Past Year Paper, Mathematics (Set - 3),Delhi, 2014, Class 12, Maths | Additional Study Material for JEE PDF Download

 Page 1


Series : OSR/I
*l*,. 
6snt3
+s+t err-gk*,f *-W-gua
T{ 
siErqq ffi r
Candidates must write the Code on
the title page of the answer-book.
1afr+o-av 
ait: Ioo
I 
Maximum Marks : 100
riiri.
Roll No.
o ge-erqlqordt+'gq +Ekfrlps t r
o rrsr-rr-jr { Erf6+ ilq ftt ek lqq rqqilsqq{*} 
om wc-gfiror *5e-Y*'n fui} r
o ge'fi dqmdfm 
gsvFt-rrif 
zg strt r
o 
Srrrft 
rFt ul utrfaqqrvJsur+ + v6d, 
gl':rul 
rqia rqvqftrd t
o 
{n 
}rfi-wr si Wi 
+ frq 15 flffie $T uttit fEqrrqr t r mq-.Ir or f+mor 
WRT 
{ tO.tS 
q$
fuqr 
qr&n 
I 10.15 
qq 
t 10.30 {q ird6-trFT !5qqT 
qfi-q{ q}+ 
31Y{ 
gs erqtr + d{lr * str-
EksTw*ttstnrdifudn 
I
o 
Please check that this question paper contains 8 printed pages.
o 
Code number given on the right hand side of the question paper should be written on the
title page of the answer-book by the candidate.
o 
Please check that this question paper contains 29 questions.
o 
Please write dovm the Serial Number of the question before attempting it.
o 
15 minutes time has been allotted to read this question paper. The question paper will be
distributed at 10.15 a.m. From 10.15 a.m. to 10.30 a.m., the students will read the
question paper only and will not write any answer on the answer-book during this period.
MATIISMATICS
futifu€rtzt : 3 w4
Time allowed: 
j 
hours 
l
wqrqfrTtvr;
(i) az,fr sfr erfuri 
t r
(ii) 
qsyw-wilZg 
erlfiEiahazsiC f$rtfuifi; 
4 
dtwrv tu,0'€erif tO swf Fil?+
drdr 
Vqqaw* 
I urrsad 12 ev;rf,F77+ SrdF Errdqw * I soc sl7 Yfr
Se# tsdrrl 
;iawt r
(iii) s€ 
-rr 
f 
pr# 
mr+ # rar 
<ra 
Vm 
W 
qTqq 
swet snr dt 
afiq?zrqdr argcr fr<' tr smd'
fr
(iv) 
Wf 
sTa-wtfu?E?f rfurfiarriiotad t rY++ wn* 
efud;qrd'2 wif t
amftrfu* r 
0d 
srfryqitd+orqni Wdlfqaarermi 
I
(v) *ffi?der# 
yqlrr*+ 
Wqfu 
%f+ 
t fle*zraarrglqr 
ewdylvralqsTvftinart* r
651u3
tP.T.O.
Page 2


Series : OSR/I
*l*,. 
6snt3
+s+t err-gk*,f *-W-gua
T{ 
siErqq ffi r
Candidates must write the Code on
the title page of the answer-book.
1afr+o-av 
ait: Ioo
I 
Maximum Marks : 100
riiri.
Roll No.
o ge-erqlqordt+'gq +Ekfrlps t r
o rrsr-rr-jr { Erf6+ ilq ftt ek lqq rqqilsqq{*} 
om wc-gfiror *5e-Y*'n fui} r
o ge'fi dqmdfm 
gsvFt-rrif 
zg strt r
o 
Srrrft 
rFt ul utrfaqqrvJsur+ + v6d, 
gl':rul 
rqia rqvqftrd t
o 
{n 
}rfi-wr si Wi 
+ frq 15 flffie $T uttit fEqrrqr t r mq-.Ir or f+mor 
WRT 
{ tO.tS 
q$
fuqr 
qr&n 
I 10.15 
qq 
t 10.30 {q ird6-trFT !5qqT 
qfi-q{ q}+ 
31Y{ 
gs erqtr + d{lr * str-
EksTw*ttstnrdifudn 
I
o 
Please check that this question paper contains 8 printed pages.
o 
Code number given on the right hand side of the question paper should be written on the
title page of the answer-book by the candidate.
o 
Please check that this question paper contains 29 questions.
o 
Please write dovm the Serial Number of the question before attempting it.
o 
15 minutes time has been allotted to read this question paper. The question paper will be
distributed at 10.15 a.m. From 10.15 a.m. to 10.30 a.m., the students will read the
question paper only and will not write any answer on the answer-book during this period.
