Mathematics Optional Paper 1- 2015 | UPSC Previous Year Question Papers and Video Analysis PDF Download

 Page 1


^ CS (M ain) Exam:20l5
T lf^ T rT
I
MATHEMATICS 
Paper—I
3ff&Fm 3T¥ : 250 
Maximum Marks : 250
% frftr s f j& t
f i m j m f ^ 3r7T P n = r f^ m X cfcz w f t v r # y/Hg«f+ :
& J Z (8) wt $ * ? t it w*sf fit* n fv fd ' $ mr f&ft $h mfrft if if $ wt i t
v f i f f l f f # < £ o f w fF £ \Jr7T ^ # I
o
jo t f r ^ ? r 1 3ftr 5 yfowf i mr wi¥t # ' # # wr-i-*m w z ft jm rr f ^ f rffr
sn pff £ zm
3 c * fa > WFT/'m £ 3f¥ far t j j t f /
STF# £ 3 W /cT# W $ W #^ ftm > T JF&W 3 fN $ 7#W-W #' ^77 W #, 3 fk fff
w sw w ? w w vw -w -zm : (*%.&%.) yf&mi $ j w - t o ? t 3 ^ t i t few
W F T T W!%m j/rc?/&<? * fT & l* T $ tfRtft'ffi 3FU M W tm $ fcfW W SWT TT 3R? ^ I
z# s jjc R f f lc f i $tf $ srM & f m w f mr foffw tftftnri
m zfrrrf& ti n it, w$w mr mmcff w^m m f # jtjw # /
m t' £ swtf ^ mm m i$ m ¥f wrfti vft mzi *iit it, tit wm $ vm ¥f mm & f w rw ff
w£ $ f > zm mm: few mr w t i wrr-w-zm yf&ivr #‘ mft ®ter f sr r m ; ? w £ m # ? w
wv # to t mm wfeqi
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. I and 5 are compulsory and out of the remaining, THREE are to be attempted choosing at 
least ONE 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 a medium other than the authorized one.
Assume suitable data, if considered necessary, and indicate the same clearly.
Unless and otherwise indicated, symbols and notations carry their usual standard meaning.
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.
PrvfFfrr w : #=r w i
Time Allowed: Three Hours
G-AVZ-O-NBUA
V-&T°Z~0~s£B7&Z 1
+
Page 2


^ CS (M ain) Exam:20l5
T lf^ T rT
I
MATHEMATICS 
Paper—I
3ff&Fm 3T¥ : 250 
Maximum Marks : 250
% frftr s f j& t
f i m j m f ^ 3r7T P n = r f^ m X cfcz w f t v r # y/Hg«f+ :
& J Z (8) wt $ * ? t it w*sf fit* n fv fd ' $ mr f&ft $h mfrft if if $ wt i t
v f i f f l f f # < £ o f w fF £ \Jr7T ^ # I
o
jo t f r ^ ? r 1 3ftr 5 yfowf i mr wi¥t # ' # # wr-i-*m w z ft jm rr f ^ f rffr
sn pff £ zm
3 c * fa > WFT/'m £ 3f¥ far t j j t f /
STF# £ 3 W /cT# W $ W #^ ftm > T JF&W 3 fN $ 7#W-W #' ^77 W #, 3 fk fff
w sw w ? w w vw -w -zm : (*%.&%.) yf&mi $ j w - t o ? t 3 ^ t i t few
W F T T W!%m j/rc?/&<? * fT & l* T $ tfRtft'ffi 3FU M W tm $ fcfW W SWT TT 3R? ^ I
z# s jjc R f f lc f i $tf $ srM & f m w f mr foffw tftftnri
m zfrrrf& ti n it, w$w mr mmcff w^m m f # jtjw # /
m t' £ swtf ^ mm m i$ m ¥f wrfti vft mzi *iit it, tit wm $ vm ¥f mm & f w rw ff
w£ $ f > zm mm: few mr w t i wrr-w-zm yf&ivr #‘ mft ®ter f sr r m ; ? w £ m # ? w
wv # to t mm wfeqi
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. I and 5 are compulsory and out of the remaining, THREE are to be attempted choosing at 
least ONE 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 a medium other than the authorized one.
