Past Year Paper, Mathematics (Set - 2),Delhi, 2014, Class 12, Maths | Additional Study Material for JEE PDF Download

 Page 1


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651112
fiR.
rmclEt[iTflaril 
r
Candidates must write the Code on
the title page of the answer-book.
st6
o 
!ffircfr-qo-ri{fu WyFr-Eri[EBRysS t r
o 
trFr-rrr { 
qri6} 
Erq *t er}r 
fqqlq*ts;rqt *} 6H sr{-gfitr*,r *.w-g*wfed r
o 
!ffir 
dq m'ri{ toss 
ylrT-Er il 
zg }rrt 
t r
o grFfl[FrmvrrfrqarvJ6.Fr++rr6d, gyq6TF:qi6. 
wq*rftrd r
. 
Es 
qF[-wt 
si Td+ + fuq 15 frqe 
Eil 
{Frq felqr 
.rqr 
t r 
yw-q* 
*i f+otq 
WEq 
,t 
tO.rS 
qi
fuqr 
qrtn 
I 10.15 
qq 
t 10.30 
qQ 
iro on iqa IF{-rEr 
qtn 
sir gs srqRr 
+ 
qt{q 
i rrt_
gflersTw*ttr+tqa1ffiri 
r
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.
r 
Please check that this question paper contains 29 questions.
e 
Please write dorvn 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.
MATHEMATICS
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(ii) 
W,rrr-wif zg vwtqi#7ezsldfur&a*: atddutTFr tqvgsrif tO 
ywfeqC;
drdqr 
vaiiaw d' r srsac 12 sqrsF'{+ drdfr 
qtroirwi 
r srs vlz 
yw
tffisyaqoai6wi r
(iii) sus erf Hd'q'F# #arr 
frffi 
qr qrqq 
wnr wr 
qT 
ena?qqdTqwr&r' 
Er fiqrd
fr
(iv) 
Yf 
vw-wtfu 
?Ef'# rfurftEreiot' ard eswltdzrT* siqf 
qrd 
z 
qwl+
etmrTrfu* 
r 
dd 
s*srifcc,lrqni 
wd?futTqn1* 
t
(v) eryrizr #vqhr 
q? 
avqfu 
aaf t r ailqpqqir 
qrl 
w eilq dyqdtq sr{sft 
qfur 
rsr# f, r
tP.T.O.
Page 2


Series : OSR/I -f--i
;ff*" 
651112
fiR.
rmclEt[iTflaril 
r
Candidates must write the Code on
the title page of the answer-book.
st6
o 
!ffircfr-qo-ri{fu WyFr-Eri[EBRysS t r
o 
trFr-rrr { 
qri6} 
Erq *t er}r 
fqqlq*ts;rqt *} 6H sr{-gfitr*,r *.w-g*wfed r
o 
!ffir 
dq m'ri{ toss 
ylrT-Er il 
zg }rrt 
t r
o grFfl[FrmvrrfrqarvJ6.Fr++rr6d, gyq6TF:qi6. 
wq*rftrd r
. 
Es 
qF[-wt 
si Td+ + fuq 15 frqe 
Eil 
{Frq felqr 
.rqr 
t r 
yw-q* 
*i f+otq 
WEq 
,t 
tO.rS 
qi
fuqr 
qrtn 
I 10.15 
qq 
t 10.30 
qQ 
iro on iqa IF{-rEr 
qtn 
sir gs srqRr 
+ 
qt{q 
i rrt_
gflersTw*ttr+tqa1ffiri 
r
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.
r 
Please check that this question paper contains 29 questions.
e 
Please write dorvn 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.
MATHEMATICS
tirilri.
