Mathematics Optional Paper 2- 2015 | UPSC Previous Year Question Papers and Video Analysis PDF Download

 Page 1


C -A V Z -O -N B U B
WW : 3T^ : 250
^ ^ fiRfafisTfl fiM ^ f^ n 'ffra q i^ * ^ )
^ $ f^ 3 H 3 O T l^ *iT T f*ft %<$ ^ 3t M ^ f I
auJkcM ^ ^ % 3 t R ^ 11
TO W s9T 1 3fK 5 3?tof I cm i 3T^t ro t 3 $ T T f^ q fJ W ^ $ H cfi WR cffa TO! % 3tR
ilr^«h OT/^FT % ‘f^ T c T 3R> "33% W & 1^ ^ f I
t o ! % -m 3#iftT *n«ra ^ % ! *rA t e s i 3#a w& id m -m 3 t o ^wr I, afa w T mzm ^r t o
TO -W -^tR ( ^ o # o T*o) ^ f ^ T % Tg&fS ^ M s W R ^ t o ¦ 5 T R 1 Tnf^fcf 3R?
f^ b ’tfl "RIW ? ^ R r tts l ^ 3tR ^ «$| 3^> "P-R ^tl
-qfe 3 m w z it, e ft w ^ tf^ ?«n vrnrn M s ^ f ^ i
< r a s ^ ^%?i ? n *n 3 r^fcf?r 3 t«t ! 3 sp> f i
3ERt % M m l ~ £ \ ¥H 1 O T ^ K ^ f t l ^ l^ i T [ T 7 3^ft % 3rR ^ ^ ^PTcH ^ *n x >rft 3$ T O ^ !
^ # i 3 w ft ^ ^ ^ &m\ y® % vft Tgk: wiz <?tf^i
MATHEMATICS (PAPER-II)
Time Allowed : Three Hours Maximum Marks : 250
QUESTION PAPER SPECIFIC INSTRUCTIONS
(Please read each of the following instructions carefully before attempting questions)
There are EIGHT questions divided in two Sections and printed both in HINDI and 
in ENGLISH.
Candidate has to attempt FIVE questions in all.
Question Nos. 1 and 5 are compulsory and out of the remaining, THREE are to be attempted 
choosing at least ONE question from each Section.
The number of marks carried by a question/part is indicated against it.
Answers must be written in the medium authorized in the Admission Certificate which must 
be stated clearly on the cover of this Question-cum-Answer (QCA) Booklet in the space 
provided. No marks will be given for answers written in medium other than the authorized
Assume suitable data, if considered necessary, and indicate the same clearly.
Unless otherwise indicated, symbols and notations carry their usual standard meanings. 
Attempts of questions shall be counted in sequential order. Unless struck off, attempt of a 
question shall be counted even if attempted partly. Any page or portion of the page left blank 
in the Question-cum-Answer Booklet must be clearly struck off.
e-ffVX-0-RB8B/35 1 [ P.T.O.
Page 2


C -A V Z -O -N B U B
WW : 3T^ : 250
^ ^ fiRfafisTfl fiM ^ f^ n 'ffra q i^ * ^ )
^ $ f^ 3 H 3 O T l^ *iT T f*ft %<$ ^ 3t M ^ f I
auJkcM ^ ^ % 3 t R ^ 11
TO W s9T 1 3fK 5 3?tof I cm i 3T^t ro t 3 $ T T f^ q fJ W ^ $ H cfi WR cffa TO! % 3tR
ilr^«h OT/^FT % ‘f^ T c T 3R> "33% W & 1^ ^ f I
t o ! % -m 3#iftT *n«ra ^ % ! *rA t e s i 3#a w& id m -m 3 t o ^wr I, afa w T mzm ^r t o
TO -W -^tR ( ^ o # o T*o) ^ f ^ T % Tg&fS ^ M s W R ^ t o ¦ 5 T R 1 Tnf^fcf 3R?
f^ b ’tfl "RIW ? ^ R r tts l ^ 3tR ^ «$| 3^> "P-R ^tl
-qfe 3 m w z it, e ft w ^ tf^ ?«n vrnrn M s ^ f ^ i
< r a s ^ ^%?i ? n *n 3 r^fcf?r 3 t«t ! 3 sp> f i
3ERt % M m l ~ £ \ ¥H 1 O T ^ K ^ f t l ^ l^ i T [ T 7 3^ft % 3rR ^ ^ ^PTcH ^ *n x >rft 3$ T O ^ !