Assume suitable data, if considered necessary, and indicate the same clearly.
Unless and otherwise indicated, symbols and notations carry their usual standard meaning.
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.
PrvfFfrr w : #=r w i
Time Allowed: Three Hours
G-AVZ-O-NBUA
V-&T°Z~0~s£B7&Z 1
+
< p |W 5 — 3 T
SECTION—A
Q. 1(a) fti* in? V, « (1, 1, 2, 4), V2 - (2, -1 , -5 , 2), V3 - (1, -1 , -4, 0) cW T
v 4 = (2, l, l, 6) tfti+a : FRfa # i m vs -m t ? ^trc % w $f w # i
Q. 1(b) PlHfcHfed 3 WftcT 3^T dr^M I^ f^ lfc R :
"1 2 3 4"
2 1 4 5
1 5 5 7 -
8 1 14 17_
Reduce the following matrix to row echelon form and hence find its rank :
"1 2 3 4 '
2 1 4 5
The vectors Vj - (1, 1, 2, 4), V2 - (2, - I , -5 , 2), V3 = (1, -1 , -4 , 0) and 
V4 = (2, 1, 1 ,6 ) are linearly independent. Is it true ? Justify your answer. 10
1 5 5 7
8 1 14 17
10
Q. 1(c) frRfeffefl tffaT *TR f^TicR :
?
\ * •/
Evaluate the following lim it:
10
Q. 1(d) PlHlelfifl'tf HHWiA 7 T P T :
Vsinx
Vsinx + Vcosx cosx
Evaluate the following integral:
I t
Vsinx
10
%-&PZ-0-tf35r&i
2
Page 3


^ CS (M ain) Exam:20l5
T lf^ T rT
I
MATHEMATICS 
Paper—I
3ff&Fm 3T¥ : 250 
Maximum Marks : 250
% frftr s f j& t
f i m j m f ^ 3r7T P n = r f^ m X cfcz w f t v r # y/Hg«f+ :
& J Z (8) wt $ * ? t it w*sf fit* n fv fd ' $ mr f&ft $h mfrft if if $ wt i t
v f i f f l f f # < £ o f w fF £ \Jr7T ^ # I
o
jo t f r ^ ? r 1 3ftr 5 yfowf i mr wi¥t # ' # # wr-i-*m w z ft jm rr f ^ f rffr
sn pff £ zm
3 c * fa > WFT/'m £ 3f¥ far t j j t f /
STF# £ 3 W /cT# W $ W #^ ftm > T JF&W 3 fN $ 7#W-W #' ^77 W #, 3 fk fff
w sw w ? w w vw -w -zm : (*%.&%.) yf&mi $ j w - t o ? t 3 ^ t i t few
W F T T W!%m j/rc?/&<? * fT & l* T $ tfRtft'ffi 3FU M W tm $ fcfW W SWT TT 3R? ^ I
z# s jjc R f f lc f i $tf $ srM & f m w f mr foffw tftftnri
m zfrrrf& ti n it, w$w mr mmcff w^m m f # jtjw # /
m t' £ swtf ^ mm m i$ m ¥f wrfti vft mzi *iit it, tit wm $ vm ¥f mm & f w rw ff
w£ $ f > zm mm: few mr w t i wrr-w-zm yf&ivr #‘ mft ®ter f sr r m ; ? w £ m # ? w
wv # to t mm wfeqi
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. I and 5 are compulsory and out of the remaining, THREE are to be attempted choosing at 
least ONE 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 a medium other than the authorized one.
Assume suitable data, if considered necessary, and indicate the same clearly.
Unless and otherwise indicated, symbols and notations carry their usual standard meaning.
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.
