Syllabus and Course Scheme
Academic year 2014-15

Bachelor of Science / Arts - Mathematics
Exam.- 2015
UNIVERSITY OF KOTA
MBS Marg, Swami Vivekanand Nagar,
Kota - 324 005, Rajasthan, India
Website: uok.ac.in

1

B.A. / B.Sc. Part – I Examination – 2015
MATHEMATICS
Paper
I
II
III

Nomenclature

Teaching Hr./Week

Exam. Duration

Abstract Algebra
3
3 Hrs.
Practical
2
Advanced Calculus
4
3 Hrs.
Vector Calculus and
4
3 Hrs.
Co-ordinate Geometry
Total Max. Marks (Theory / Practical)
Total Min. Pass Marks (Theory / Practical)

Max. Marks
Sci.
Arts
50
44
25
22
75
66
75
68
200/25
72 / 9

178 / 22
64 / 8

B.A./ B.Sc. Part-II Exam. – 2015
Paper

Nomenclature

Teaching Hr./Week

Exam. Duration

I
II

Real Analysis
4
3 Hrs
Differential Equations
3
3 Hrs.
Practical
2
III
Mechanics
4
3 Hrs.
Total Max. Marks (Theory / Practical)
Total Min. Pass Marks (Theory / Practical)
Note : Common paper will be set for both faculties i.e., Arts and Science

Max. Marks
Sci.
Arts
75
66
50
44
25
22
75
68
200/25
178 / 22
72 / 9
64 / 8

B.A./B.Sc. Part III Examination - 2015
Paper
I
II
III

Nomenclature

Teaching Hr./Week

Linear Algebra & complex analysis
Mathematical Statistics and Linear
programming
Numerical Analysis & C-Programming
Practical

Exam. Duration

4
4

3 Hrs.
3 Hrs.

3
2

3 Hrs.

Total Max. Marks (Theory / Practical)
Total Min. Pass Marks (Theory / Practical)
Note : Common paper will be set for both faculties i.e., Arts and Science

Max. Marks
Sci.
Arts
75
66
75
66
50
25
200/25
72 / 9

46
22
178 / 22
64 / 8

Innovation and Employability of the Maths course The syllabus of Mathematics includes three papers per year in the 3 year degree program of
the UG (B.Sc.). The syllabus is at par with the syllabi of prestigious universities and
institutes of India and follows the UGC model curriculum. The syllabus includes essential
part of Operation Research and Mathematical Statistics also, which is very helpful for the
employability of the students in industries various department related to statistics
2

B.A. / B.Sc. Part – I Examination – 2015
MATHEMATICS
Scheme :
Paper Nomenclature

Teaching Hr./Week

Exam. Duration

I

Max. Marks
Sci.
Arts
50
44
25
22
75
66
75
68

Abstract Algebra
3
3 Hrs.
Practical
2
II
Advanced Calculus
4
3 Hrs.
III
Vector Calculus and
4
3 Hrs.
Co-ordinate Geometry
Total Max. Marks (Theory / Practical)
200/25
Total Min. Pass Marks (Theory / Practical) 72 / 9
Note : Common paper will be set for both faculties i.e., Arts and Science

178 / 22
64 / 8

Paper I – Abstract Algebra
Time : 3 Hrs
Max Marks : Science : 50 / Arts : 44
Note : The question paper will contain three sections as under –
Section-A :
One compulsory question with 10 parts, having 2 parts from each unit, short answer
in 20 words for each part.
Total marks :05 (Science / Arts)
Section-B : 10 questions, 2 questions from each unit, 5 questions to be attempted, taking one from
each unit, answer approximately in 250 words.
Total marks : 25 (Science / Arts)
Section-C : 04 questions (question may have sub division) covering all units but not more than
one question from each unit, descriptive type, answer in about 500 words, 2 questions
to be attempted.
Total marks : (Science -20 / Arts- 14)
Unit – I
Binary operation (Composition). Addition and multiplication modulo operations. Definition of a
group with examples and simple properties (including its alternate definitions).
Permutation group, cycle, transpositions, even and odd permutations and alternating group. Order of
an element of a group and its properties.
Unit – II
Subgroups of a group with its properties,Cyclic groups and their properties, Cosets decomposition.
Index of a subgroup, Lagrange’s theorem and its applications, Fermat’s and Euler’s theorems.
Unit – III
Normal subgroups with properties. Simple groups, Quotient groups.
Group homomorphism with its kernel and properties. Isomorphism, Cayle’s theorem, automorphism,
Fundamental theorem of homomorphism.
Unit – IV
Rings, Zero divisors, integral domains and fields. Characteristic of a ring, Subrings, subfield, prime
field, ring homomorphism and isomorphism. Imbedding of an integral domain in a field, Field of quotients.
Unit – V
Ideals and their properties. Principal ideals and principal ideal ring. Prime ideal. Maximal ideal.
Fundamental theorem of ring homomorphism.
3