^ # i 3 w ft ^ ^ ^ &m\ y® % vft Tgk: wiz <?tf^i
MATHEMATICS (PAPER-II)
Time Allowed : Three Hours Maximum Marks : 250
QUESTION PAPER SPECIFIC INSTRUCTIONS
(Please read each of the following instructions carefully before attempting questions)
There are EIGHT questions divided in two Sections and printed both in HINDI and 
in ENGLISH.
Candidate has to attempt FIVE questions in all.
Question Nos. 1 and 5 are compulsory and out of the remaining, THREE are to be attempted 
choosing at least ONE question from each Section.
The number of marks carried by a question/part is indicated against it.
Answers must be written in the medium authorized in the Admission Certificate which must 
be stated clearly on the cover of this Question-cum-Answer (QCA) Booklet in the space 
provided. No marks will be given for answers written in medium other than the authorized
Assume suitable data, if considered necessary, and indicate the same clearly.
Unless otherwise indicated, symbols and notations carry their usual standard meanings. 
Attempts of questions shall be counted in sequential order. Unless struck off, attempt of a 
question shall be counted even if attempted partly. Any page or portion of the page left blank 
in the Question-cum-Answer Booklet must be clearly struck off.
e-ffVX-0-RB8B/35 1 [ P.T.O.
A / Section—A
1. (a) ' (i) ^ 8 ^rn2?n'#f^T|
How many generators are there of the cyclic group G of order 8? Explain.
(ix) 4%^^PJp {e, a, b, c} e dr^M^ (snfifefe) «H i5 ?^
" 3 1 ? fe> t H lfl 1 » I
Taking a group {e, a, b, c} of order 4, where e is the identity, construct 
composition tables showing that one is cyclic while the other is not.
5+5=10
(b) ^ w i te s t m m % t o 1 1
Give an example of a ring having identity but a subring of this having a 
different identity. 10
(c) W ^ ( - i ) n + 1 - J L _ % vfivm vfasm i
n =1 n +1
oo ^
Test the convergence and absolute convergence of the series V (-l)rl + 1 —-----. 10
n =1 ^ +1
(d) % W l v{x, ijj- \n{x2 +T/2) + x + iy M ^ t ^1 W f u[x, W
/(2} = u + iy z % ^ ^ ^ T cl ^TTT|
Show that the function v{x, ]J \ = ln(x2 + y2) + x + y is harmonic. Find its
conjugate harmonic function u{x, ly ). Also, find the corresponding analytic 
function f{z!\ = u + iv in terms of z, 10
(e) ftnfafed w m i T& M t ^ % frR wi :
Solve the following assignment problem to maximize the sales : 10
Territories ($fc)
1 2 7 m N V 7
A 3 4 5 6 7
B 4 15 13 7 6
C 6 13 12 5 11
D 7 12 15 8 5
E 8 13 10 6 9
C-7IVX-0-RBHB/35 2
Page 3


C -A V Z -O -N B U B
WW : 3T^ : 250
^ ^ fiRfafisTfl fiM ^ f^ n 'ffra q i^ * ^ )
^ $ f^ 3 H 3 O T l^ *iT T f*ft %<$ ^ 3t M ^ f I
auJkcM ^ ^ % 3 t R ^ 11
TO W s9T 1 3fK 5 3?tof I cm i 3T^t ro t 3 $ T T f^ q fJ W ^ $ H cfi WR cffa TO! % 3tR
ilr^«h OT/^FT % ‘f^ T c T 3R> "33% W & 1^ ^ f I
t o ! % -m 3#iftT *n«ra ^ % ! *rA t e s i 3#a w& id m -m 3 t o ^wr I, afa w T mzm ^r t o
TO -W -^tR ( ^ o # o T*o) ^ f ^ T % Tg&fS ^ M s W R ^ t o ¦ 5 T R 1 Tnf^fcf 3R?
f^ b ’tfl "RIW ? ^ R r tts l ^ 3tR ^ «$| 3^> "P-R ^tl
-qfe 3 m w z it, e ft w ^ tf^ ?«n vrnrn M s ^ f ^ i
< r a s ^ ^%?i ? n *n 3 r^fcf?r 3 t«t ! 3 sp> f i
3ERt % M m l ~ £ \ ¥H 1 O T ^ K ^ f t l ^ l^ i T [ T 7 3^ft % 3rR ^ ^ ^PTcH ^ *n x >rft 3$ T O ^ !