PrvfFfrr w : #=r w i
Time Allowed: Three Hours
G-AVZ-O-NBUA
V-&T°Z~0~s£B7&Z 1
+
< p |W 5 — 3 T
SECTION—A
Q. 1(a) fti* in? V, « (1, 1, 2, 4), V2 - (2, -1 , -5 , 2), V3 - (1, -1 , -4, 0) cW T
v 4 = (2, l, l, 6) tfti+a : FRfa # i m vs -m t ? ^trc % w $f w # i
Q. 1(b) PlHfcHfed 3 WftcT 3^T dr^M I^ f^ lfc R :
"1 2 3 4"
2 1 4 5
1 5 5 7 -
8 1 14 17_
Reduce the following matrix to row echelon form and hence find its rank :
"1 2 3 4 '
2 1 4 5
The vectors Vj - (1, 1, 2, 4), V2 - (2, - I , -5 , 2), V3 = (1, -1 , -4 , 0) and 
V4 = (2, 1, 1 ,6 ) are linearly independent. Is it true ? Justify your answer. 10
1 5 5 7
8 1 14 17
10
Q. 1(c) frRfeffefl tffaT *TR f^TicR :
?
\ * •/
Evaluate the following lim it:
10
Q. 1(d) PlHlelfifl'tf HHWiA 7 T P T :
Vsinx
Vsinx + Vcosx cosx
Evaluate the following integral:
I t
Vsinx
10
%-&PZ-0-tf35r&i
2
Q . 1 (e ) ‘a’ ^ ERTr^ £ f c T X r , ax - 2y + z + 1 2 = 0, ^tcR ?
x2 + y 2 + z2 — 2x - 4 y + 2z - 3 = 0 F re f T O T T t l F T ? f T O t
For w hat positive value of a, the plane ax - 2 y + z + 1 2 = 0 touches th e sphere 
x2 + y2 + z2 - 2 x - 4 y + 2z - 3 = 0 and hence find the point of contact. 1 0
Q . 2 (a) lift 3T T ^ A =
1 0 0 
1 0 1 
0 1 0
e ra 3nc^ a3 0 to tffair
If m atrix A =
1 0 0 
1 0 1 
0 1 0
then find A 3 0 .
12
Q . 2(b) ^ 0 i« ra i+ K Z'z ^ ^ ^ R c T T % I t'Z $ ^ J T rT R C F T P T T ?t, e f t
3TOR f^TT 3TJTO I
A conical tent is of given cap acity . F or the least am ount of C anvas required, for it, find 
the ratio of its height to the radius of its b ase. 1 3
Q . 2 (c) P rn fc lfiN d 3 lT < a ^ % 3 ? T ^ F T ttftT V? 3 f T ^ 7 H TO :
1 1 3“
1 5 1
3 1 1 _
F ind th e eigen values and eigen vectors of the m atrix :
1 1 3
1 5 1
3 1 1
1 2
Q . 2 (d ) ^ 5 y z - 8 zx - 3 x y = 0 4 < tH < cF^c( ^ ^ W W
6 x = 3 y = 2z it, < r a 3 p ir it i
If 6 x = 3 y = 2z represents o n e of th e three m utually perpendicular generators of the cone 
5yt - * 8zx - 3 x y = 0 then obtain th e equations of the other tw o generators. 1 3
Q . 3 (a) V ® R 3 < T O T T e A (V ) « I F T a{ , A (V ) * I1 nft
T(ap a^ a3 ) = (2aj + 5aj + a 3, -3at + % - a3 , -a, + 2 a 2 + 3 a 3 )
^itt 11 era 3na r<
V ' = (lr 0, 1 ) V 2 = (-1, 2 , 1 ) V 3 = (3 , -1, 1 )
3n^jF T t o i
'Q -2 E ya ZX>-< $&l&t
3
+
Page 4


^ CS (M ain) Exam:20l5
T lf^ T rT
I
MATHEMATICS 
Paper—I
3ff&Fm 3T¥ : 250 
Maximum Marks : 250
% frftr s f j& t
f i m j m f ^ 3r7T P n = r f^ m X cfcz w f t v r # y/Hg«f+ :
& J Z (8) wt $ * ? t it w*sf fit* n fv fd ' $ mr f&ft $h mfrft if if $ wt i t
v f i f f l f f # < £ o f w fF £ \Jr7T ^ # I
o
jo t f r ^ ? r 1 3ftr 5 yfowf i mr wi¥t # ' # # wr-i-*m w z ft jm rr f ^ f rffr
sn pff £ zm
3 c * fa > WFT/'m £ 3f¥ far t j j t f /
STF# £ 3 W /cT# W $ W #^ ftm > T JF&W 3 fN $ 7#W-W #' ^77 W #, 3 fk fff
w sw w ? w w vw -w -zm : (*%.&%.) yf&mi $ j w - t o ? t 3 ^ t i t few
W F T T W!%m j/rc?/&<? * fT & l* T $ tfRtft'ffi 3FU M W tm $ fcfW W SWT TT 3R? ^ I
z# s jjc R f f lc f i $tf $ srM & f m w f mr foffw tftftnri
m zfrrrf& ti n it, w$w mr mmcff w^m m f # jtjw # /
m t' £ swtf ^ mm m i$ m ¥f wrfti vft mzi *iit it, tit wm $ vm ¥f mm & f w rw ff
w£ $ f > zm mm: few mr w t i wrr-w-zm yf&ivr #‘ mft ®ter f sr r m ; ? w £ m # ? w
wv # to t mm wfeqi
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. I and 5 are compulsory and out of the remaining, THREE are to be attempted choosing at 
least ONE 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 a medium other than the authorized one.