Euclidean ring and its properties. Polynomial over a ring. Polynomial ring. Polynomial over an
integral domain and over a field. Division algorithm.
Books Recommended for reference:1.
I. N. Herstien, Topics in Algebra, Wiley Eastern Ltd., New Delhi, 1975.
2.
Nathan Jacohson, Lectures in abstract Algebra Vol. I, W. H. freeman, 1980 (also published by
Hindustan Publishing Company).
3.
Shanti Narayan, A text book of Modern Abstract Algebra, S. Chand and Co. New Delhi.
4.
Surjeet Singh and Qazi Zameeruddin, Vikas Publishing House, Pvt. Ltd., Delhi
5.
A, R, Vasishtha, Modern Algebra, Krishna Prakashan Mandir, Meerut
List of Practicals for B.A./B.Sc. (Pt.-I)
B.A. (Pt-I) (Total Marks- 22) Teaching Hr./ Week
Record
05
Practical -1 06
2 Pd./ Week
Practical -II 06
Viva-Voce
05

Practical Marks-25 Science / 22 Arts
B.Sc. (Pt-I) (Total Marks- 25)
Record
05
Practical -1 07
Practical -II 07
Viva-Voce
06

Name of Practicals
1. Graphs of algebraic polynomials of degree four and above.
2. Simplification of logic circuits with the help of boolean Algebra.
3. formation of truth table of Boolean functions.
4. Curve tracing of plane curves.
5. Application of fundamental theorem on morphism of gruoups.
6. Construction of composition tables for some special operations.
7. Find roots of algebraic equation by graphical method.
8. Application of Lagrange theorem.
9. Problems related to permutations and permutation groups.
10. Problems related to ring.

Paper II – Advanced Calculus
Time : 3 Hrs

Max Marks: Science : 75 / Arts : 66

Note : The question paper will contain three sections as under –
Section-A :

One compulsory question with 10 parts, having 2 parts from each unit, short answer
in 20 words for each part.
Total marks : 10 (Science) 5 (Arts)

Section-B :

10 questions, 2 questions from each unit, 5 questions to be attempted, taking one from
each unit, answer approximately in 250 words.

Section-C :

Total marks : 35 (Science / Arts)

04 questions (question may have sub division) covering all units but not more than
one question from each unit, descriptive type, answer in about 500 words, 2 questions
to be attempted.

Total marks : 30 (Science) (Arts-26)

Unit – I

4

Polar coordinates, angle between radius vector and tangent, polar sub tangent and subnormal.
Perpendicular from pole on tangent. Pedal equation of a curve. Derivative of length of an arc in cartesian
and polar coordinates.
Curvature, Radius of curvature and its formula in various forms. Centre of curvature, chord of
curvature.
Unit – II
Partial differential coefficients of a function of two or more variables. Total differential coefficient.
Composite function, Euler’s theorem on homogeneous functions of two, three and m-variables. First and
second differential coefficients of an implicit function. Taylor’s theorem for a function of two variables.
Jacobians with properties. Maxima, minima and saddle points of functions of two and three
variables. Lagrange’s method of undetermined multipliers.
Unit – III
Asymptotes, envelopes and evolutes.
Test for points of inflexion and multiple points. Test for concavity and convexity. Tracing of curves
in cartesian and polar coordinates.
Unit – IV
Quadrature, Rectification, Volumes and surfaces of solids of revolution. Differentiation under the
sign of integration.
Unit – V
Beta and Gamma functions. Double integrals and their evaluation by change of order and changing
into polar coordinates.
Triple integrals, Dirichlet’s double and triple integrals with their Liouville’s extension.
Books Recommended for reference :1.
Gorakh Prasad, Differential calculus, Pothishala Private Ltd., Allahabad.
2.
Gorakh Prasad, Integral calculus, Pothishala Private Ltd., Allahabad

Paper III – Vector Calculus and Coordinate Geometry
Time : 3 Hrs
Max Marks: Science : 75 / Arts : 68
Note : The question paper will contain three sections as under –
Section-A :

One compulsory question with 10 parts, having 2 parts from each unit, short answer
in 20 words for each part.
Total marks : (10 Science) (Arts- 05)

Section-B :

10 questions, 2 questions from each unit, 5 questions to be attempted, taking one from
each unit, answer approximately in 250 words.