^ # i 3 w ft ^ ^ ^ &m\ y® % vft Tgk: wiz <?tf^i
MATHEMATICS (PAPER-II)
Time Allowed : Three Hours Maximum Marks : 250
QUESTION PAPER SPECIFIC INSTRUCTIONS
(Please read each of the following instructions carefully before attempting questions)
There are EIGHT questions divided in two Sections and printed both in HINDI and 
in ENGLISH.
Candidate has to attempt FIVE questions in all.
Question Nos. 1 and 5 are compulsory and out of the remaining, THREE are to be attempted 
choosing at least ONE question from each Section.
The number of marks carried by a question/part is indicated against it.
Answers must be written in the medium authorized in the Admission Certificate which must 
be stated clearly on the cover of this Question-cum-Answer (QCA) Booklet in the space 
provided. No marks will be given for answers written in medium other than the authorized
Assume suitable data, if considered necessary, and indicate the same clearly.
Unless otherwise indicated, symbols and notations carry their usual standard meanings. 
Attempts of questions shall be counted in sequential order. Unless struck off, attempt of a 
question shall be counted even if attempted partly. Any page or portion of the page left blank 
in the Question-cum-Answer Booklet must be clearly struck off.
e-ffVX-0-RB8B/35 1 [ P.T.O.
A / Section—A
1. (a) ' (i) ^ 8 ^rn2?n'#f^T|
How many generators are there of the cyclic group G of order 8? Explain.
(ix) 4%^^PJp {e, a, b, c} e dr^M^ (snfifefe) «H i5 ?^
" 3 1 ? fe> t H lfl 1 » I
Taking a group {e, a, b, c} of order 4, where e is the identity, construct 
composition tables showing that one is cyclic while the other is not.
5+5=10
(b) ^ w i te s t m m % t o 1 1
Give an example of a ring having identity but a subring of this having a 
different identity. 10
(c) W ^ ( - i ) n + 1 - J L _ % vfivm vfasm i
n =1 n +1
oo ^
Test the convergence and absolute convergence of the series V (-l)rl + 1 —-----. 10
n =1 ^ +1
(d) % W l v{x, ijj- \n{x2 +T/2) + x + iy M ^ t ^1 W f u[x, W
/(2} = u + iy z % ^ ^ ^ T cl ^TTT|
Show that the function v{x, ]J \ = ln(x2 + y2) + x + y is harmonic. Find its
conjugate harmonic function u{x, ly ). Also, find the corresponding analytic 
function f{z!\ = u + iv in terms of z, 10
(e) ftnfafed w m i T& M t ^ % frR wi :
Solve the following assignment problem to maximize the sales : 10
Territories ($fc)
1 2 7 m N V 7
A 3 4 5 6 7
B 4 15 13 7 6
C 6 13 12 5 11
D 7 12 15 8 5
E 8 13 10 6 9
C-7IVX-0-RBHB/35 2
2. (a) ^ R T& I, 1^FT 1 % < W \ R STF^Tc^ /?' (j? onto R ') ^
Whiten (j) t, c T t /?' clrH M * 3 ^ ? 0(1) 11
If i? is a ring with unit element 1 and < J > is a homomorphism of R onto R', 
prove that $(1) is the unit element of R'. 15
( b) W i
/W =
_ I n * n + 1 n
0, x = 0
ftRH I? e ft £ / {x )d x ^ I^ H P
Is the function
\ —^~r < X < -i-
/ ( * ) = " n+1 n
[0, x = 0
Riemann integrable? If yes, obtain the value of [\f{x)dx.
J Q 15
(c) w i / ( z ) = — 2 z 3 - ~ % 2 = 0 % c M 3 3 m r t f ^ t o
z 2 -3 z + 2
Find all possible Taylor’s and Laurent’s series expansions of the function 
2 z-3
/(z) - ------- about the point z - 0.
z -3 z + 2 20
3. (a) 3*=nfa W 1 ^ ^
r e 2 +1
z{z + l){z-~i)‘
dz; C:\z\-2
*TR te#pr|
State Cauchy’s residue theorem. Using it, evaluate the integral 
r 6 2 +1
f ---- g- +1 — dz; C: | z| = 2
Jc z{z + \ ){z-i)2 15
oo
(fy M V ---- —----% fe T C ,
n=i (1 + n2x 2)
nx
Test the series of functions V  -----—— for uniform convergence. 15
n=l (1 + n X )
C-?TVK-0-RBaB/35 3 ( P.T.O.