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Time allowed: 3 hours 
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Maximum Marlts : 100
HFTTATFiltST:
(, nfrwerW* r
(ii) 
W,rrr-wif zg vwtqi#7ezsldfur&a*: atddutTFr tqvgsrif tO 
ywfeqC;
drdqr 
vaiiaw d' r srsac 12 sqrsF'{+ drdfr 
qtroirwi 
r srs vlz 
yw
tffisyaqoai6wi r
(iii) sus erf Hd'q'F# #arr 
frffi 
qr qrqq 
wnr wr 
qT 
ena?qqdTqwr&r' 
Er fiqrd
fr
(iv) 
Yf 
vw-wtfu 
?Ef'# rfurftEreiot' ard eswltdzrT* siqf 
qrd 
z 
qwl+
etmrTrfu* 
r 
dd 
s*srifcc,lrqni 
wd?futTqn1* 
t
(v) eryrizr #vqhr 
q? 
avqfu 
aaf t r ailqpqqir 
qrl 
w eilq dyqdtq sr{sft 
qfur 
rsr# f, r
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 iomprises of 10 questions of one mark each, Section 
- 
B comprises of
72 questions of 
four 
mirks iach 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, intemal choice has been provided in 4
questions of 
four 
marles each and 2 questions of six marla each. You have to atternpt
only one of the alternatives in all such questions'
(v) 
(Jse 
of calculators is not permitted. You may askfor logarithmic tables, if required'
EIrg- 3[
SECTION. A
Irfi {qr 1 t 10 Hqtr+fi'YFT 1 
eim 
qr 
t r
Question 
numbers I to 10 carry l mark each'
t. {rRn i + 3i +ztmefu1 2i-ri * otwqq}qaro+tkq 
t
Find the projection of the vector t + :i + 7t on the vector Zi 
- 
li + Ot.
Z. e-q TrkttT sr 
qtq$ q+e'rur 
frTil +itqq + fiE (a, b, c) t tt*t 
qdr 
t Eelr 
qIm?T
?.(l+i+fl1=2$qqi6E$ 
1
Write the vector equation of the plane, passing through t[re point (a, b, c) and parallel
to the plane ?' 
(l +i + [1 
= 
z.
3. 
(r.n 
. 
# 
* vfr-effisffE frftqq r
write the antiderivatir. or 
(l.E. 
fJ
4 frr[: 
i].[l 1]=[ ; !].d(,-y)Hrq''ffe*lrsrq
,',[:i].[ 
l1]=[ ; l],o'u(x-v)
65t112
Page 3


Series : OSR/I -f--i
;ff*" 
651112
fiR.
rmclEt[iTflaril 
r
Candidates must write the Code on
the title page of the answer-book.
st6
o 
!ffircfr-qo-ri{fu WyFr-Eri[EBRysS t r
o 
trFr-rrr { 
qri6} 
Erq *t er}r 
fqqlq*ts;rqt *} 6H sr{-gfitr*,r *.w-g*wfed r
o 
!ffir 
dq m'ri{ toss 
ylrT-Er il 
zg }rrt 
t r
o grFfl[FrmvrrfrqarvJ6.Fr++rr6d, gyq6TF:qi6. 
wq*rftrd r
. 
Es 
qF[-wt 
si Td+ + fuq 15 frqe 
Eil 
{Frq felqr 
.rqr 
t r 
yw-q* 
*i f+otq 
WEq 
,t 
tO.rS 
qi
fuqr 
qrtn 
I 10.15 
qq 
t 10.30 
qQ 
iro on iqa IF{-rEr 
qtn 
sir gs srqRr 
+ 
qt{q 
i rrt_
gflersTw*ttr+tqa1ffiri 
r
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.
r 
Please check that this question paper contains 29 questions.
e 
Please write dorvn 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.
MATHEMATICS
tirilri.
Roll No.
fulfuHrtzt: 3 
qr*l
Time allowed: 3 hours 
l
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ef6: 
too
I 
Maximum Marlts : 100
HFTTATFiltST:
(, nfrwerW* r
(ii) 
W,rrr-wif zg vwtqi#7ezsldfur&a*: atddutTFr tqvgsrif tO 
ywfeqC;
drdqr 
vaiiaw d' r srsac 12 sqrsF'{+ drdfr 
qtroirwi 
r srs vlz 
yw
tffisyaqoai6wi r
(iii) sus erf Hd'q'F# #arr 
frffi 
qr qrqq 
wnr wr 
qT 
ena?qqdTqwr&r' 
Er fiqrd
fr
(iv) 
Yf 
vw-wtfu 
?Ef'# rfurftEreiot' ard eswltdzrT* siqf 
qrd 
z 
qwl+
etmrTrfu* 
r 
dd 
s*srifcc,lrqni 
wd?futTqn1* 
t
(v) eryrizr #vqhr 
q? 
