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Unique Paper Code : 12271102
Name of Paper :Mathematical Methods for Economics -I
Name of Coyrse :B.A. (Hons.) Economics

Semester :I : 1

Duration :3 hours
Maximum Marks :75

(Write your Roll No. on the top immediately on receipt of this question paper.) L

(rrrmwtMlamf~dhdfwrsnm.~irr~afdd~)

NOTE:- Answers may be written either in English or in Hindi; but the same medium
should be used throughout the paper.
\

There are six questions in all.

All questions are compulsory.
A simple calculator can be used.

pras;riP/

t n ~ m o r ~ ~ ~ -~ ~ m ~ ~ i

All parts of a question should be answered together.

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 This question paper contains
16+3 printed pages]
1s121
Roll No.

7447
S. No. of Question Paper
nique Paper Code 12271102 IC

Name of the Paper MathematicalMethods for Economies-I

Name of the Course B.A. (Hons.) Economics

ICBCS C-2, Corel

Semeste

Duration: 3 Hours Maximum Marks 75

(Write your Roll No. on the 1op immediately on receipt of this question paper.)

Note Answers. may be written either in English or in Hindi;

but the same medium should be used throughout the

paper.

There are six questions in alI.

All questions are compulsory.

A simple calculator can be used.

P.T.O
 (2)

Answer any wo of the following

25-10
(A) Determine the domain and range of the follówino

inequality

y ft) = In[In(e - 1)] < 0.

(B) Find allx such that

y=4-2x 2x2 -2

ii)
(x-0.5) (In(l -x) 0.
3/2

(C)Determine
C) the direction of logical conclusion (P Q

or Q P or P Q) in case of the following

propositions

() P:fx) has a local extremum at x =a, where f"(a)

exists.

ais a stationary point of f(x), i.e.

S(a) =0.

(i) P :x satisfies the inequality: ( 1 -x) 20(x +20

Q:r lies in the open interval from 5 to 1.

 7447
(3)

tiHT (range)
(A) famfatan rtfHh1 1 TH(domain)

0.
y-ft) =
In[In(e - 1)] <

(B)

) y= 4 -21 2-2

(i) y=
(x-0.5)(in(-) 0.
3/2

f
C)f e r f e n 7 z fsfA (logical
Q or Q P or P Q) i t RAT
conclusion) (P »

(local extremum) TEÍJsa) fTETHTT I

TRR T stationary point)

Q:x =

a,fx) 1 T
0.
37247q f'(a) =

0 HTE R I EI
()
(i) P:x, 37H41

P.T.O.
 (4) 7447
Answer any three of the
following 3x5-15
A) Draw in the same
graph the regions represented by the
following two sets

S =
{(x, y): r* +s 25}
T
{(, y): y2 12
In each case, plot coordinates of all points where the
graphs intersect each other or intersect the coordinate

axes. Determine if sets S and T are disjoint.

(B) For the function defined as follows:

2 x, 0Sx<2
S(x) = {Vx, 2 SxS4

1,x>4

Plot the function. Verify continuity and differentiability

of the function at x =
2 and x =
4.

C)Find the asymptotes of the rectangular hypèrbola

(k-2)x+k -4
y=
(k -6)x +k -3

given that the asymptotes intersect at a point that lies

on the straight line y = 2x -

7.
 7447
(5)
kiosk at the university
(D) Harish runs a 'rent a bicycle"
D)
10 per
station. He currently charges a price of Rs.
metro

demand is of 100 bicycles

at which the average
bicycle
estimates that, each time the
per day. An industry expert

increases by Rs. 5 per bicycle, the average

rental price
Express the rental
demand drops by ten bicycles per day.
the method of
quadratic function and
use
income as a

to determine (i) the rental

completing the squares'

that maximises your income from renting bicycles

price
income.
and (i) maximum

(A)

S {, y): 7 +y s 25}

T {r, y): y 2 12}

ufaofea (intersect) 7Ta fITE%

v7
(coordinate axes) 5 ufd 7 I IT

P.T.O.
 (6)
47
(B)