MATIISMATICS
futifu€rtzt : 3 w4
Time allowed: 
j 
hours 
l
wqrqfrTtvr;
(i) az,fr sfr erfuri 
t r
(ii) 
qsyw-wilZg 
erlfiEiahazsiC f$rtfuifi; 
4 
dtwrv tu,0'€erif tO swf Fil?+
drdr 
Vqqaw* 
I urrsad 12 ev;rf,F77+ SrdF Errdqw * I soc sl7 Yfr
Se# tsdrrl 
;iawt r
(iii) s€ 
-rr 
f 
pr# 
mr+ # rar 
<ra 
Vm 
W 
qTqq 
swet snr dt 
afiq?zrqdr argcr fr<' tr smd'
fr
(iv) 
Wf 
sTa-wtfu?E?f rfurfiarriiotad t rY++ wn* 
efud;qrd'2 wif t
amftrfu* r 
0d 
srfryqitd+orqni Wdlfqaarermi 
I
(v) *ffi?der# 
yqlrr*+ 
Wqfu 
%f+ 
t fle*zraarrglqr 
ewdylvralqsTvftinart* r
651u3
tP.T.O.
General Instructions :
(i) All questions are compulsory
(ii) The question paper consists of 29 questions divided into three sections A, B and C.
Section 
- 
A comprises of 10 questions of one mark each, Section 
- 
B comprises of
72 questions of 
four 
marks each and Section 
- 
C comprises of 7 questions of six
marks each.
(iii) All questions in Section 
- 
A are to be answered in one word, one sentence or as per
the exact requirement of the question.
(iv) There is no overall choice. However, internal choice has been provided in 4
questions offour marks each and 2 questions of six marks each. You have to attempt
only one of the alternatives in all such questions.
(v) 
Use of calculators is not permitted. You may askfor logarithmic tables, if required.
,rffioi1o
$F {qr 1 t 10 iro'qdq, sF r eis'sT 
t r
Question 
numbers L to 10 carry 1 mark each.
1 *r[ 
; 1 ].[ ; I 
]=[ ; l ]*d(,-y)EFrrTr{vro+1tuq 
r
[: 41 l-r vl I t ot
If 2Ls 
, J*Lo i 
l=L ,o ,l,Rna(x-y).
2. frq etq5v$q.qor*)x*floqre*tfqq:, 
[x 
1]
Solve the following matrix equation forx:, 
[x 
I
l- r 0l
l-, o l=o'
l- r 0l
)l-r 
o.l=o'
3. 
qk
lzx5l
tt-
l8 xl
bc 5ll
s ,l=l
, 
write the value of x.
i i l*d"rsf 
qrqfrtuq 
r
6
7
-2
3
4.
(r^n 
. 
U)* 
sfr-srqsirq frfiqq r
Write the antiderivative 
"t 
(rrF 
. 
1f,)
65m3
Page 3


Series : OSR/I
*l*,. 
6snt3
+s+t err-gk*,f *-W-gua
T{ 
siErqq ffi r
Candidates must write the Code on
the title page of the answer-book.
1afr+o-av 
ait: Ioo
I 
Maximum Marks : 100
riiri.