Page 4


C -A V Z -O -N B U B
WW : 3T^ : 250
^ ^ fiRfafisTfl fiM ^ f^ n 'ffra q i^ * ^ )
^ $ f^ 3 H 3 O T l^ *iT T f*ft %<$ ^ 3t M ^ f I
auJkcM ^ ^ % 3 t R ^ 11
TO W s9T 1 3fK 5 3?tof I cm i 3T^t ro t 3 $ T T f^ q fJ W ^ $ H cfi WR cffa TO! % 3tR
ilr^«h OT/^FT % ‘f^ T c T 3R> "33% W & 1^ ^ f I
t o ! % -m 3#iftT *n«ra ^ % ! *rA t e s i 3#a w& id m -m 3 t o ^wr I, afa w T mzm ^r t o
TO -W -^tR ( ^ o # o T*o) ^ f ^ T % Tg&fS ^ M s W R ^ t o ¦ 5 T R 1 Tnf^fcf 3R?
f^ b ’tfl "RIW ? ^ R r tts l ^ 3tR ^ «$| 3^> "P-R ^tl
-qfe 3 m w z it, e ft w ^ tf^ ?«n vrnrn M s ^ f ^ i
< r a s ^ ^%?i ? n *n 3 r^fcf?r 3 t«t ! 3 sp> f i
3ERt % M m l ~ £ \ ¥H 1 O T ^ K ^ f t l ^ l^ i T [ T 7 3^ft % 3rR ^ ^ ^PTcH ^ *n x >rft 3$ T O ^ !
^ # i 3 w ft ^ ^ ^ &m\ y® % vft Tgk: wiz <?tf^i
MATHEMATICS (PAPER-II)
Time Allowed : Three Hours Maximum Marks : 250
QUESTION PAPER SPECIFIC INSTRUCTIONS
(Please read each of the following instructions carefully before attempting questions)
There are EIGHT questions divided in two Sections and printed both in HINDI and 
in ENGLISH.
Candidate has to attempt FIVE questions in all.
Question Nos. 1 and 5 are compulsory and out of the remaining, THREE are to be attempted 
choosing at least ONE question from each Section.
The number of marks carried by a question/part is indicated against it.
Answers must be written in the medium authorized in the Admission Certificate which must 
be stated clearly on the cover of this Question-cum-Answer (QCA) Booklet in the space 
provided. No marks will be given for answers written in medium other than the authorized
Assume suitable data, if considered necessary, and indicate the same clearly.
Unless otherwise indicated, symbols and notations carry their usual standard meanings. 
Attempts of questions shall be counted in sequential order. Unless struck off, attempt of a 
question shall be counted even if attempted partly. Any page or portion of the page left blank 
in the Question-cum-Answer Booklet must be clearly struck off.
e-ffVX-0-RB8B/35 1 [ P.T.O.
A / Section—A
1. (a) ' (i) ^ 8 ^rn2?n'#f^T|
How many generators are there of the cyclic group G of order 8? Explain.
(ix) 4%^^PJp {e, a, b, c} e dr^M^ (snfifefe) «H i5 ?^
" 3 1 ? fe> t H lfl 1 » I
Taking a group {e, a, b, c} of order 4, where e is the identity, construct 
composition tables showing that one is cyclic while the other is not.
