of
Samuelson's multiplier-accelerator interaction
model.

3. (a) Give examples of problems where you can

use Poisson distribution. Does it have a
probability density function ? Why or why
not ? Discuss your answer in the context
of mean and variance of the Poisson
distribution.

(b) In a sequence of 4 trials with probability of

1
success p = 3— , what is the probability that

there will be exactly 2 successes ?

4. (a) If )7 is sample mean, prove E( ), the

expected value of x equals the
population mean.

(b) Describe the process of testing hypothesis

about population proportion of a given
attribute.

MEC-003 2
 SECTION B

Answer any five questions from this section. 5x12=60

5. Suppose a die is rolled. You are told that the
number is even. What is the probability that it
is 4 ?

6. Find the inverse of the matrix

1 3 2
A= 3 10 6
2 5 5

7. Estimate the regression equation of x on y from

the data given below :

x 5 8 3 10 5
y 8 6 11 -8

8. Explain the method of maximum likelihood for

estimating the value of a population parameter.

9. Solve the following linear programming model in

x1 and x2 :
Maximize y = 45x1 + 55x2
subject to
6x1 + 4x2 120
180 10x2 + 3x1
x >— 0
x2 0.

MEC-003 3 P .T.O.
10. (a) Find when
dx
(i) y = log (ex + 3)
1
y=
Vx 2 a2

3
L 2
(b) Let q = – — + 2L + 12L is a production
3
function, where L = labour. Find the
maximum L beyond which return from L
starts diminishing.

What is sampling distribution ?

State the properties of point estimators.

State the central limit theorem.

12. Suppose x has the following probability

distribution :

x 0 1 2 3 4

P(n) 0-2 0.2 0.1 0.3 0.2

Find the mean and variance of the distribution.

MEC-003 4
No. of Printed Pages : 8 I MEC-003

MASTER OF ARTS (ECONOMICS)

Term-End Examination

June, 2018
CIE 9

MEC-003 : QUANTITATIVE TECHNIQUES

Time : 3 hours Maximum Marks : 100

Note : Answer the questions from each section as directed.

SECTION A

Answer any two questions from this section. 2x20=40

1. Consider a Cobb-Douglas production function

ALa Kl-a, a > 0

Prove that
(a) It is a homogeneous function of degree 1.
(b) The average and marginal productivities of
L and K, the two inputs, depend on the
ratios of the two inputs.
(c) Elasticity of substitution is unity.
M EC-003 1 P.T.O.
2. (a) If x is the sample mean, prove that
E( x) = 11, where E( x) is the expected value
of x and p. = population mean.

(b) Describe the process of testing hypothesis

about population proportion of a given
attribute.

3. A revenue maximising monopolist requires

a profit of at least 1500. His cost function is
C = 500 + 4q + 8q2 and his demand function is
P = 304 — 2q.

(a) Determine his output level (q) and price

(P).

(b) Contrast these values with those that

would be achieved under profit
maximisation.

(a) Consider the aggregate production function

Q = Ka L", where Q, K and L are all
functions of time. Depict and solve for the
time path of capital output ratio.

(b) Solve (t + 2y) dy + (y + 3t 2) dt = 0.

MEC-003 2
 SECTION B

Answer any five questions from this section. 5x12=60

5. Explain the method of maximum likelihood for

estimating the value of a population parameter.

6. Find the inverse of the following matrix :

1 3 2
A= 3 10 6
2 5 5

7. Suppose you roll a die and the outcome is an odd

number. What is the probability that it is 5 ?

8. Solve the following linear programming model in

xi and x2 :
Maximize z = 45x1 + 55x2
6x1 + 4x2 — 120subjecto 5_ 0
3x1 + 10x2 180
xl, X2 0.

9. Write short notes on the following :

(a) Kuhn-Tucker Condition
(b) Taylor's Expansion
MEC-003 3 P.T.O.
10. For 150 beams of a particular type, the mean and
standard deviation of breadth were found to be
respectively 8.5 mm and 0.5 mm. Test if the
observed mean differs significantly from 8 mm.