Assume suitable data, if considered necessary, and indicate the same clearly.
Unless and otherwise indicated, symbols and notations carry their usual standard meaning.
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.
PrvfFfrr w : #=r w i
Time Allowed: Three Hours
G-AVZ-O-NBUA
V-&T°Z~0~s£B7&Z 1
+
< p |W 5 — 3 T
SECTION—A
Q. 1(a) fti* in? V, « (1, 1, 2, 4), V2 - (2, -1 , -5 , 2), V3 - (1, -1 , -4, 0) cW T
v 4 = (2, l, l, 6) tfti+a : FRfa # i m vs -m t ? ^trc % w $f w # i
Q. 1(b) PlHfcHfed 3 WftcT 3^T dr^M I^ f^ lfc R :
"1 2 3 4"
2 1 4 5
1 5 5 7 -
8 1 14 17_
Reduce the following matrix to row echelon form and hence find its rank :
"1 2 3 4 '
2 1 4 5
The vectors Vj - (1, 1, 2, 4), V2 - (2, - I , -5 , 2), V3 = (1, -1 , -4 , 0) and 
V4 = (2, 1, 1 ,6 ) are linearly independent. Is it true ? Justify your answer. 10
1 5 5 7
8 1 14 17
10
Q. 1(c) frRfeffefl tffaT *TR f^TicR :
?
\ * •/
Evaluate the following lim it:
10
Q. 1(d) PlHlelfifl'tf HHWiA 7 T P T :
Vsinx
Vsinx + Vcosx cosx
Evaluate the following integral:
I t
Vsinx
10
%-&PZ-0-tf35r&i
2
Q . 1 (e ) ‘a’ ^ ERTr^ £ f c T X r , ax - 2y + z + 1 2 = 0, ^tcR ?
x2 + y 2 + z2 — 2x - 4 y + 2z - 3 = 0 F re f T O T T t l F T ? f T O t
For w hat positive value of a, the plane ax - 2 y + z + 1 2 = 0 touches th e sphere 
x2 + y2 + z2 - 2 x - 4 y + 2z - 3 = 0 and hence find the point of contact. 1 0
Q . 2 (a) lift 3T T ^ A =
1 0 0 
1 0 1 
0 1 0
e ra 3nc^ a3 0 to tffair
If m atrix A =
1 0 0 
1 0 1 
0 1 0
then find A 3 0 .