2- x, 0Sx<2
sx)= {Vr, 2SxS4
+1, x>4

4 TR TIT (continuity) 7TachTHT4AT

=
r

(differentiability) FATYT I

C) AT27ATER TaTACA (rectangular hyperbola)

y=2)r +k -4
k-6)x+k -3
1 31-TaÍYTAÍ (asymptotes) 77 tfg, ztë TE AT

A&f 7THÍYtT FTM y =2x 7 T PA

(D)

10. f 3 a H sfafkT 100 ITSf6C

 7447
(7)

5qtaATA hTA (quadratic function)

Answer any three of the following 3x5-15

3.

series has its second term a,

=
-48 and
(A) A geometricc
fifth term a 6.

Find the first term and common ratio of the series.

()

Find the to infinity of the series.

(ii) sum

Show that the magnitude of the difference between

(iii)
the sum of first n terms of the series and its sum

to infinity is given by 26

Given the following approximation for small values

(B)

ofx

(1 t axy" = I - 24x + 270x

P.T.O.
 (8)
7447
Where n is an integer greater than 1 :

Find thevalues of n and a.

i) Use the values ofn and a and a suitable value

of x to obtain an
approximate value of (0.9985)6

C)Evaluate the following limits

() y= lim
+1
x 5 +x*

2
(i) y= lim y+6
8n

D) Suppose in a given city with n

individuals, total market

demand X =

2-X, where x, is the demand of the

ith consumer. The inverse demand

function is given

by x, f(P), where P is the market price. If half the

consumers with 75% share in total

demand, have price
elasticity of demand |E,| =
2 and the
remaining have price
elasticity of demand |E,= 1.5, estimate the price elasticity
of demand of all
consumers taken
together.
 7447
(9)

(A) v at7 JaI (geometric series) 1 THT

=
- 48 Y7a Ta, =
6 EI
a,

ratio) l|

(i)

(ii)

TRAT 26 - n I

(B)x HA EG fAHataT fTAA (approximation)

(1 +
ar" 1 -
24x + 270x,

TEn, 1 À I U5 U5 (integer)

() n a H

(i)

7E4 (0.9985)6 HHHE HA (approximate

value) T
P.T.O.
 (10)
7447
(C)

() y= lim +l

(in) y= lim (+6

+6
8n

(D)

PT X 2,
HTACTA (inverse) HTT HTAx, ={f(P) E, TEi P IR

T H 4a E,| =2 a

(elasticity of demand of all consumers taken together)

Answer any three of the following 3x5-15

(A) Graph the following function and verify that it is

one-to-one

In(x); 0<x<l
sr) =

-1;x21
 (1) 7447

Find the inverse functionf(r) and identify its domain

and range. Draw f(r) in the same graph and comment

on the nature of symmetry in graphs of f(r) and f (r).

Plot the coordinates of all points where the graphs intersect

the coordinate axes.

(B) The Coconut Farmers' Association in India estimated

that the value V() of coconut produce (in lakh

rupees) increases over time according to the following

function

V()=avbre

where a, b, c> 0, a > e and t is time for coconuts to

ripen.

Assuming that the discount rate is r

( Find optimal time f for the producers to pick

coconuts such that the present value of the harvest

is maximized (second order condition for optimum

need not be verified).

(i) How does a change in discount rate change the

optimal time of picking the coconuts ?

P.T.O.
 (12)
7447
Given the function

7-(16 if x* 0;
s(x)={1+(16)
7 if x = 0.

Prove that there is a point in the open interval (2, 4) in

which the function f(x) has a value of 1.

(D) If the function

f x ) = axe-x, a >0

has a local maximum at point (2, 10), then find a and b.

Find the point of inflection of f(x).