Roll No.
o ge-erqlqordt+'gq +Ekfrlps t r
o rrsr-rr-jr { Erf6+ ilq ftt ek lqq rqqilsqq{*} 
om wc-gfiror *5e-Y*'n fui} r
o ge'fi dqmdfm 
gsvFt-rrif 
zg strt r
o 
Srrrft 
rFt ul utrfaqqrvJsur+ + v6d, 
gl':rul 
rqia rqvqftrd t
o 
{n 
}rfi-wr si Wi 
+ frq 15 flffie $T uttit fEqrrqr t r mq-.Ir or f+mor 
WRT 
{ tO.tS 
q$
fuqr 
qr&n 
I 10.15 
qq 
t 10.30 {q ird6-trFT !5qqT 
qfi-q{ q}+ 
31Y{ 
gs erqtr + d{lr * str-
EksTw*ttstnrdifudn 
I
o 
Please check that this question paper contains 8 printed pages.
o 
Code number given on the right hand side of the question paper should be written on the
title page of the answer-book by the candidate.
o 
Please check that this question paper contains 29 questions.
o 
Please write dovm the Serial Number of the question before attempting it.
o 
15 minutes time has been allotted to read this question paper. The question paper will be
distributed at 10.15 a.m. From 10.15 a.m. to 10.30 a.m., the students will read the
question paper only and will not write any answer on the answer-book during this period.
MATIISMATICS
futifu€rtzt : 3 w4
Time allowed: 
j 
hours 
l
wqrqfrTtvr;
(i) az,fr sfr erfuri 
t r
(ii) 
qsyw-wilZg 
erlfiEiahazsiC f$rtfuifi; 
4 
dtwrv tu,0'€erif tO swf Fil?+
drdr 
Vqqaw* 
I urrsad 12 ev;rf,F77+ SrdF Errdqw * I soc sl7 Yfr
Se# tsdrrl 
;iawt r
(iii) s€ 
-rr 
f 
pr# 
mr+ # rar 
<ra 
Vm 
W 
qTqq 
swet snr dt 
afiq?zrqdr argcr fr<' tr smd'
fr
(iv) 
Wf 
sTa-wtfu?E?f rfurfiarriiotad t rY++ wn* 
efud;qrd'2 wif t
amftrfu* r 
0d 
srfryqitd+orqni Wdlfqaarermi 
I
(v) *ffi?der# 
yqlrr*+ 
Wqfu 
%f+ 
t fle*zraarrglqr 
ewdylvralqsTvftinart* r
651u3
tP.T.O.
General Instructions :
(i) All questions are compulsory
(ii) The question paper consists of 29 questions divided into three sections A, B and C.
Section 
- 
A comprises of 10 questions of one mark each, Section 
- 
B comprises of
72 questions of 
four 
marks each and Section 
- 
C comprises of 7 questions of six
marks each.
(iii) All questions in Section 
- 
A are to be answered in one word, one sentence or as per
the exact requirement of the question.
(iv) There is no overall choice. However, internal choice has been provided in 4
questions offour marks each and 2 questions of six marks each. You have to attempt
only one of the alternatives in all such questions.
(v) 
Use of calculators is not permitted. You may askfor logarithmic tables, if required.
,rffioi1o
$F {qr 1 t 10 iro'qdq, sF r eis'sT 
t r
Question 
numbers L to 10 carry 1 mark each.
1 *r[ 
; 1 ].[ ; I 
]=[ ; l ]*d(,-y)EFrrTr{vro+1tuq 
r
[: 41 l-r vl I t ot
If 2Ls 
, J*Lo i 
l=L ,o ,l,Rna(x-y).
2. frq etq5v$q.qor*)x*floqre*tfqq:, 
[x 
1]
Solve the following matrix equation forx:, 
[x 
I
l- r 0l
l-, o l=o'
l- r 0l
)l-r 
o.l=o'
3. 
qk
lzx5l
tt-
l8 xl
bc 5ll
s ,l=l
, 
write the value of x.
i i l*d"rsf 
qrqfrtuq 
r
6
7
-2
3
4.
(r^n 
. 
U)* 
sfr-srqsirq frfiqq r
Write the antiderivative 
"t 
(rrF 
. 
1f,)
65m3
5. {fr sin 
(ra-'}*.or-'r)= 
, 
t 
ni, $TrtFtirmeifqq I
If sin (sin-t 
f 
* ,or-' ,) 
= 
,, then find the value of x.