12
Q . 2(b) ^ 0 i« ra i+ K Z'z ^ ^ ^ R c T T % I t'Z $ ^ J T rT R C F T P T T ?t, e f t
3TOR f^TT 3TJTO I
A conical tent is of given cap acity . F or the least am ount of C anvas required, for it, find 
the ratio of its height to the radius of its b ase. 1 3
Q . 2 (c) P rn fc lfiN d 3 lT < a ^ % 3 ? T ^ F T ttftT V? 3 f T ^ 7 H TO :
1 1 3“
1 5 1
3 1 1 _
F ind th e eigen values and eigen vectors of the m atrix :
1 1 3
1 5 1
3 1 1
1 2
Q . 2 (d ) ^ 5 y z - 8 zx - 3 x y = 0 4 < tH < cF^c( ^ ^ W W
6 x = 3 y = 2z it, < r a 3 p ir it i
If 6 x = 3 y = 2z represents o n e of th e three m utually perpendicular generators of the cone 
5yt - * 8zx - 3 x y = 0 then obtain th e equations of the other tw o generators. 1 3
Q . 3 (a) V ® R 3 < T O T T e A (V ) « I F T a{ , A (V ) * I1 nft
T(ap a^ a3 ) = (2aj + 5aj + a 3, -3at + % - a3 , -a, + 2 a 2 + 3 a 3 )
^itt 11 era 3na r<
V ' = (lr 0, 1 ) V 2 = (-1, 2 , 1 ) V 3 = (3 , -1, 1 )
3n^jF T t o i
'Q -2 E ya ZX>-< $&l&t
3
+
Let V = R3 and T € A(V), for all a} e A(V), be defined by
T(ap a^ a3) = (2a{ + 5 ^ + ^ - 3 s l ] + % - a3, -a j + 2a2 + 3a3)
What is the matrix T relative to the basis
V, - (1, 0, 1) V2 = (-1, 2, 1) v 3 - (3, -1, 1) ? 12
Q. 3(b) x2 + y2 + z2 — 1 f^ ^ Ic T (2, 1,3)^ 3 rf£ j< t> d H 1 ? I
Which point of the sphere x2 + y2 + z2 = 1 is at the maximum distance from the point 
(2, 1, 3) ? 13
Q. 3(c) (i) Wrier fr'+lfcTi' (2, 3, 1) ^ (4, -5, 3) % t * T
x - m $ w ^ r r 1 1
Obtain the equation of the plane passing through the points (2, 3 ,1 ) and (4, -5, 3) 
parallel to x-axis. 6
(ii) f in ite :
x ~ a + d _ y - a _ z - a - d x - b + c = y ~ b = z - b - c
a - 8 a a + 5 P~Y P P + Y
1 1 ^ ft, ^ ^m cfcT
ftw 1 1
Verify if the lines :
x - a + d _ y - a z - a - d x - b + c _ y - b _ z - b - c
a -5 a a + 5 a n C * P~Y P P + Y
are coplanar. If yes, then find the equation of the plane in which they lie. 7
Q. 3(d) f ^ T SHPficH ^ ;
JJ ( x - y ) 2cos2(x + y)dxdy
R
t, ittf W 5W T (7 1 , 0) (2 7 1 , 7 t) (7 1 , 2 7 1 ) (0, 7 l) t !
Evaluate the integral
JJ (x - y)2 cos2(x + y) dx dy
R
where R is the rhombus with successive vertices as (n, 0) (2?t, 71) (71, 27t) (0, 7t). 12
V -P TP Z-'O -'& B xm
4
+
Page 5


^ CS (M ain) Exam:20l5
T lf^ T rT
I
MATHEMATICS 
Paper—I
3ff&Fm 3T¥ : 250 
Maximum Marks : 250
% frftr s f j& t
f i m j m f ^ 3r7T P n = r f^ m X cfcz w f t v r # y/Hg«f+ :
& J Z (8) wt $ * ? t it w*sf fit* n fv fd ' $ mr f&ft $h mfrft if if $ wt i t
v f i f f l f f # < £ o f w fF £ \Jr7T ^ # I
o
jo t f r ^ ? r 1 3ftr 5 yfowf i mr wi¥t # ' # # wr-i-*m w z ft jm rr f ^ f rffr
sn pff £ zm
3 c * fa > WFT/'m £ 3f¥ far t j j t f /
STF# £ 3 W /cT# W $ W #^ ftm > T JF&W 3 fN $ 7#W-W #' ^77 W #, 3 fk fff
w sw w ? w w vw -w -zm : (*%.&%.) yf&mi $ j w - t o ? t 3 ^ t i t few
W F T T W!%m j/rc?/&<? * fT & l* T $ tfRtft'ffi 3FU M W tm $ fcfW W SWT TT 3R? ^ I
z# s jjc R f f lc f i $tf $ srM & f m w f mr foffw tftftnri
m zfrrrf& ti n it, w$w mr mmcff w^m m f # jtjw # /
m t' £ swtf ^ mm m i$ m ¥f wrfti vft mzi *iit it, tit wm $ vm ¥f mm & f w rw ff
w£ $ f > zm mm: few mr w t i wrr-w-zm yf&ivr #‘ mft ®ter f sr r m ; ? w £ m # ? w
wv # to t mm wfeqi
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. I and 5 are compulsory and out of the remaining, THREE are to be attempted choosing at 
least ONE 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 a medium other than the authorized one.