(A)

f47 UG-7-U (one-to-one)

In(x); 0<x<l
s)=
x-1x21
 13) 7447

H51 TmA (inverse) 7 x) FI7 T

HTHAT (symmetry) zta T feuuit tfTuI

(B)

3FH (Ta . T) V), HH4 H

V()= nbt +c

TE a, b, c>0, a> e aA TAt T5 7 TA

7EH f7 i 7 (discount rate) r

(present value) YhAA E (SZ04tUI

P.T.O.
 14)
7447
cfamta 4 (second order) 51 Td HAa

(i)

C)

7-(16 if x * 0;
Sr)={I+(16
if x = 0.

a faqy fay f q RIT (2, 4) VT T

(D)

S) =

axe, a>0

71 fa 2, 10) U T4TTA 3f(ocal maximum)

(point of inflection) TT 9 7
 7447
15)

Answer any two of the following 2x6-12

Consider the function f(r) =

2r3 + 3x -
12x + 24,
(A)

defined for all x e R.

Find the stationary point(s) ofy=fr) and determine

is maximum or
whether each stationary point a

minimum point.

Plot the curve f(x) depicting clearly the

y

and the extreme values attained

stationary points

at these points.

(ii) State the set of values of k for which the equation

fx) =
k has three solutions.

def+ned the
(B) Given the function f(x) =

6x* -

3x over

interval[-1, 1]:

() Find the global maximum and minimum values of

(i) Find the interval(s) in which the function increases

(i)
and/or decreases.

P.T.O.
 (16)
7447
(ii) Find the interval(s) in which the function is concave
and/or convex.

C)A function f (r) is known to be

continuous and

differentiable for all x. Find f'(r) where

Find all stationary points of f(x) and classify each ass a

local maximum, a local minimum, or neither.

(A)
(A) 4txe R URHTTUA HTMf) =
23 + 3x2 -

12r

+24 faR

(maximum) I AT3 (minimum)I

(i)
 (17) 7447

B) 7RTM -I, 1] T YRATYA A f(r) =6r 3r

(B)

(i)

THT (increasing) T1/Terqi HHTA

(decreasing) I

(ii)

HATT (concave) 7/37yaT 3HT (convex) I

C)

ro-P(2-3}
d
0+t' 0-7)|a.

P.T.O.
 18)
7447
Answer ali the questions
2x4-8
(A) Find the area of the region bounded vertically by
y and y =
6 +x and bounded horizontally by
x = 0 and x = 5.

(B) Consider the two-sector model :

Y,-C,+
C, 0.75Y,-1 + 400

I, 200

() Find the difference equation in Y, generated by this

model.

i) Solve the difference equation for Y, and determine

whether the solution path of Y, is convergent or

divergent.

(i) Find the value of C, given that Y 4,000.

(A) 7 4 7 Fra 9 7 y =A y = 6 +x
3TRI GER yfTHTf4a gx = 0 tax = 5 GRI F - R

qfeiferr
 (19) 7447

(B)

Y, - C, + ,

+ 400
C,
=

0.75Y,-

200

(i)
T4 (solution path
a ifyg fs Y, T
HTHRT (divergent) I
37fTHRT (convergent)

(ii)
f5 Y4,0001

19
3,200
7447
This question paper contains 19 printed pages]

Roll No.

S. No. of Question Paper 3547

Unique Paper Code :12271102 JC

Name ofthe Paper Mathematical Methods for Economics-I

Name of the Course B.A. (Hons.) Economics

Semester :

Maximum Marks: 755

Duration:3 Hours
(Write your Roll No. on the top immediately on receipt of this question paper)

Note Answers may be written either in English or in Hindi;

but the same medium should be used throughout the

paper.

There are five questions in all.

All questions are compulsory.

A simple calculator can be used.

P.TO.
 2 3547
3) 3547
1. Answer any nwo
of the following 2x4-8
C) Determine the direction of logical conclusion (P > Q
(A) Let fx) be a function with domain (-2, 3] and range or Q> P or P Q) in case of the following
[0, 8]. What are the domains and
ranges of the followingg propositions
functions ?

S * -1) 0 P: The series , is convergent.

(in 4 - ) + 1. Q: lim,-+ , .

(B) The given figure shows the graph of the function (in P ? > 16
y 8) = px< +
qx + r.
Q:r>4
( Check which of the constants
p, q and r are

>0, = 0, or < 0.

(A) i f g fs S), TIF (domain) [-2. 3] a R

(range)[0, 8] 3TT A7 fA-fafaa i

TTT a ?