O. 
q$ yf+ilqr6t{s{5qt3t*'qgffi 
d, 
qmr * goffi*rfr 
{kqr 
t 
*{-t a, b e R 
- {0}
+kq a 
* 
b 
= 
$* 
rEf, t I 
qF( 
2 
*' (x 
* 
5) 
= 
10 
t 
nt, $T 
qH 
ffa dkq r
Let 
x 
be a binary operation, on the set of all non-zero real numbers, given by a 
* 
b 
= 
+
forall a,b € R- 
t0).Findthevalueof 
x, given that2 
x (x 
* 
5) 
= 
10.
7. 
qtqvr 
I + 3j + ztor 
qk{r 
zi 
- 
:3 * of 
qtsq)q 
ffo +lflt{q r
Find the projection of the vector i + :i + 7t on the vector Zi 
- 
li + Ot.
g. 
s{r 
qrkrcr 
eT 
qlqvr 
{r+ffir {rer +tFre' 
qt 
t{S (a, b, c) t tlor 
qrer 
t Hsrt 
qtrtrcT
?.ti+3+tl=2*'qqimt r
Write the vector equation of the plane, passing through the point (a, b, c) and parallel
to the plane ? 
. 
fi 
+i + t; 
= 
z.
ril?
g. qFT 
{rf, ftikq , 
I 
g(rin 
x 
- 
cos x) dx.
0
fi/2
f
Evaluate : 
I 
er(sin.r 
- 
cos r) d,r.
J
0
10. 
qi{vit 
A 
=2i*i-stst{ 
B 
=zi*i-zt+qtrmm+q$rTqs'qrf,m{rtffrflafuq 
r
Write a unit vector in the direction of the sum of the vectors i 
= 
Zi * 
i 
* 
5t and
iAAA
t5=2i+j-7k.
6sfit3 
3 IP.T.O.
Page 4


Series : OSR/I
*l*,. 
6snt3
+s+t err-gk*,f *-W-gua
T{ 
siErqq ffi r
Candidates must write the Code on
the title page of the answer-book.
1afr+o-av 
ait: Ioo
I 
Maximum Marks : 100
riiri.
Roll No.
o ge-erqlqordt+'gq +Ekfrlps t r
o rrsr-rr-jr { Erf6+ ilq ftt ek lqq rqqilsqq{*} 
om wc-gfiror *5e-Y*'n fui} r
o ge'fi dqmdfm 
gsvFt-rrif 
zg strt r
o 
Srrrft 
rFt ul utrfaqqrvJsur+ + v6d, 
gl':rul 
rqia rqvqftrd t
o 
{n 
}rfi-wr si Wi 
+ frq 15 flffie $T uttit fEqrrqr t r mq-.Ir or f+mor 
WRT 
{ tO.tS 
q$
fuqr 
qr&n 
I 10.15 
qq 
t 10.30 {q ird6-trFT !5qqT 
qfi-q{ q}+ 
31Y{ 
gs erqtr + d{lr * str-
EksTw*ttstnrdifudn 
I
o 
Please check that this question paper contains 8 printed pages.
o 
Code number given on the right hand side of the question paper should be written on the
title page of the answer-book by the candidate.
o 
Please check that this question paper contains 29 questions.
o 
Please write dovm the Serial Number of the question before attempting it.
o 
15 minutes time has been allotted to read this question paper. The question paper will be
distributed at 10.15 a.m. From 10.15 a.m. to 10.30 a.m., the students will read the
question paper only and will not write any answer on the answer-book during this period.