Section-C :

Total marks : 35 (Science / Arts)

04 questions (question may have sub division) covering all units but not more than
one question from each unit, descriptive type, answer in about 500 words, 2 questions
to be attempted.
Total marks : (30-Science) (Arts- 28)

Unit – I
5

Vector differentiation and integration, Gradient, divergence and curl. Vector identities, Line and
surface integrals.
Theorems of Gauss, Green, Stokes(without proof) and problems based on these.
Unit – II
Parabola : Standard equation, parametric co-ordinates, length of chord, tangent, normal and its
properties, two tangents from a point, chord of contact, polar, pole, chord with a given middle point,
diameter and three normals from a point.
Ellipse : standard equation, auxiliary circle, eccentric angle, tangent, normal, two tangents from
point, chord of contact, pole, polar, chord whose mid point given, diameter, conjugate diameters and four
normals from a point.
Unit – III
Hyperbola : Standard equation, parametric co-ordinates, asymptotes, equation referred to asymptotes
as axes, conjugate diameters and rectangular hyperbola.
Polar Equation : Standard equation, directrix, tangent, normal, polar and asymptotes.
Unit – IV
Sphere : standard equations in various forms, plane section, sphere through the circle of intersection
of two spheres, power of a point, tangent plane, polar plane, polar line, angle of intersection of two spheres,
length of tangent, radical plane, radical axis, co-axial system of spheres and limiting points.
Cone : Homogeneous equation in x, y, z, cone with a given vertex and given base, enveloping cone,
condition for the general equation to represent a cone, tangent plane, reciprocal cone, angle between the two
lines, in which a plane cuts a cone, three mutually perpendicular generators and right circular cone.
Cylinder : Right circular cylinder and enveloping cylinder.
Unit – V
Central Conicoids : Standard equation, tangent plane, condition of tangency, director sphere, polar
plane, polar lines, section with a given center, enveloping cone, enveloping cylinder.
Ellipsoid : Normal, six normals from a point, cone through six normals, conjugate diameters and
their properties.
Books Recommended for Reference:1.
Shanti Narain, A Test Book of vector calculus, S. Chand and Co., New Delhi.
2.
Murray R. Spiegel, Vector Analysis, Schaum Publishing Company, New York.
3.
J. N. Sharma & A. R. Vasishtha, Vector Calculus, Krishna Prakashan Mandir, Meerut.
4.
S. L. Loney, the elements of coordinate Geometry, Macmillan and Company, London.
5.
Gorakh Prasad and H. C. Gupta, Text Book of Coordinate Geometry, Pothishala Pvt. Ltd.,
Allahabad.
6.
R. J. T. Bell, Elementary Treatise on Coordinate Geometry of Three dimension Macmillan India
Ltd., 1994.
7.
Shanti Narayan, Solid Geometry, S. Chand and Company, New Delhi.
8.
M. Ray & S. S. Seth, Differential calculus, students, friends & Co. Agra.
9.
M. Ray & S. S. Seth, Integral calculus, students, friends & Co. Agra.