avqfu 
aaf t r ailqpqqir 
qrl 
w eilq dyqdtq sr{sft 
qfur 
rsr# f, r
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 iomprises of 10 questions of one mark each, Section 
- 
B comprises of
72 questions of 
four 
mirks iach 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, intemal choice has been provided in 4
questions of 
four 
marles each and 2 questions of six marla each. You have to atternpt
only one of the alternatives in all such questions'
(v) 
(Jse 
of calculators is not permitted. You may askfor logarithmic tables, if required'
EIrg- 3[
SECTION. A
Irfi {qr 1 t 10 Hqtr+fi'YFT 1 
eim 
qr 
t r
Question 
numbers I to 10 carry l mark each'
t. {rRn i + 3i +ztmefu1 2i-ri * otwqq}qaro+tkq 
t
Find the projection of the vector t + :i + 7t on the vector Zi 
- 
li + Ot.
Z. e-q TrkttT sr 
qtq$ q+e'rur 
frTil +itqq + fiE (a, b, c) t tt*t 
qdr 
t Eelr 
qIm?T
?.(l+i+fl1=2$qqi6E$ 
1
Write the vector equation of the plane, passing through t[re point (a, b, c) and parallel
to the plane ?' 
(l +i + [1 
= 
z.
3. 
(r.n 
. 
# 
* vfr-effisffE frftqq r
write the antiderivatir. or 
(l.E. 
fJ
4 frr[: 
i].[l 1]=[ ; !].d(,-y)Hrq''ffe*lrsrq
,',[:i].[ 
l1]=[ ; l],o'u(x-v)
65t112
s. frq enryq*+tur 
+t x*.ft(ra 
dftrq ,, 
tu \l 
\ 
I 
]= 
,
Solve rhe following 
mafrix equation forx:, 
k Ul 
\ 
I 
]= 
"
6 
*l? 
;l=
,l? 
;l=l
6 
-z 
l.
7 , ltHl 
.rrnrrmfufuq 
r
6-2
73
, 
write the value of x.
7. ek sin 
(r*-, 
f 
* .or-, ,) 
= 
, t, nt, sT sFr um *tflqq r
rf sin 
(sin-, 
| 
* .or-, ,) 
= 
,, then find the value of x.
8. ett {f+fieT*k'srf*,.if 
*wg@ 
t, 
,rnr * gokemrft 
{Bmr 
t, 
qt 
H-t a, b e R 
_ 
{0}
Sfaeu* 
U=*rnrqffit 
rqkz x 
(x* 
5) 
= 
t0t ai*6rqparo+1fuq 
r
Let 
* 
be a binary operation, 
on the set of all non-zero real numbers, given 
by a 
x 
b 
= 
+
for all a, b € R 
- {0}.Find the value of x, given that2x (x x 
5) 
= 
10.
g. qH 
Erfr +1iqq , 
J.or-, 
(sin 
x) d.r.
f
Evaluate 
: 
J 
cos-l (sin 
x) d.r.
10. 
qkntn 
? 
q\r 
u'eqqffi{+tfq.l?l= 
3, 
lb,l=3* 
? r. u,\rs.q:rs.nPflrt, 
H}
? sil{ 
E' ++q or *iur fufuq r
If vectors 
? and E' *r such rhat, 
I 
? 
I = 
3, 
I 
b' 
| 
=l 
*a? x u, is a unit vector, then
write the angle between 
? anA U'.