5+5=10
(b) ^ w i te s t m m % t o 1 1
Give an example of a ring having identity but a subring of this having a 
different identity. 10
(c) W ^ ( - i ) n + 1 - J L _ % vfivm vfasm i
n =1 n +1
oo ^
Test the convergence and absolute convergence of the series V (-l)rl + 1 —-----. 10
n =1 ^ +1
(d) % W l v{x, ijj- \n{x2 +T/2) + x + iy M ^ t ^1 W f u[x, W
/(2} = u + iy z % ^ ^ ^ T cl ^TTT|
Show that the function v{x, ]J \ = ln(x2 + y2) + x + y is harmonic. Find its
conjugate harmonic function u{x, ly ). Also, find the corresponding analytic 
function f{z!\ = u + iv in terms of z, 10
(e) ftnfafed w m i T& M t ^ % frR wi :
Solve the following assignment problem to maximize the sales : 10
Territories ($fc)
1 2 7 m N V 7
A 3 4 5 6 7
B 4 15 13 7 6
C 6 13 12 5 11
D 7 12 15 8 5
E 8 13 10 6 9
C-7IVX-0-RBHB/35 2
2. (a) ^ R T& I, 1^FT 1 % < W \ R STF^Tc^ /?' (j? onto R ') ^
Whiten (j) t, c T t /?' clrH M * 3 ^ ? 0(1) 11
If i? is a ring with unit element 1 and < J > is a homomorphism of R onto R', 
prove that $(1) is the unit element of R'. 15
( b) W i
/W =
_ I n * n + 1 n
0, x = 0
ftRH I? e ft £ / {x )d x ^ I^ H P
Is the function
\ —^~r < X < -i-
/ ( * ) = " n+1 n
[0, x = 0
Riemann integrable? If yes, obtain the value of [\f{x)dx.
J Q 15
(c) w i / ( z ) = — 2 z 3 - ~ % 2 = 0 % c M 3 3 m r t f ^ t o
z 2 -3 z + 2
Find all possible Taylor’s and Laurent’s series expansions of the function 
2 z-3
/(z) - ------- about the point z - 0.
z -3 z + 2 20
3. (a) 3*=nfa W 1 ^ ^
r e 2 +1
z{z + l){z-~i)‘
dz; C:\z\-2
*TR te#pr|
State Cauchy’s residue theorem. Using it, evaluate the integral 
r 6 2 +1
f ---- g- +1 — dz; C: | z| = 2
Jc z{z + \ ){z-i)2 15
oo
(fy M V ---- —----% fe T C ,
n=i (1 + n2x 2)
nx
Test the series of functions V  -----—— for uniform convergence. 15
n=l (1 + n X )
C-?TVK-0-RBaB/35 3 ( P.T.O.
(c) TtoPH wm\ T K :
"^ tf^ n r z = Xj + 2x2 - 3 x 3 + 4x4
f t
x\ + x 2 + 2x3 + 3 x 4 =12 
x2 + 2x3 + x 4 = 8
xl> *2> X3> J C 4 
Consider the following linear programming problem :
Maximize Z = x{ + 2x2 -3 x 3 + 4x4 
subject to
x: + x2 + 2x3 + 3 x 4 =12 
x2 + 2x3 + x4 = 8 
Xj, x2, x3, x4 £ 0
I 3?R 3 T O 3 W t fS F T cT W I?
Using the definition, find its all basic solutions. Which of these are
degenerate basic feasible solutions and which are non-degenerate basic
feasible solutions?
(a) smnfl^*Fra
I/ I?
Without solving the problem, show that it has an optimal solution. Which 
of the basic feasible solution(s) is/are optimal? 20
4. (a) W fWeffecI Wpm W I c T C T T T T r f f? ^ F T T % , c fr ^cTT^
% ^ 1 ? :
Do the following sets form integral domains with respect to ordinary addition 
and multiplication? If so, state if they are fields : 5+6+4=15
(i) b42 % ^ ^ b ^
The set of numbers of the form bj2 with b rational
(ii) m
The set of even integers
(iii) ^ fH ti^<<
The set of positive integers
(b) x 2 +2 y2 < 1 ^ W I /(x, £ / ) = x 2 +3 y2 -zy% TO ftndH iIRt
TO ^ tf^ l
Find the absolute maximum and minimum values of the function 
/(*> y )~ x2 +3y2 - y over the region x 2 +2y2 <1. 15
C-ffVZ-0-RBHB / 35
f
4
Page 5


C -A V Z -O -N B U B
WW : 3T^ : 250
^ ^ fiRfafisTfl fiM ^ f^ n 'ffra q i^ * ^ )
^ $ f^ 3 H 3 O T l^ *iT T f*ft %<$ ^ 3t M ^ f I
auJkcM ^ ^ % 3 t R ^ 11
TO W s9T 1 3fK 5 3?tof I cm i 3T^t ro t 3 $ T T f^ q fJ W ^ $ H cfi WR cffa TO! % 3tR
ilr^«h OT/^FT % ‘f^ T c T 3R> "33% W & 1^ ^ f I
t o ! % -m 3#iftT *n«ra ^ % ! *rA t e s i 3#a w& id m -m 3 t o ^wr I, afa w T mzm ^r t o
TO -W -^tR ( ^ o # o T*o) ^ f ^ T % Tg&fS ^ M s W R ^ t o ¦ 5 T R 1 Tnf^fcf 3R?