6

B.A. / B.Sc. Part – I Examination – 2015
;kstuk
iz'ui=
A

'kh"kZd

dkyka'k&lIrkg

vof/k

iw.kkZad
foKku
50
25
75
75

dyk
44
22
66
68

vewÙkZ cht xf.kr
3
3 ?k.Vsa
izk;ksfxd
2
AA
mPp dyu
4
3 ?k.Vsa
AAA
lfn'k dyu ,ao
4
3 ?k.Vsa
funsZ'kkad T;kfefr
dqy vad ¼lS)kfUrd @ izk;ksfxd½
200@25
178@22
U;wure mÙkh.kkZad ¼lS)kfUrd @ izk;ksfxd½
72@9
64@8
Note : Common paper will be set for both faculties i.e., Arts and Science
iz'u i= & I - vewÙkZ cht xf.kr
le; % 3 ?kaVs
vf/kdre vad% foKku%50 @dyk%44
uksV %
bl iz'u i= esa 03 [k.M fuEu izdkj gksaxsa %
[k.M v %
bl [k.M esa ,d vfuok;Z iz’u ftlesa izR;sd bdkbZ ls 02 y?kq iz'u ysrs gq, dqy 10 y?kq
iz'u gksaxs A izR;sd y?kq iz'u dk mÙkj yxHkx 20 'kCnksa esa gks Adqy vad % 05¼foKku@dyk½
[k.M c %
bl [k.M esa izR;sd bdkbZ ls 02 iz'u ysrs gq, dqy 10 iz'u gksxsa s A izR;sd bdkbZ ls ,d
iz'u dk p;u djrs gq, dqy 05 iz'uksa ds mÙkj nsus gksaxs A izR;sd iz'u dk mÙkj yxHkx
250 'kCnksa esa gks A
dqy vad%25 ¼foKku@dyk½
[k.M l %
bl [k.M esa 04 iz'u o.kZukREkd gksaxs ¼iz'u esa Hkkx Hkh gks ldrs gS½tks lHkh bdkbZ;ksa esa ls
fn, tkosaxs] fdUrq ,d bdkbZ ls ,d ls vf/kd iz'u ugha gksxk A nks iz'uksa ds mÙkj fn;s tkus
gSaAizR;sd iz'u dk mÙkj yxHkx 500 'kCnksa esa gks A
dqy vad% ¼foKku&20@dyk&14½
bdkbZ & I
f}vk/kkjh lafØ;k] ;ksx xq.ku eksM~;wyks lafØ;k] lewg dh ifjHkk"kk] mnkgj.k ,oa lkekU; xq.k/keZ ¼lewg
dh oSdfYid ifjHkk"kk lfgr½ Øep; lewg] pØ] i{kkUrj.k] le ,oa fo"ke Øep; ,dkUrj lewg] lewg ds
vo;o dh dksfV rFkk xq.k/keZA
bdkbZ & II
milewg rFkk mlds xq.k] pØh; lewg ,oa mlds xq.k/keZ] lgleqPp;] milewg dk lwpdkad] ysxzkat
izes; ,oa blds vuqiz;ksx] QesZaV ,oa vk;yj izes;A
bdkbZ & III
izlkekU; milewg ,oa mlds xq.k/keZ] ljy lewg rFkk foHkkx lewg] lewg lekdkfjrk] lekdkfjrk dh
vf"V vkSj xq.k] rqY;dkfjrk] dsyh izes;] Lodkfjrk] lekdkfjrk dh ewy izes;A
bdkbZ & IV
oy;] 'kwU; ds Hkktd] iw.kkZadh; izkUr rFkk {ks=] oy; rFkk iw.kkZadh; izkUr dk vfHky{k.k] mioy;]
mi{ks=] vHkkT; {ks=] oy;] lekdkfjrk rFkk rqY;dkfjrk] iw.kkZd
a h; izkUr dk {ks= es vUr% LFkkiu] foHkkx {ks=A
bdkbZ & V
xq.ktkofy;¡k ,oa xq.k/keZ] eq[; xq.ktkoyh] eq[; xq.ktkoyh oy;] vHkkT; xq.ktkoyh] mfPp"B
xq.ktkoyh] oy; lekdkfjrk ij ewyHkwr izes;] ;wfDyfM;u oy; ,oa xq.k/keZ] cgqin oy; iw.kkZdh; izkUr ,oa
{ks= ij cgqin oy;] fMfotuy ,Yxksjn~e A
7

ch-,-@ch-,l-lh- ¼Hkkx&izFke½ ds iz;ksxksa dh lwph
iz;ksfxd vad% foKku %25@dyk% 22
ch-,-&Hkkx izFke (Total Marks- 22)
Record
Practical -1
Practical -II
Viva-Voce

-

05
06
06
05

B.Sc. (Pt-I) (Total Marks- 25)
Record
05
Practical -1 07
Practical -II 07
Viva-Voce
06