MATIISMATICS
futifu€rtzt : 3 w4
Time allowed: 
j 
hours 
l
wqrqfrTtvr;
(i) az,fr sfr erfuri 
t r
(ii) 
qsyw-wilZg 
erlfiEiahazsiC f$rtfuifi; 
4 
dtwrv tu,0'€erif tO swf Fil?+
drdr 
Vqqaw* 
I urrsad 12 ev;rf,F77+ SrdF Errdqw * I soc sl7 Yfr
Se# tsdrrl 
;iawt r
(iii) s€ 
-rr 
f 
pr# 
mr+ # rar 
<ra 
Vm 
W 
qTqq 
swet snr dt 
afiq?zrqdr argcr fr<' tr smd'
fr
(iv) 
Wf 
sTa-wtfu?E?f rfurfiarriiotad t rY++ wn* 
efud;qrd'2 wif t
amftrfu* r 
0d 
srfryqitd+orqni Wdlfqaarermi 
I
(v) *ffi?der# 
yqlrr*+ 
Wqfu 
%f+ 
t fle*zraarrglqr 
ewdylvralqsTvftinart* r
651u3
tP.T.O.
General Instructions :
(i) All questions are compulsory
(ii) The question paper consists of 29 questions divided into three sections A, B and C.
Section 
- 
A comprises of 10 questions of one mark each, Section 
- 
B comprises of
72 questions of 
four 
marks each and Section 
- 
C comprises of 7 questions of six
marks each.
(iii) All questions in Section 
- 
A are to be answered in one word, one sentence or as per
the exact requirement of the question.
(iv) There is no overall choice. However, internal choice has been provided in 4
questions offour marks each and 2 questions of six marks each. You have to attempt
only one of the alternatives in all such questions.
(v) 
Use of calculators is not permitted. You may askfor logarithmic tables, if required.
,rffioi1o
$F {qr 1 t 10 iro'qdq, sF r eis'sT 
t r
Question 
numbers L to 10 carry 1 mark each.
1 *r[ 
; 1 ].[ ; I 
]=[ ; l ]*d(,-y)EFrrTr{vro+1tuq 
r
[: 41 l-r vl I t ot
If 2Ls 
, J*Lo i 
l=L ,o ,l,Rna(x-y).
2. frq etq5v$q.qor*)x*floqre*tfqq:, 
[x 
1]
Solve the following matrix equation forx:, 
[x 
I
l- r 0l
l-, o l=o'
l- r 0l
)l-r 
o.l=o'
3. 
qk
lzx5l
tt-
l8 xl
bc 5ll
s ,l=l
, 
write the value of x.
i i l*d"rsf 
qrqfrtuq 
r
6
7
-2
3
4.
(r^n 
. 
U)* 
sfr-srqsirq frfiqq r
Write the antiderivative 
"t 
(rrF 
. 
1f,)
65m3
5. {fr sin 
(ra-'}*.or-'r)= 
, 
t 
ni, $TrtFtirmeifqq I
If sin (sin-t 
f 
* ,or-' ,) 
= 
,, then find the value of x.
O. 
q$ yf+ilqr6t{s{5qt3t*'qgffi 
d, 
qmr * goffi*rfr 
{kqr 
t 
*{-t a, b e R 
- {0}
+kq a 
* 
b 
= 
$* 
rEf, t I 
qF( 
2 
*' (x 
* 
5) 
= 
10 
t 
nt, $T 
qH 
ffa dkq r
Let 
x 
be a binary operation, on the set of all non-zero real numbers, given by a 
* 
b 
= 
+
forall a,b € R- 
t0).Findthevalueof 
x, given that2 
x (x 
* 
5) 
= 
10.
7. 
qtqvr 
I + 3j + ztor 
qk{r 
zi 
- 
:3 * of 
qtsq)q 
ffo +lflt{q r
Find the projection of the vector i + :i + 7t on the vector Zi 
- 
li + Ot.
g. 
s{r 
qrkrcr 
eT 
qlqvr 
{r+ffir {rer +tFre' 
qt 
t{S (a, b, c) t tlor 
qrer 
t Hsrt 
qtrtrcT
?.ti+3+tl=2*'qqimt r
Write the vector equation of the plane, passing through the point (a, b, c) and parallel
to the plane ? 
. 
fi 
+i + t; 
= 
z.
ril?
g. qFT 
{rf, ftikq , 
I 
g(rin 
x 
- 
cos x) dx.
0
fi/2
f
Evaluate : 
I 
er(sin.r 
- 
cos r) d,r.