651112 
' 
,*'T'o'
Page 4


Series : OSR/I -f--i
;ff*" 
651112
fiR.
rmclEt[iTflaril 
r
Candidates must write the Code on
the title page of the answer-book.
st6
o 
!ffircfr-qo-ri{fu WyFr-Eri[EBRysS t r
o 
trFr-rrr { 
qri6} 
Erq *t er}r 
fqqlq*ts;rqt *} 6H sr{-gfitr*,r *.w-g*wfed r
o 
!ffir 
dq m'ri{ toss 
ylrT-Er il 
zg }rrt 
t r
o grFfl[FrmvrrfrqarvJ6.Fr++rr6d, gyq6TF:qi6. 
wq*rftrd r
. 
Es 
qF[-wt 
si Td+ + fuq 15 frqe 
Eil 
{Frq felqr 
.rqr 
t r 
yw-q* 
*i f+otq 
WEq 
,t 
tO.rS 
qi
fuqr 
qrtn 
I 10.15 
qq 
t 10.30 
qQ 
iro on iqa IF{-rEr 
qtn 
sir gs srqRr 
+ 
qt{q 
i rrt_
gflersTw*ttr+tqa1ffiri 
r
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.
r 
Please check that this question paper contains 29 questions.
e 
Please write dorvn 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.
MATHEMATICS
tirilri.
Roll No.
fulfuHrtzt: 3 
qr*l
Time allowed: 3 hours 
l
Setfama+ 
ef6: 
too
I 
Maximum Marlts : 100
HFTTATFiltST:
(, nfrwerW* r
(ii) 
W,rrr-wif zg vwtqi#7ezsldfur&a*: atddutTFr tqvgsrif tO 
ywfeqC;
drdqr 
vaiiaw d' r srsac 12 sqrsF'{+ drdfr 
qtroirwi 
r srs vlz 
yw
tffisyaqoai6wi r
(iii) sus erf Hd'q'F# #arr 
frffi 
qr qrqq 
wnr wr 
qT 
ena?qqdTqwr&r' 
Er fiqrd
fr
(iv) 
Yf 
vw-wtfu 
?Ef'# rfurftEreiot' ard eswltdzrT* siqf 
qrd 
z 
qwl+
etmrTrfu* 
r 
dd 
s*srifcc,lrqni 
wd?futTqn1* 
t
(v) eryrizr #vqhr 
q? 
avqfu 
aaf t r ailqpqqir 
qrl 
w eilq dyqdtq sr{sft 
qfur 
rsr# f, r
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 iomprises of 10 questions of one mark each, Section 
- 
B comprises of
72 questions of 
four 
mirks iach 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, intemal choice has been provided in 4
questions of 
four 
marles each and 2 questions of six marla each. You have to atternpt
only one of the alternatives in all such questions'
(v) 
(Jse 
of calculators is not permitted. You may askfor logarithmic tables, if required'
EIrg- 3[
SECTION. A
Irfi {qr 1 t 10 Hqtr+fi'YFT 1 
eim 
qr 
t r
Question 
numbers I to 10 carry l mark each'
t. {rRn i + 3i +ztmefu1 2i-ri * otwqq}qaro+tkq 
t
Find the projection of the vector t + :i + 7t on the vector Zi 
- 
li + Ot.
Z. e-q TrkttT sr 
qtq$ q+e'rur 
frTil +itqq + fiE (a, b, c) t tt*t 
qdr 
t Eelr 
qIm?T
?.(l+i+fl1=2$qqi6E$ 
1
Write the vector equation of the plane, passing through t[re point (a, b, c) and parallel
to the plane ?' 
(l +i + [1 
= 
z.
3. 
(r.n 
. 
# 
* vfr-effisffE frftqq r
write the antiderivatir. or 
(l.E. 
fJ
4 frr[: 
i].[l 1]=[ ; !].d(,-y)Hrq''ffe*lrsrq
,',[:i].[ 
l1]=[ ; l],o'u(x-v)
65t112
s. frq enryq*+tur 
+t x*.ft(ra 
dftrq ,, 
tu \l 
\ 
I 
]= 
,
Solve rhe following 
mafrix equation forx:, 
k Ul 
\ 
I 
]= 
"
6 
*l? 
;l=
,l? 
;l=l
6 
-z 
l.
7 , ltHl 
.rrnrrmfufuq 
r
6-2
73
, 
write the value of x.