(0 -1)

(in 4 ) + 1.
y 8) =

pr* + qx+r
*
(B) fetea f HTA y =gir) = px- qx
()The graph is symmetric about the line x k.

Find k.

P.T.O
 4) 3547
3547

Answer any three of the following 34-12

a61-7R > 0, =0 7 <0
(A) Find the equations of the tangent ines to the curve

x-1
4Parallel
+]
to the line x - 2y 2.

(B) Find the asymptotes of the following functions

(0 y=el+

x-1
y gtx) = px* + qx +r

Evaluate the following limit lim,

(in) 7 3TTE TEx =

k Y FHfA (symmetric)
i) Consider the infinite series

C) fffer . For what values of

f AT (logical
conclusion) EM (P >
Q Q P. P Q)
x does the series converge ? Find the sum of the

series if x = 1.2.

it Approximate the function x ) = x'8 by a Taylor

() P:
, tYHRT (convergent) D)

polynomial of degree 2 at x 8. Use it to find an

Q: lim,-+0 .

approximate value of 93, Find an upper bound for the

(i) P > 16
error of approximation corresponding to the result
Qx> 4
obtained.
P.T.O.
 6 3547 3547
. 7

3. Answer any four of the following 4x5-20

(A) 4 y tar-2y 24H-RT (paralle!) (A) Graph the function g(x) and check its continuity at

x = 1 and x = -1

B) ffnfen i 3AHNT (asymptotes) FTd

2r-1 if x<-1
gx)={r?+I if -1SrS
(0
y=el x+I ifx >1l
(in
(B) If yy+1 = log (x + Vx2 +1) show that
C) IAI lim ,0
r2
dy
(a)(r2 +1)+y -1 =0

(6++y-0
2 dx

3ATTT (converge) at ? zx =

1.2 (i) Find the point(s) of inflection of the function

D) hT A) =x3 S)= xe*

x 8 7ife (degree) 2
z T (Taylor polynomial) fofea (C) () Let f be twice differentiable on [0, 2], show that
(approximate) fyi H HKTTT 9l3 if o) =
0, A1) =
2, A2) =
4, then there is an
HfH2 TA (approximate value) TT IMYI TT
* o ( 0 , 2) such that f"x) = 0.
TfUTHHTM H A zë (error of approximation)
F A T (upper bond) aT i I (i) Graph the
function y = -

1|.

P.T.O.
 8 3547
9) 3547
(D) () Find the expression for elasticity of h(r) =

fx)
in terms of
E, and E, the elasticities of fx) and
(A)
s) w.r.t x
respectively.
(ii) Prove that (x) x = -i F (continuity) fu
=
ev -3 has a
unique solution
in the interval (1, 2x-1 if x<-1
4)
E) (a)
gx)=r+1 if -1Sxs1
Suppose that the interest rate 'r such that the x+1 if x > 1 |
present value of receiving Rs. Az in t2 years
from
B) a yl*?+1= log (x +r? +1), dI, z7ÍS
now is the same as
receiving Rs. A in
f
years from now, given that 2 1 Assuming
interest is compounded annually (o)r+1 1=0

() Show that A
A
(i) Show that the present value of
a)+y-0
receiving
(in h f(x)= xe-, A^ fgi (points of
Rs. Az. (2 +
k) years from is
now equal to
inflection) 7 FIT f I
the present value of receiving Rs. A, ( +k)
C) ( HI TMY f5f [o, 2] TT aR 37AHTHTY
years from now.

(6) The equation 3xe» -2y =3x2 +y2 defines y as

(differentiable)diITSY zA0) =0.
A1) 2, 2 ) = 4, Y TT x e (0. 2)
a differentiable function of about the
x
point
x, y) =
(1, 0). What is the linear foH fY S"() = 0
approximation
to y about x? (i 7T y = - 1| 1 . ATE THII

P.T.O.
 10
3547
(D) ) 3547
-fu) zi aT (elasticity) d
Hx)
A) gtr) (b)HtrI Jxe»
HTd -2y =3z? +y?, y f
, F41: E, E (x,y) =
(1. 0) 3TE-T YAA
(i) f fy f )=e -3 1
3ATH-T eiy Hfa (linear
approximation)
3RTM
(1. 4) y HGia (unique) 7 4. Answer any
E) (a)
two
of the following 27.5-15
Tiify f5 TT f (A) ( For fr) 3x(r 4)
=
+
find the global extreme

points on the interval [-5, -1].