f^ b ’tfl "RIW ? ^ R r tts l ^ 3tR ^ «$| 3^> "P-R ^tl
-qfe 3 m w z it, e ft w ^ tf^ ?«n vrnrn M s ^ f ^ i
< r a s ^ ^%?i ? n *n 3 r^fcf?r 3 t«t ! 3 sp> f i
3ERt % M m l ~ £ \ ¥H 1 O T ^ K ^ f t l ^ l^ i T [ T 7 3^ft % 3rR ^ ^ ^PTcH ^ *n x >rft 3$ T O ^ !
^ # i 3 w ft ^ ^ ^ &m\ y® % vft Tgk: wiz <?tf^i
MATHEMATICS (PAPER-II)
Time Allowed : Three Hours Maximum Marks : 250
QUESTION PAPER SPECIFIC INSTRUCTIONS
(Please read each of the following instructions carefully before attempting questions)
There are EIGHT questions divided in two Sections and printed both in HINDI and 
in ENGLISH.
Candidate has to attempt FIVE questions in all.
Question Nos. 1 and 5 are compulsory and out of the remaining, THREE are to be attempted 
choosing at least ONE question from each Section.
The number of marks carried by a question/part is indicated against it.
Answers must be written in the medium authorized in the Admission Certificate which must 
be stated clearly on the cover of this Question-cum-Answer (QCA) Booklet in the space 
provided. No marks will be given for answers written in medium other than the authorized
Assume suitable data, if considered necessary, and indicate the same clearly.
Unless otherwise indicated, symbols and notations carry their usual standard meanings. 
Attempts of questions shall be counted in sequential order. Unless struck off, attempt of a 
question shall be counted even if attempted partly. Any page or portion of the page left blank 
in the Question-cum-Answer Booklet must be clearly struck off.
e-ffVX-0-RB8B/35 1 [ P.T.O.
A / Section—A
1. (a) ' (i) ^ 8 ^rn2?n'#f^T|
How many generators are there of the cyclic group G of order 8? Explain.
(ix) 4%^^PJp {e, a, b, c} e dr^M^ (snfifefe) «H i5 ?^
" 3 1 ? fe> t H lfl 1 » I
Taking a group {e, a, b, c} of order 4, where e is the identity, construct 
composition tables showing that one is cyclic while the other is not.
5+5=10
(b) ^ w i te s t m m % t o 1 1
Give an example of a ring having identity but a subring of this having a 
different identity. 10
(c) W ^ ( - i ) n + 1 - J L _ % vfivm vfasm i
n =1 n +1
oo ^
Test the convergence and absolute convergence of the series V (-l)rl + 1 —-----. 10
n =1 ^ +1
(d) % W l v{x, ijj- \n{x2 +T/2) + x + iy M ^ t ^1 W f u[x, W
/(2} = u + iy z % ^ ^ ^ T cl ^TTT|
Show that the function v{x, ]J \ = ln(x2 + y2) + x + y is harmonic. Find its
conjugate harmonic function u{x, ly ). Also, find the corresponding analytic 
function f{z!\ = u + iv in terms of z, 10
(e) ftnfafed w m i T& M t ^ % frR wi :
Solve the following assignment problem to maximize the sales : 10
Territories ($fc)
1 2 7 m N V 7
A 3 4 5 6 7
B 4 15 13 7 6
C 6 13 12 5 11
D 7 12 15 8 5
E 8 13 10 6 9
C-7IVX-0-RBHB/35 2
2. (a) ^ R T& I, 1^FT 1 % < W \ R STF^Tc^ /?' (j? onto R ') ^
Whiten (j) t, c T t /?' clrH M * 3 ^ ? 0(1) 11
If i? is a ring with unit element 1 and < J > is a homomorphism of R onto R', 
prove that $(1) is the unit element of R'. 15
( b) W i
/W =
_ I n * n + 1 n
0, x = 0
ftRH I? e ft £ / {x )d x ^ I^ H P
Is the function
\ —^~r < X < -i-
/ ( * ) = " n+1 n
[0, x = 0
Riemann integrable? If yes, obtain the value of [\f{x)dx.