7. ek sin 
(r*-, 
f 
* .or-, ,) 
= 
, t, nt, sT sFr um *tflqq r
rf sin 
(sin-, 
| 
* .or-, ,) 
= 
,, then find the value of x.
8. ett {f+fieT*k'srf*,.if 
*wg@ 
t, 
,rnr * gokemrft 
{Bmr 
t, 
qt 
H-t a, b e R 
_ 
{0}
Sfaeu* 
U=*rnrqffit 
rqkz x 
(x* 
5) 
= 
t0t ai*6rqparo+1fuq 
r
Let 
* 
be a binary operation, 
on the set of all non-zero real numbers, given 
by a 
x 
b 
= 
+
for all a, b € R 
- {0}.Find the value of x, given that2x (x x 
5) 
= 
10.
g. qH 
Erfr +1iqq , 
J.or-, 
(sin 
x) d.r.
f
Evaluate 
: 
J 
cos-l (sin 
x) d.r.
10. 
qkntn 
? 
q\r 
u'eqqffi{+tfq.l?l= 
3, 
lb,l=3* 
? r. u,\rs.q:rs.nPflrt, 
H}
? sil{ 
E' ++q or *iur fufuq r
If vectors 
? and E' *r such rhat, 
I 
? 
I = 
3, 
I 
b' 
| 
=l 
*a? x u, is a unit vector, then
write the angle between 
? anA U'.
651112 
' 
,*'T'o'
Etug-Et
SECTION - 
B
,rfi {reqr 11 t 22 rou*osq{ 
C 
ei6.qr t r
Question 
numbers 
lLto22 
cwry 4 marks each'
11. 
qa 
oirqrm flfi 
qi&q 
Ht 
qot 
f(x) 
= 
3# 
- 
+f 
- 
tz* + s
(a) fq{etq$iqlat 
I
(b) ftirr 6rgqm 
| 1
sIqET
elF,, 
= 
a sin30 
dsrT y 
= 
a cos30 
+M' e 
= 
fr 
wwi 
tot aqT 
e${Tiq *qrfu'qur frrd
dfqq r
Find the intervals 
in which the function 
f(x) 
= 
3f 
- 
+x3 
- 
12* + 5 is
(a) strictlYincreasing
(b) strict$ decreasing
OR
Find the equations 
of the tangent and normal to the curve x 
= 
a sin30 and 
y 
= 
a cos30 at
lt
0 
=7.
+
\
I
t2. 
q,-nrnqm{q 
J##"
eNrat
qmnrdqitq 
,!a-:1Pffia'
f sin6x + cos6x 
'
Evatuate: 
J 
ffiorr, 
o,
OR
' 
, 
-ghp+ 
3.r 
- 
18 dr
Evaluate: 
J 
(r
t3. tTq 
effitr€ 
q++tq 
*} re 
qfrkg 
:
G-D**"=h'
Solve the following 
differential 
equation 
:
(i-r)ff.rn=h'
65lLl2
Page 5


Series : OSR/I -f--i
;ff*" 
651112
fiR.
rmclEt[iTflaril 
r
Candidates must write the Code on
the title page of the answer-book.
st6
o 
!ffircfr-qo-ri{fu WyFr-Eri[EBRysS t r
o 
trFr-rrr { 
qri6} 
Erq *t er}r 
fqqlq*ts;rqt *} 6H sr{-gfitr*,r *.w-g*wfed r
o 
!ffir 
dq m'ri{ toss 
ylrT-Er il 
zg }rrt 
t r
o grFfl[FrmvrrfrqarvJ6.Fr++rr6d, gyq6TF:qi6. 
wq*rftrd r
. 
Es 
qF[-wt 
si Td+ + fuq 15 frqe 
Eil 
{Frq felqr 
.rqr 
t r 
yw-q* 
*i f+otq 
WEq 
,t 
tO.rS 
qi
fuqr 
qrtn 
I 10.15 
qq 
t 10.30 
qQ 
iro on iqa IF{-rEr 
qtn 
sir gs srqRr 
+ 
qt{q 
i rrt_
gflersTw*ttr+tqa1ffiri 
r
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.
r 
Please check that this question paper contains 29 questions.
e 
Please write dorvn 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.