TT aHT H
(present value) 4 (i) If f is a one-to-one twice differentiable function

with inverse g. show that 8 "(x) =

-J 8)
Se(x))
Show that iff is increasing and concave its inverse

is convex.
(Compounded annually) BT
( zs (B) ( Show that the function
f Az> A flx) =
ar + bx +
c is

concave if a 0 and convex if a 2 0 without

using derivatives.

(in Let
J)=
( ) = _ e 2

Find the intervals on which

the function is
increasing and/or decreasing.

PTO
 12 3547 13 3547

C) The curve
C passes through the origin in the

r-y plane and its gradient is 37TT |-5, -1) d f a

given by (A) Ax) =
429
3rtr +

(elobal)7H f (extreme points) A I

=x(1-x?)e-*á, Find its stationary points and
(i) f q-À-T (one-to-one). 17

classify them as maximum or minimum points. TaTri A f e 1 y f a (inverse)

(i) A coin and stamp dealer estimates that the value (S'(e(x))*

of V of his collection (in lakhs of YITER fs afx /aAA (increasing) T 34

rupees)
(concave) F51 T 4 3H7 (convex)
increases over time
according to the
following
runction
B)
V() =1000eV . If rate of interest is 8%
fx)= a + bx + c HAAT a a 0 TT

compounded annually, find optimal time for the

e
coin and stamp dealer to sell his
collection such
4+e

that the present value of the collection f H À TE 61 GT (increasing)

is

maximised T/37T THHT (decreasing)

(second order condition for
optimum

need not be C) a C xyHTET À f (origin) TTAT

verified). How does a change in the

T dy
discount rate changs the optimal time r ?
HT GT (gradient) x(-x)e**
P.T.O.
 14 ) 3547 15) 3547

Hfe fagi (stationary 5. Answer any four of the following

points)1 45-20

t7T 33T (maximum) T Ht3 (A) () Show that for any wo n xn matrices A and

(minimum) aitga fsy B, tr(AB)-u(BA), where tr(A) denotes the trace

of a n xn matrix A.

(in Find the rank of the following matrix for all values

of the parameter :

3 5 7-A

fT 3r9 HAE aa 1TaA (optimal) (B) ( Solve the following system of equations
X - Y + Z = 0

X + 2Y - Z =0

2X Y + 3Z 0

TET )I 7(discount rate) YfTAA (i) What are degrees of freedom ? Determine the

number of degrees of freedom of the above

system
of equations.

P.T.O.
 16
3547 17 3547

(C) Given that {u, v. w) is a

linearly independent set of
vectors in some vector space V,
prove that
( the set {4, v} is (A) (0 ZTE f5 f i an xn A TB
linearly independent.
(in the set {u, tr(AB) r(BA), T6 tr(A), n *n f a A
u
+v) is
linearly independent.
(iii) the set {u *
v, + (trace) RI FI
v
w). is
linearly independent.
D) Consider the following system of equations (i) T (parameter) F a Afan
-mX+ y = b
f fe (rank) frg
-mX*y= b2
4
Prove that if m, *
m2, then the system of equations 7-
has exactly one solution. Find the solution.

(i) Suppose that m,

m2. Then under what « ditions
will the system of (B) (O. fafcifiem ithRT H (system of equations)
equations be consistent ?

(E) ( Let v be any vector of

length 3. Le A
ifau
=
(v, 2v, 3v)
be the 3 x 3 matrix with columns v, 2v, 3v, Prove X Y + Z 0
that A is singular.
(i) Find equation of the line formed at X + 2Y Z =0
intersections
of the two planes:
2X+Y+3Z =0
X 5Y + 32 =
II and 3X + 2Y -

2Z =
-7.

P.T.O.