1- pkj ;k pkj ls vf/kd ?kkr okys chth; cgqinksa dk ys[kk fp= vkjs[k.kA
2- cwyh; cht xf.kr dh lgk;rk ls rdZ ifjiFkksa dk ljyhdj.k djukA
3- cwyh; Qyuksa ds fy, lR;rk lkj.kh cukukA
4- leryh; oØksa dk vuqjs[k.k djukA
5- lewg lekdkfjrk dh ewy izes; ds vuqiz;ksxA
6- dqN fo”ks’k lafØ;kvksa ds fy, lafØ;k lkj.kh dk fuekZ.k djukA
7- xzkQh; fof/k ls chth; lehdj.kksa ds ewy Kkr djukA
8- ysxzkat izes; ds vuqiz;ksxA
9- Øep; ,oa Øep; lewg ls lEcfU/kr leL;k,aA
10- oy; ls lEcfU/kr leL;k,aA

iz'u i= & II- mPp dyu
le; % 3 ?kaVs
vf/kdre vad% foKku% 75@dyk% 66
uksV %
bl iz'u i= esa 03 [k.M fuEu izdkj gksaxsa %
[k.M v %
bl [k.M esa ,d vfuok;Z iz'u ftlesa izR;sd bdkbZ ls 02 y?kq iz'u ysrs gq, dqy 10 y?kq
iz'u gksaxs A izR;sd y?kq iz'u dk mÙkj yxHkx 20 'kCnksa esa gksA dqy vad% foKku%10@dyk%5
[k.M c %
bl [k.M esa izR;sd bdkbZ ls 02 iz'u ysrs gq, dqy 10 iz'u gksxsa s A izR;sd bdkbZ ls ,d
iz'u dk p;u djrs gq, dqy 05 iz'uksa ds mÙkj nsus gksaxs A izR;sd iz'u dk mÙkj yxHkx
250 'kCnksa esa gksA
dqy vad% foKku@dyk% 35
[k.M l %
bl [k.M esa 04 iz'u o.kZukREkd gksaxs ¼iz'u esa Hkkx Hkh gks ldrs gS½tks lHkh bdkbZ;ksa esa ls
fn, tkosaxs] fdUrq ,d bdkbZ ls ,d ls vf/kd iz'u ugha gksxkAnks iz'uksa ds mÙkj fn;s tkus
gSaAizR;sd iz'u dk mÙkj yxHkx 500 'kCnksa esa gks A
dqy vad% foKku% 30@dyk% 26
bdkbZ & I
/kzqoh funsZ'kkad] /kzqokUrj js[kk ,oa Li'kZ js[kk ds e/; dks.k] /kzqoh; v/k%Li'khZ ,oa v/kksyEc] Li'kZ js[kk
ij /kzqo ls yEc dh yEckbZA oØ dk ikfnd lehdj.k pki dh yEckbZ dk vkdyu ¼dkfrZ; ,oa /kzqoh;
funsZ'kkadks esa½ oØrk f=T;k ,oa fofHkUu lw=] oØrk dsUnz] oØrk thokA
bdkbZ & II
nks o nks ls vf/kd pjksa ds vkaf'kd vodyu] lEiw.kZ vodyu xq.kkad nks] rhu rFkk m pjksa ds
le?kkr Qyuksa ds fy, vk;yj izes;] la;qDr Qyu] vLi"V Qyuksa ds fy;s izFke o f}rh; vody xq.kkadA
nks pjksa ds Qyu ds fy;s Vsyj izes;] tsdksfc;u ,oa muds xq.k/keZ] nks o rhu pjksa ds fy;s mfPp"B] fufEu"B
,oa iY;k.k fcUnq] vfu/kk;Z xq.kkadks dh ykxzkat fof/kA
8

bdkbZ & III
vuUr Lif’kZ;ka] vUokyksi rFkk dsUnzt ufr ifjorZu fcUnw ,oa cgqy fcUnqvksa] mÙkyrk o voryrk gsrq
ijh{k.kA dkrhZ; ,oa /kzqoh; oØks dk vuqjs[k.kA
bdkbZ & IV
{kS=dyu] pkidyu ifjØe.k ?kukd`fr;ksa dk vk;ru ,oa i`"Bh; {ks=Qy] lekdyu fpUg ds vUrxZr
vodyuA
bdkbZ & V
chVk ,oa xkek Qyu] f} lekdyu Kkr djuk lekdyu ds Øe esa ifjorZu djuk ,oa /kzqoh;
funsZ'kkdksa esa ifjofrZr djukA