MATHEMATICS
tirilri.
Roll No.
fulfuHrtzt: 3 
qr*l
Time allowed: 3 hours 
l
Setfama+ 
ef6: 
too
I 
Maximum Marlts : 100
HFTTATFiltST:
(, nfrwerW* r
(ii) 
W,rrr-wif zg vwtqi#7ezsldfur&a*: atddutTFr tqvgsrif tO 
ywfeqC;
drdqr 
vaiiaw d' r srsac 12 sqrsF'{+ drdfr 
qtroirwi 
r srs vlz 
yw
tffisyaqoai6wi r
(iii) sus erf Hd'q'F# #arr 
frffi 
qr qrqq 
wnr wr 
qT 
ena?qqdTqwr&r' 
Er fiqrd
fr
(iv) 
Yf 
vw-wtfu 
?Ef'# rfurftEreiot' ard eswltdzrT* siqf 
qrd 
z 
qwl+
etmrTrfu* 
r 
dd 
s*srifcc,lrqni 
wd?futTqn1* 
t
(v) eryrizr #vqhr 
q? 
avqfu 
aaf t r ailqpqqir 
qrl 
w eilq dyqdtq sr{sft 
qfur 
rsr# f, r
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 iomprises of 10 questions of one mark each, Section 
- 
B comprises of
72 questions of 
four 
mirks iach 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, intemal choice has been provided in 4
questions of 
four 
marles each and 2 questions of six marla each. You have to atternpt
only one of the alternatives in all such questions'
(v) 
(Jse 
of calculators is not permitted. You may askfor logarithmic tables, if required'
EIrg- 3[
SECTION. A
Irfi {qr 1 t 10 Hqtr+fi'YFT 1 
eim 
qr 
t r
Question 
numbers I to 10 carry l mark each'
t. {rRn i + 3i +ztmefu1 2i-ri * otwqq}qaro+tkq 
t
Find the projection of the vector t + :i + 7t on the vector Zi 
- 
li + Ot.
Z. e-q TrkttT sr 
qtq$ q+e'rur 
frTil +itqq + fiE (a, b, c) t tt*t 
qdr 
t Eelr 
qIm?T
?.(l+i+fl1=2$qqi6E$ 
1
Write the vector equation of the plane, passing through t[re point (a, b, c) and parallel
to the plane ?' 
(l +i + [1 
= 
z.
3. 
(r.n 
. 
# 
* vfr-effisffE frftqq r
write the antiderivatir. or 
(l.E. 
fJ
4 frr[: 
i].[l 1]=[ ; !].d(,-y)Hrq''ffe*lrsrq
,',[:i].[ 
l1]=[ ; l],o'u(x-v)
65t112
s. frq enryq*+tur 
+t x*.ft(ra 
dftrq ,, 
tu \l 
\ 
I 
]= 
,
Solve rhe following 
mafrix equation forx:, 
k Ul 
\ 
I 
]= 
"
6 
*l? 
;l=
,l? 
;l=l
6 
-z 
l.
7 , ltHl 
.rrnrrmfufuq 
r
6-2
73
, 
write the value of x.
7. ek sin 
(r*-, 
f 
* .or-, ,) 
= 
, t, nt, sT sFr um *tflqq r
rf sin 
(sin-, 
| 
* .or-, ,) 
= 
,, then find the value of x.
8. ett {f+fieT*k'srf*,.if 
*wg@ 
t, 
,rnr * gokemrft 
{Bmr 
t, 
qt 
H-t a, b e R 
_ 
{0}
Sfaeu* 
U=*rnrqffit 
rqkz x 
(x* 
5) 
= 
t0t ai*6rqparo+1fuq 
r
Let 
* 
be a binary operation, 
on the set of all non-zero real numbers, given 
by a 
x 
b 
= 
+
for all a, b € R 
- {0}.Find the value of x, given that2x (x x 
5) 
= 
10.
g. qH 
Erfr +1iqq , 
J.or-, 
(sin 
x) d.r.
f
Evaluate 
: 
J 
cos-l (sin 
x) d.r.