J Q 15
(c) w i / ( z ) = — 2 z 3 - ~ % 2 = 0 % c M 3 3 m r t f ^ t o
z 2 -3 z + 2
Find all possible Taylor’s and Laurent’s series expansions of the function 
2 z-3
/(z) - ------- about the point z - 0.
z -3 z + 2 20
3. (a) 3*=nfa W 1 ^ ^
r e 2 +1
z{z + l){z-~i)‘
dz; C:\z\-2
*TR te#pr|
State Cauchy’s residue theorem. Using it, evaluate the integral 
r 6 2 +1
f ---- g- +1 — dz; C: | z| = 2
Jc z{z + \ ){z-i)2 15
oo
(fy M V ---- —----% fe T C ,
n=i (1 + n2x 2)
nx
Test the series of functions V  -----—— for uniform convergence. 15
n=l (1 + n X )
C-?TVK-0-RBaB/35 3 ( P.T.O.
(c) TtoPH wm\ T K :
"^ tf^ n r z = Xj + 2x2 - 3 x 3 + 4x4
f t
x\ + x 2 + 2x3 + 3 x 4 =12 
x2 + 2x3 + x 4 = 8
xl> *2> X3> J C 4 
Consider the following linear programming problem :
Maximize Z = x{ + 2x2 -3 x 3 + 4x4 
subject to
x: + x2 + 2x3 + 3 x 4 =12 
x2 + 2x3 + x4 = 8 
Xj, x2, x3, x4 £ 0
I 3?R 3 T O 3 W t fS F T cT W I?
Using the definition, find its all basic solutions. Which of these are
degenerate basic feasible solutions and which are non-degenerate basic
feasible solutions?
(a) smnfl^*Fra
I/ I?
Without solving the problem, show that it has an optimal solution. Which 
of the basic feasible solution(s) is/are optimal? 20
4. (a) W fWeffecI Wpm W I c T C T T T T r f f? ^ F T T % , c fr ^cTT^
% ^ 1 ? :
Do the following sets form integral domains with respect to ordinary addition 
and multiplication? If so, state if they are fields : 5+6+4=15
(i) b42 % ^ ^ b ^
The set of numbers of the form bj2 with b rational
(ii) m
The set of even integers
(iii) ^ fH ti^<<
The set of positive integers
(b) x 2 +2 y2 < 1 ^ W I /(x, £ / ) = x 2 +3 y2 -zy% TO ftndH iIRt
TO ^ tf^ l
Find the absolute maximum and minimum values of the function 
/(*> y )~ x2 +3y2 - y over the region x 2 +2y2 <1. 15
C-ffVZ-0-RBHB / 35
f
4
(c) ftofaf&s i m n m vn ^ o t jw ft left m m f^ f^ i ^ ^
wm\ ^ T T R ^ ft 3 left wm\ ^r ^ ^ :
s r te p te n z = 2x1 -4 x 2 +5x3 
%
Xj + 4x2 -2 x 3 < 2 
-x t + 2x2 + 3x3 £ 1 
Xj, x 2, x3 £0
Solve the following linear programming problem by the simplex method. Write 
its dual. Also, write the optimal solution of the dual from the optimal table of 
the given problem : 20
Maximize Z -2 x x -4 x 2 +5x3
subject to
xL +4x2 - 2x 3 <2 
-Xj +2x2 +3x3 £1 
X j, x2, x3 £0
Tgvz—B / Section—B
5. (a) ETftOT
Q / 2 + z 2 - x 2) p - 2xyq + 2xz = 0
p = — cmTg = ~ , ^ t 
dx 3y
Solve the partial differential equation
{y2 + z 2 - x 2) p - 2xyq + 2xz = 0
, 3z , dz
where p = — and q = — .
dx dy 10
(b) (D2 + D D '-2 D /2)u = e x+y ^ ^ ^tf^, ^ t j D = A ^ r £>' = — .
3x 3i/
Solve (D2 +D D ' - 2D '2) u - ex+ y, where D = ~ and D ' = A . 10
3x 3i/
fc/ ((paq) — » r) v ({pAq) -» -r) % frR cffa p, g, r 3 (3 T «ren
f^fr^fcr ^ ^rm fan ^ t t ^ M u t 1 %
Find the principal (or canonical) disjunctive normal form in three variables 
p, q, r for the Boolean expression ((p a q) -» r) v ((p a q) -» -r). Is the given 
Boolean expression a contradiction or a tautology? 10
C-HV2-0-RBHB/35 5 [ P.T.O.