J
0
10. 
qi{vit 
A 
=2i*i-stst{ 
B 
=zi*i-zt+qtrmm+q$rTqs'qrf,m{rtffrflafuq 
r
Write a unit vector in the direction of the sum of the vectors i 
= 
Zi * 
i 
* 
5t and
iAAA
t5=2i+j-7k.
6sfit3 
3 IP.T.O.
,rffio*1,
ym 
riqr rr t zz H*r*oqm 
I 
oi+.qr 
t r
Question 
numbers ll to 22 carry 4 marks each.
11. fritt+{qRvri 
a, B, A+fuqfuq*leqiq,
[a 
* u', E' * ?, ? * ?] 
= 
zfi,B, A]
3IW{t
n+J
nRvr ?, u' aqr ? 
tS tfu' ? * B + ? 
= 
d aqr 
I 
? 
I = 
3, 
I 
ts' 
| 
- s ireTr 
I 
? 
I = 
z t r ? nqr B
++qorqiqaraqfr&q 
r
Prove that, for any three vectors ?, bt, ?
[? 
* B, B * ?, ? * ?] 
=zli,ts, 
?]
OR
vectors ?, b' ana d ur. such that ? * B + ? 
= 
d ano 
l?l = 
3, 
lBl = 
5 ano 
l?l = 
z.
Find the angle between ?anO Ut.
FTq €rqsm 
qfurur 
*t co +lfuq,
(*'*t)*.r*=*.
Solve the following differential equation :
.) 
-.dv 
2
(x- 
- 
t) 
-.o* 
Zry 
= 
il t.
qnnrnsliqq, 
[W*
J 
Sm-x. cos-.r
qelitt
qFT 
Ern *1iqq , 
! 
A- 
:;1ffi, 
- 
rs a,
Evaluate' 
[W*
J 
slnir. cos._r
OR
?_
Evaluate , 
) 
@- 
l1@rs o*
12.
13.
65t1t3
Page 5


Series : OSR/I
*l*,. 
6snt3
+s+t err-gk*,f *-W-gua
T{ 
siErqq ffi r
Candidates must write the Code on
the title page of the answer-book.
1afr+o-av 
ait: Ioo
I 
Maximum Marks : 100
riiri.
Roll No.
o ge-erqlqordt+'gq +Ekfrlps t r
o rrsr-rr-jr { Erf6+ ilq ftt ek lqq rqqilsqq{*} 
om wc-gfiror *5e-Y*'n fui} r
o ge'fi dqmdfm 
gsvFt-rrif 
zg strt r
o 
Srrrft 
rFt ul utrfaqqrvJsur+ + v6d, 
gl':rul 
rqia rqvqftrd t
o 
{n 
}rfi-wr si Wi 
+ frq 15 flffie $T uttit fEqrrqr t r mq-.Ir or f+mor 
WRT 
{ tO.tS 
q$
fuqr 
qr&n 
I 10.15 
qq 
t 10.30 {q ird6-trFT !5qqT 
qfi-q{ q}+ 
31Y{ 
gs erqtr + d{lr * str-
EksTw*ttstnrdifudn 
I
o 
Please check that this question paper contains 8 printed pages.
o 
Code number given on the right hand side of the question paper should be written on the
title page of the answer-book by the candidate.
o 
Please check that this question paper contains 29 questions.
o 
Please write dovm the Serial Number of the question before attempting it.
o 
15 minutes time has been allotted to read this question paper. The question paper will be
distributed at 10.15 a.m. From 10.15 a.m. to 10.30 a.m., the students will read the
question paper only and will not write any answer on the answer-book during this period.