10. 
qkntn 
? 
q\r 
u'eqqffi{+tfq.l?l= 
3, 
lb,l=3* 
? r. u,\rs.q:rs.nPflrt, 
H}
? sil{ 
E' ++q or *iur fufuq r
If vectors 
? and E' *r such rhat, 
I 
? 
I = 
3, 
I 
b' 
| 
=l 
*a? x u, is a unit vector, then
write the angle between 
? anA U'.
651112 
' 
,*'T'o'
Etug-Et
SECTION - 
B
,rfi {reqr 11 t 22 rou*osq{ 
C 
ei6.qr t r
Question 
numbers 
lLto22 
cwry 4 marks each'
11. 
qa 
oirqrm flfi 
qi&q 
Ht 
qot 
f(x) 
= 
3# 
- 
+f 
- 
tz* + s
(a) fq{etq$iqlat 
I
(b) ftirr 6rgqm 
| 1
sIqET
elF,, 
= 
a sin30 
dsrT y 
= 
a cos30 
+M' e 
= 
fr 
wwi 
tot aqT 
e${Tiq *qrfu'qur frrd
dfqq r
Find the intervals 
in which the function 
f(x) 
= 
3f 
- 
+x3 
- 
12* + 5 is
(a) strictlYincreasing
(b) strict$ decreasing
OR
Find the equations 
of the tangent and normal to the curve x 
= 
a sin30 and 
y 
= 
a cos30 at
lt
0 
=7.
+
\
I
t2. 
q,-nrnqm{q 
J##"
eNrat
qmnrdqitq 
,!a-:1Pffia'
f sin6x + cos6x 
'
Evatuate: 
J 
ffiorr, 
o,
OR
' 
, 
-ghp+ 
3.r 
- 
18 dr
Evaluate: 
J 
(r
t3. tTq 
effitr€ 
q++tq 
*} re 
qfrkg 
:
G-D**"=h'
Solve the following 
differential 
equation 
:
(i-r)ff.rn=h'
65lLl2
t4.
15.
qk 
y 
= 
,r t, ni frr6 +ifrTq 16 
# 
i 
(*91 
f 
= 
o.
rr y 
= 
.rr, prove ,nurff-+ 
eI- 
) 
= 
o.
n A' A a x-) J
ffi drr €ffi 7, b,a + rdq' isd drftilq t-*-
[? 
* u', B + ?, ? * ?] 
=zli,ts, 
d]
sTqqt
qfur 
?, dnqr?tttt+.?* B +? 
=daqr lAl 
= 
3, 
IBI 
=5ilqt 
lAl =z 
t r ?mr B
+eiqmr+lurxradfrq r
Prove that, for any three vectors ?, B, ?
[? 
* B, B * ?, ? * ?] 
=zli,B, 
?]
OR
:-J
vectors E, b and d are such that ? * d + ? 
= 
d ana 
l?l 
= 
3, 
I 
b'l 
= 
5 ana 
l?l 
= 
z.
Find the angle between danO d.
16.
=f;*'(0,';
fls€*ltw tr 2 tan-t(}) * r".-' 
ffi 
+ z tarrtft) 
= 
f;
pro'ethatcorrf@
$it+slnx-1I-sm"r.
=l; 
*' (0,'o)
Prove that2,rr-' 
(,l 
* r".-' 
ffi 
+ 2 
'urrt(,l 
= 
i
17. 
qIil 
A 
= {!,2,3,.....,gldelr 
AxAi[ R\rs'{,+qt, 
ail 
Rx Aif 
1a, 
b), (c,0+ft{q
(a, b) R (c, d) uk a + d 
= 
b + c Ent 
qfuflfsd 
t r fucqifqqfu 
pqs'lFqilr 
{-{q t I (€trf,r
q,i 
I(2, ,l 
$ n6 eifvu r
Let A 
= {1, 
2, 3,....., 9} 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 (2,5)1.
6snt2 5
OR
lP.T.O.