Assume suitable data, if considered necessary, and indicate the same clearly.
Unless and otherwise indicated, symbols and notations carry their usual standard meaning.
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.
PrvfFfrr w : #=r w i
Time Allowed: Three Hours
G-AVZ-O-NBUA
V-&T°Z~0~s£B7&Z 1
+
< p |W 5 — 3 T
SECTION—A
Q. 1(a) fti* in? V, « (1, 1, 2, 4), V2 - (2, -1 , -5 , 2), V3 - (1, -1 , -4, 0) cW T
v 4 = (2, l, l, 6) tfti+a : FRfa # i m vs -m t ? ^trc % w $f w # i
Q. 1(b) PlHfcHfed 3 WftcT 3^T dr^M I^ f^ lfc R :
"1 2 3 4"
2 1 4 5
1 5 5 7 -
8 1 14 17_
Reduce the following matrix to row echelon form and hence find its rank :
"1 2 3 4 '
2 1 4 5
The vectors Vj - (1, 1, 2, 4), V2 - (2, - I , -5 , 2), V3 = (1, -1 , -4 , 0) and 
V4 = (2, 1, 1 ,6 ) are linearly independent. Is it true ? Justify your answer. 10
1 5 5 7
8 1 14 17
10
Q. 1(c) frRfeffefl tffaT *TR f^TicR :
?
\ * •/
Evaluate the following lim it:
10
Q. 1(d) PlHlelfifl'tf HHWiA 7 T P T :
Vsinx
Vsinx + Vcosx cosx
Evaluate the following integral:
I t
Vsinx
10
%-&PZ-0-tf35r&i
2
Q . 1 (e ) ‘a’ ^ ERTr^ £ f c T X r , ax - 2y + z + 1 2 = 0, ^tcR ?
x2 + y 2 + z2 — 2x - 4 y + 2z - 3 = 0 F re f T O T T t l F T ? f T O t
For w hat positive value of a, the plane ax - 2 y + z + 1 2 = 0 touches th e sphere 
x2 + y2 + z2 - 2 x - 4 y + 2z - 3 = 0 and hence find the point of contact. 1 0
Q . 2 (a) lift 3T T ^ A =
1 0 0 
1 0 1 
0 1 0
e ra 3nc^ a3 0 to tffair
If m atrix A =
1 0 0 
1 0 1 
0 1 0
then find A 3 0 .