MATIISMATICS
futifu€rtzt : 3 w4
Time allowed: 
j 
hours 
l
wqrqfrTtvr;
(i) az,fr sfr erfuri 
t r
(ii) 
qsyw-wilZg 
erlfiEiahazsiC f$rtfuifi; 
4 
dtwrv tu,0'€erif tO swf Fil?+
drdr 
Vqqaw* 
I urrsad 12 ev;rf,F77+ SrdF Errdqw * I soc sl7 Yfr
Se# tsdrrl 
;iawt r
(iii) s€ 
-rr 
f 
pr# 
mr+ # rar 
<ra 
Vm 
W 
qTqq 
swet snr dt 
afiq?zrqdr argcr fr<' tr smd'
fr
(iv) 
Wf 
sTa-wtfu?E?f rfurfiarriiotad t rY++ wn* 
efud;qrd'2 wif t
amftrfu* r 
0d 
srfryqitd+orqni Wdlfqaarermi 
I
(v) *ffi?der# 
yqlrr*+ 
Wqfu 
%f+ 
t fle*zraarrglqr 
ewdylvralqsTvftinart* r
651u3
tP.T.O.
General Instructions :
(i) All questions are compulsory
(ii) The question paper consists of 29 questions divided into three sections A, B and C.
Section 
- 
A comprises of 10 questions of one mark each, Section 
- 
B comprises of
72 questions of 
four 
marks each and Section 
- 
C comprises of 7 questions of six
marks each.
(iii) All questions in Section 
- 
A are to be answered in one word, one sentence or as per
the exact requirement of the question.
(iv) There is no overall choice. However, internal choice has been provided in 4
questions offour marks each and 2 questions of six marks each. You have to attempt
only one of the alternatives in all such questions.
(v) 
Use of calculators is not permitted. You may askfor logarithmic tables, if required.
,rffioi1o
$F {qr 1 t 10 iro'qdq, sF r eis'sT 
t r
Question 
numbers L to 10 carry 1 mark each.
1 *r[ 
; 1 ].[ ; I 
]=[ ; l ]*d(,-y)EFrrTr{vro+1tuq 
r
[: 41 l-r vl I t ot
If 2Ls 
, J*Lo i 
l=L ,o ,l,Rna(x-y).
2. frq etq5v$q.qor*)x*floqre*tfqq:, 
[x 
1]
Solve the following matrix equation forx:, 
[x 
I
l- r 0l
l-, o l=o'
l- r 0l
)l-r 
o.l=o'
3. 
qk
lzx5l
tt-
l8 xl
bc 5ll
s ,l=l
, 
write the value of x.
i i l*d"rsf 
qrqfrtuq 
r
6
7
-2
3
4.
(r^n 
. 
U)* 
sfr-srqsirq frfiqq r
Write the antiderivative 
"t 
(rrF 
. 
1f,)
65m3
5. {fr sin 
(ra-'}*.or-'r)= 
, 
t 
ni, $TrtFtirmeifqq I
If sin (sin-t 
f 
* ,or-' ,) 
= 
,, then find the value of x.
O. 
q$ yf+ilqr6t{s{5qt3t*'qgffi 
d, 
qmr * goffi*rfr 
{kqr 
t 
*{-t a, b e R 
- {0}
+kq a 
* 
b 
= 
$* 
rEf, t I 
qF( 
2 
*' (x 
* 
5) 
= 
10 
t 
nt, $T 
qH 
ffa dkq r
Let 
x 
be a binary operation, on the set of all non-zero real numbers, given by a 
* 
b 
= 
+
forall a,b € R- 
t0).Findthevalueof 
x, given that2 
x (x 
* 
5) 
= 
10.
7. 
qtqvr 
I + 3j + ztor 
qk{r 
zi 
- 
:3 * of 
qtsq)q 
ffo +lflt{q r
Find the projection of the vector i + :i + 7t on the vector Zi 
- 
li + Ot.
g. 
s{r 
qrkrcr 
eT 
qlqvr 
{r+ffir {rer +tFre' 
qt 
t{S (a, b, c) t tlor 
qrer 
t Hsrt 
qtrtrcT
?.ti+3+tl=2*'qqimt r
Write the vector equation of the plane, passing through the point (a, b, c) and parallel
to the plane ? 
. 
fi 
+i + t; 
= 
z.
ril?
g. qFT 
{rf, ftikq , 
I 
g(rin 
x 
- 
cos x) dx.
0
fi/2
f
Evaluate : 
I 
er(sin.r 
- 
cos r) d,r.