11. A sub-committee of 6 has to be formed out of a

group of 7 men and 4 ladies. Calculate the
probability that it will consist of
(a) at least 2 ladies, and
(b) exactly 2 ladies.

12. Suppose x has the following probability

distribution :

x 0 1 2 3 4
P(x) 0.2 0.2 0.1 0.3 0.2

Find the mean and variance of the distribution.

M EC-003
 No. of Printed Pages : 11
I MEC-003 I
MASTER OF ARTS (ECONOMICS)

Term-End Examination
December, 2011

MEC-003 : QUANTITATIVE TECHNIQUES

Time : 3 hours Maximum Marks : 100
SECTION - A
Attempt any two questions from this section.
1. A revenue maximising monopolist, requires a
profit of at least 1500. Her demand and cost
functions are : 2x20=40
P=304 -- 2q and
C=500 +4q+8q 2
Determine her output level and price. Contrast
these values with those that would be achieved
under profit maximisation.

2. (a) Given the supply and demand for the

cobweb model as :

CD Qst 61Y--5-t
CD
0 Qdt = 19 —6 Pt
OD Find the inter temporal equilibrium price
CO
and determine whether the equilibrium is
stable.
(b) Establish the stability condition of
samuelson's multiplier-accelerator
interaction model.

MEC-003 1 P.T.O.
3. (a) Given the marginal propensity to import
M' (y) = 0.1 and the information that M=20
when y=0, find import function M(y).
(b) Marginal propensity to consume
C'(y) =0.8 + 0.1 y -112 and information that
C = y when y =100, find consumption
function C(y)

4. (a) If x i , x2, --, x n is a random sample from a

normal population N(p,,1). Show that

1 ,2
X - L A.i is an unbiased estimator of
11 • =.
11

µ2 + 1.
(b) Derive the least squares normal equation for
fitting a parabolic curve. What change will
be necessary if there was a change of origin
for the X data only ?

MEC-003 2


SECTION B

Answer any 5 questions from this section. 5x12=60

5. A bag contains 8 red balls and 5 white balls. Two

successive draws of 3 balls are made without
replacement. Find the probability that the first
drawing will give 3 white balls and the second
3 red balls.

6. Consider the matrices :

—1 —2 3 - 1 1 --1 -
A= 6 12 6
and B = 2 —3 4
5 10 5 3 —2 3

Find the rank of matrices A, B, A + B, AB, BA.

7. Ten competitors in a musical contest were ranked

by 3 judges A, B, C in the following order :

Contestant 1 2 3 4 5 6 7 8 9 10
Rank by A 1 6 5 10 3 2 4 9 7 8
Rank by B 3 5 8 4 7 10 2 1 6 9
Rank by C 6 4 9 8 1 2 3 10 5 7

Use rank correlation method to discuss which pair

of judges has the nearest approach to common
liking in music.

MEC-003
3 P.T.O.
8. Marks obtained by 12 students in college test (x)
and university test (y) are as follows :

x 41 45 50 68 47 77 90 100 80 100 40 43
y 60 63 60 48 85 56 53 91 74 98 65 43

What is your estimate of the marks a student

would have got in the university test, if he got 60
in the college test but was ill for the university
test.

9. What is a Poisson Distribution ? Find the mean

and variance of a poisson distribution.

10. (a) Solve the following problem graphically :

Min C = 0.6x 1 +x 2
Sub to 10x 1 + 4x2 20
5x1 + 5x2 20
2x1 + 6x 2 > 12
x1 , and x2 > 0.
(b) Why does the solution occur at a corner
point only ? Give reasons.

11. Given the input-output matrix :

0.1 0.3 0.1

A = 0.0 0.2 0.2
0.0 0.0 0.3

MEC-003 4
 and final demands are F1 , F 2 and F3
. Find the
output level consistent with the model what would
be the output levels if :
F1 =20, F 2 = 0, F3 100.

12. (a) The demand curve for a consumer is

p — d = — where d and b are constants. Find

the price elasticity of demand.

•

(b) Find --
, when :
ax

(i) y =-- log (ex +3)

1
(ii) y=
a2

MEC-003 5 P.T.O.