12
Q . 2(b) ^ 0 i« ra i+ K Z'z ^ ^ ^ R c T T % I t'Z $ ^ J T rT R C F T P T T ?t, e f t
3TOR f^TT 3TJTO I
A conical tent is of given cap acity . F or the least am ount of C anvas required, for it, find 
the ratio of its height to the radius of its b ase. 1 3
Q . 2 (c) P rn fc lfiN d 3 lT < a ^ % 3 ? T ^ F T ttftT V? 3 f T ^ 7 H TO :
1 1 3“
1 5 1
3 1 1 _
F ind th e eigen values and eigen vectors of the m atrix :
1 1 3
1 5 1
3 1 1
1 2
Q . 2 (d ) ^ 5 y z - 8 zx - 3 x y = 0 4 < tH < cF^c( ^ ^ W W
6 x = 3 y = 2z it, < r a 3 p ir it i
If 6 x = 3 y = 2z represents o n e of th e three m utually perpendicular generators of the cone 
5yt - * 8zx - 3 x y = 0 then obtain th e equations of the other tw o generators. 1 3
Q . 3 (a) V ® R 3 < T O T T e A (V ) « I F T a{ , A (V ) * I1 nft
T(ap a^ a3 ) = (2aj + 5aj + a 3, -3at + % - a3 , -a, + 2 a 2 + 3 a 3 )
^itt 11 era 3na r<
V ' = (lr 0, 1 ) V 2 = (-1, 2 , 1 ) V 3 = (3 , -1, 1 )
3n^jF T t o i
'Q -2 E ya ZX>-< $&l&t
3
+
Let V = R3 and T € A(V), for all a} e A(V), be defined by
T(ap a^ a3) = (2a{ + 5 ^ + ^ - 3 s l ] + % - a3, -a j + 2a2 + 3a3)
What is the matrix T relative to the basis
V, - (1, 0, 1) V2 = (-1, 2, 1) v 3 - (3, -1, 1) ? 12
Q. 3(b) x2 + y2 + z2 — 1 f^ ^ Ic T (2, 1,3)^ 3 rf£ j< t> d H 1 ? I
Which point of the sphere x2 + y2 + z2 = 1 is at the maximum distance from the point 
(2, 1, 3) ? 13
Q. 3(c) (i) Wrier fr'+lfcTi' (2, 3, 1) ^ (4, -5, 3) % t * T
x - m $ w ^ r r 1 1
Obtain the equation of the plane passing through the points (2, 3 ,1 ) and (4, -5, 3) 
parallel to x-axis. 6
(ii) f in ite :
x ~ a + d _ y - a _ z - a - d x - b + c = y ~ b = z - b - c
a - 8 a a + 5 P~Y P P + Y
1 1 ^ ft, ^ ^m cfcT
ftw 1 1
Verify if the lines :
x - a + d _ y - a z - a - d x - b + c _ y - b _ z - b - c
a -5 a a + 5 a n C * P~Y P P + Y
are coplanar. If yes, then find the equation of the plane in which they lie. 7
Q. 3(d) f ^ T SHPficH ^ ;
JJ ( x - y ) 2cos2(x + y)dxdy
R
t, ittf W 5W T (7 1 , 0) (2 7 1 , 7 t) (7 1 , 2 7 1 ) (0, 7 l) t !
Evaluate the integral
JJ (x - y)2 cos2(x + y) dx dy
R
where R is the rhombus with successive vertices as (n, 0) (2?t, 71) (71, 27t) (0, 7t). 12
V -P TP Z-'O -'& B xm
4
+
#
Q. 4(a) PlHfrftekl JT F T Rf^rfcfq : 
/ / V l y - x 7] dxdy
WT R = [-1, 1 ; 0, 2].
Evaluate l i Vly-^21 dxdy
R
where R = [-1, 1 ; 0, 2]. 13
Q. 4(b) R4 ^ t f^n - m *rg^RT
{(1, 0, 0, 0), (0, 1, 0, 0), (1, 2, 0, 1), (0, 0, 0, 1)}
S T T T f^ n fe r 11 ?TcW ^ m i K
Find the dimension of the subspace of R4, spanned by the set
{(1, 0, 0, 0), (0, 1, 0, 0), (1, 2, 0, 1), (0, 0, 0, 1)}
Hence find its basis. 12
Q. 4(c) x2 + y2 = 2z ^ ^ T R c T c T ^ P R ? T c T
X = 0 ^ 'Mi?! ! ? I OT fw R T ^ T T } < sfl ^ c ft f |
Two perpendicular tangent planes to the paraboloid x2 + y2 = 2z intersect in a straight line 
in the plane x = 0. Obtain the curve to which this straight line touches. 13
Q. 4(d) t t it T tfe F T
x2- x - J y
f(x, y) =
2 (*> y) * (o, o)
x +y
o (x, y) = (0, 0)
^ f^T FRTcir ^ 3T^epfteT 
For the function
x2 - x ,/y
f(x, y) =
2 » (x, y )* (0 , 0)
x + y
0 (x, y) = (0, 0)
Examine the continuity and differentiability. 12
5
+