iz'u i= & III&lfn'k dyu ,ao funsZ'kkad T;kfefr
le; % 3 ?kaV
vf/kdre vad% foKku%75 @dyk%68
uksV %
bl iz'u i= esa 03 [k.M fuEu izdkj gksaxsa %
[k.M v %
bl [k.M esa ,d vfuok;Z iz'u ftlesa izR;sd bdkbZ ls 02 y?kq iz'u ysrs gq, dqy 10 y?kq
iz'u gksaxs A izR;sd y?kq iz'u dk mÙkj yxHkx 20 'kCnksa esa gks A dqy vad%foKku%10@dyk%5
[k.M c %
bl [k.M esa izR;sd bdkbZ ls 02 iz'u ysrs gq, dqy 10 iz'u gksxsa s A izR;sd bdkbZ ls ,d
iz'u dk p;u djrs gq, dqy 05 iz'uksa ds mÙkj nsus gksaxs A izR;sd iz'u dk mÙkj yxHkx
250 'kCnksa esa gksA
dqy vad% foKku@dyk % 35
[k.M l %
bl [k.M esa 04 iz'u o.kZukREkd gksaxs ¼iz'u esa Hkkx Hkh gks ldrs gS½tks lHkh bdkbZ;ksa esa ls
fn, tkosaxs] fdUrq ,d bdkbZ ls ,d ls vf/kd iz'u ugha gksxkAnks iz'uksa ds mÙkj fn;s tkus
gSaAizR;sd iz'u dk mÙkj yxHkx 500 'kCnksa esa gks A
dqy vad%foKku% 30@dyk% 28
bdkbZ & I
lfn'kksa dk vodyu ,oa lekdyu] xzsfM,UV] vilj.k ,oa dqary rFkk loZlfedk,a xkWl] LVksd ,oa
xzhu ds izes; ¼izek.kjfgr½ rFkk mu ij vk/kkfjr leL;k;saA
bdkbZ & II
ijoy; %& ekud lehdj.k] izkpfyd] funsZ'kkad] thok dh yEckbZ] LiZ'k js[kk vfHkyEc] xq.k/keZ] ,d
fcUnq ls nks Li'kZ js[kk,] LiZ'k thok] /kzqo] /kqzoh] e/; fcUnw ds :i esa thok] O;kl rFkk ,d fcUnw ls rhu
vfHkyEcA
nh/kZo`Ùk %& ekud lehdj.k] lgk;d o`Ùk] mRdsUnz dks.k] LiZ'k js[kk vfHkYkEc ,d fcUnw ls nks Li'kZ
js[kk,¡] LiZ'k thok] /kqzo] /kzqoh] e/; fcUnq ds :i esa thok] O;kl la;Xq eh O;kl ,oa ,d fcUnw ls pkj
vfHkyEcA
bdkbZ & III
vfr ijoy; %&ekud lehdj.k] izkpfyd] funsZ'kkad] vuUr Lif'kZ;ka vuUr Lif'kZ;ksa dks funsZ'k v{k
ekudj vfrijoy; dk lehdj.k] la;qXeh vfrjoy;] la;qXeh O;kl ds xq.k ,oaa vk;rh; vfrijoy; èkzqoh;
lehdj.k] ekud lehdj.k] fu;rk] LiZ'k js[kk vfHkyEc] /kqzoh ,oa vuUr Lif'kZ;kaA

9

bdkbZ & IV
xksyk %& fofHkUu :i ls ekud lehdj.k] leryh; ifjPNsn] nks xksyksa ds ifjPNsn ls xqtjus okys
xksys dk lehdj.k] fcUnq dh 'kfDr] Li'kZ ry] /kzqoh; ry] /kqzoh; js[kk,¡] nks ewy js[kk] lek{k xksyksa dk
fudk; rFkk lhekUr fcUnqA
'kadq] % ftldk 'kh"kZ o funsZ'kd oØ bafxr gks] vUokyksih 'kadq] f}?kkr lehdj.k }kjk ,d 'kadq dks
izznf'kZr djus dk izfrcU/k] Li'kZ ry] O;qRØe 'kadq] 'kadq dks ,d lery }kjk dkVus ij izkIr nks js[kkvksa ds
eè; dks.k] rhu ijLij ledksf.kd tud js[kkvksa dk izfrcU/k] rFkk yEc o`Ùkh; 'kadqA
csyu% yEc o`Ùkh; csyu rFkk vUokyksih csyuA
bdkbZ & V
dsUnzh; 'kkadot %& ekud lehdj.k] Li'kZ ry] Li'kZrk dk izfrcU/k] fu;ked xksyk] /kzoq h; ry]
/kzqoh; js[kk,¡] fn;s dsUnz okyk ifjPNsn] vUokyksih csyu ,oa vUokyksih 'kadq nh?kZo`rt vfHkyEc] ,d fcUnw ls
N% vfHkyEc N% vfHkyEcksa ls tkus okyk 'kadq] la;qXeh O;kl ,oa muds xq.k/keZA

10