J
0
10. 
qi{vit 
A 
=2i*i-stst{ 
B 
=zi*i-zt+qtrmm+q$rTqs'qrf,m{rtffrflafuq 
r
Write a unit vector in the direction of the sum of the vectors i 
= 
Zi * 
i 
* 
5t and
iAAA
t5=2i+j-7k.
6sfit3 
3 IP.T.O.
,rffio*1,
ym 
riqr rr t zz H*r*oqm 
I 
oi+.qr 
t r
Question 
numbers ll to 22 carry 4 marks each.
11. fritt+{qRvri 
a, B, A+fuqfuq*leqiq,
[a 
* u', E' * ?, ? * ?] 
= 
zfi,B, A]
3IW{t
n+J
nRvr ?, u' aqr ? 
tS tfu' ? * B + ? 
= 
d aqr 
I 
? 
I = 
3, 
I 
ts' 
| 
- s ireTr 
I 
? 
I = 
z t r ? nqr B
++qorqiqaraqfr&q 
r
Prove that, for any three vectors ?, bt, ?
[? 
* B, B * ?, ? * ?] 
=zli,ts, 
?]
OR
vectors ?, b' ana d ur. such that ? * B + ? 
= 
d ano 
l?l = 
3, 
lBl = 
5 ano 
l?l = 
z.
Find the angle between ?anO Ut.
FTq €rqsm 
qfurur 
*t co +lfuq,
(*'*t)*.r*=*.
Solve the following differential equation :
.) 
-.dv 
2
(x- 
- 
t) 
-.o* 
Zry 
= 
il t.
qnnrnsliqq, 
[W*
J 
Sm-x. cos-.r
qelitt
qFT 
Ern *1iqq , 
! 
A- 
:;1ffi, 
- 
rs a,
Evaluate' 
[W*
J 
slnir. cos._r
OR
?_
Evaluate , 
) 
@- 
l1@rs o*
12.
13.
65t1t3
14. Td itiil{m 
TId d&qFH+ Erm f(x) 
= 
3# 
- 
4x3 
- 
t2* + 5
(a) frtnt *iqn 
t r
(b) ffirarnqmi r
qelt[f
Effr 
.r 
= 
a sin30 irerr y 
= 
a cos30 +fu+ e 
= 
f, 
wwt tor mn 
qftdq 
*qdwrrr 
am
ffi; intervats in which the tunction f(r) 
= 
3# 
- 
+f 
- 
tz*r- 5 is
(a) 
strictlyincreasing
(b) 
srictly decreasing
OR
Find the equations of the tangent and normar to the curve x 
= 
a sin30 and y 
= 
a cos30 at
It
o 
=2.
qHr 
A 
= {1, 
2, 3,....., 9l iTen A x A { R 
gs-lirisr 
t, * e x A 
ii 
1a, 
b), (c, d) *.kq
(a, 
b) R (c, 
d) 
qk 
a + d 
= 
b + c Enr 
qlqqrfrd 
t r fqqsliqqm n 
f'oWar ffu t' I 
EEflr
q,i' 
[(2, ,] 
rfl 
aro *1&q r
Let A 
= {1, 
2, 3,....., 9I and R be the relation in A x A defined by (a, 
b) R (c, 
d) if
a + d 
= 
b + c for (a, 
b), (c, d) 
in 
A x A. Prove that R is an equivalence relation. Also
obtain the equivalence 
class 
t(2, 
5)1.
15.
16.
=|;*. (o,f)
=|; 
*. 
(0,';
irr€ft1iw tr 2 tan-t 
(f) *,..-, 
ffi 
+ 2 ,un-t(,l 
= 
f
Provethatcorll'@
Wl 
+sinx-{l-sinx
Prove that2,"r-, 
Gl 
+ sec-r 
(9 + 2 ,un-tGl 
= 
t
17. 
qky 
=,r 
t, Hrftradriqqio 
# 
+eI-I= 
o.
rry 
-rl, 
prove 
tnoff-i(#-I= 
o.
6snt3 
5
OR
lP.T.O.