CBSE Class 12 Mathematics: Question Paper for 2014 (Term-II) | Toppers Answer Sheets for Class 12 PDF Download

 Page 1


65/2 1 P.T.O. 
narjmWu H$moS >H$mo CÎma-nwpñVH$m Ho$ _wI-n¥ð 
>na Adí` {bIo§ & 
Candidates must write the Code on the 
title page of the answer-book. 
 Series OSR  
H$moS> Z§.     
65/2
 
 
Code No. 
amob Z§. 
Roll No. 
 
 
 
 
 
 
 
 
 
 
 
J{UV 
MATHEMATICS 
 
{ZYm©[aV g_` : 3 KÊQ>o   A{YH$V_ A§H$ : 100 
Time allowed : 3 hours Maximum Marks : 100 
? H¥$n`m Om±M H$a b| {H$ Bg àíZ-nÌ _o§ _w{ÐV n¥ð> 11 h¢ & 
? àíZ-nÌ _| Xm{hZo hmW H$s Amoa {XE JE H$moS >Zå~a H$mo N>mÌ CÎma-nwpñVH$m Ho$ _wI-n¥ð> na 
{bI| & 
? H¥$n`m Om±M H$a b| {H$ Bg àíZ-nÌ _| >29 àíZ h¢ & 
? H¥$n`m àíZ H$m CÎma {bIZm ewê$ H$aZo go nhbo, àíZ H$m H«$_m§H$ Adí` {bI| & 
? Bg  àíZ-nÌ  H$mo n‹T>Zo Ho$ {bE 15 {_ZQ >H$m g_` {X`m J`m h¡ &  àíZ-nÌ H$m {dVaU nydm©• 
_| 10.15 ~Oo {H$`m OmEJm &  10.15 ~Oo go 10.30 ~Oo VH$ N>mÌ Ho$db àíZ-nÌ H$mo n‹T>|Jo 
Am¡a Bg Ad{Y Ho$ Xm¡amZ do CÎma-nwpñVH$m na H$moB© CÎma Zht {bI|Jo & 
? Please check that this question paper contains 11 printed pages. 
? 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. 
? Please check that this question paper contains 29 questions. 
? Please write down the Serial Number of the question before 
attempting it. 
? 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. 
Page 2


65/2 1 P.T.O. 
narjmWu H$moS >H$mo CÎma-nwpñVH$m Ho$ _wI-n¥ð 
>na Adí` {bIo§ & 
Candidates must write the Code on the 
title page of the answer-book. 
 Series OSR  
H$moS> Z§.     
65/2
 
 
Code No. 
amob Z§. 
Roll No. 
 
 
 
 
 
 
 
 
 
 
 
J{UV 
MATHEMATICS 
 
{ZYm©[aV g_` : 3 KÊQ>o   A{YH$V_ A§H$ : 100 
Time allowed : 3 hours Maximum Marks : 100 
? H¥$n`m Om±M H$a b| {H$ Bg àíZ-nÌ _o§ _w{ÐV n¥ð> 11 h¢ & 
? àíZ-nÌ _| Xm{hZo hmW H$s Amoa {XE JE H$moS >Zå~a H$mo N>mÌ CÎma-nwpñVH$m Ho$ _wI-n¥ð> na 
{bI| & 
? H¥$n`m Om±M H$a b| {H$ Bg àíZ-nÌ _| >29 àíZ h¢ & 
? H¥$n`m àíZ H$m CÎma {bIZm ewê$ H$aZo go nhbo, àíZ H$m H«$_m§H$ Adí` {bI| & 
? Bg  àíZ-nÌ  H$mo n‹T>Zo Ho$ {bE 15 {_ZQ >H$m g_` {X`m J`m h¡ &  àíZ-nÌ H$m {dVaU nydm©• 
_| 10.15 ~Oo {H$`m OmEJm &  10.15 ~Oo go 10.30 ~Oo VH$ N>mÌ Ho$db àíZ-nÌ H$mo n‹T>|Jo 
Am¡a Bg Ad{Y Ho$ Xm¡amZ do CÎma-nwpñVH$m na H$moB© CÎma Zht {bI|Jo & 
? Please check that this question paper contains 11 printed pages. 
? 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. 
? Please check that this question paper contains 29 questions. 
? Please write down the Serial Number of the question before 
attempting it. 
? 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. 
65/2 2 
gm_mÝ` {ZX}e : 
(i) g^r àíZ A{Zdm`© h¢ & 
(ii) Bg àíZ nÌ _| 29 àíZ h¢ Omo VrZ IÊS>m| _| {d^m{OV h¢ : A, ~ VWm g &  IÊS> A _| 
10 àíZ h¢ {OZ_| go àË`oH$ EH$ A§H$ H$m h¡ & IÊS> ~ _| 12 àíZ h¢ {OZ_| go àË`oH$ Mma 
A§H$ H$m h¡ & IÊS> g _| 7 àíZ h¢ {OZ_| go àË`oH$ N>… A§H$ H$m h¡ & 
(iii) IÊS> A _| g^r àíZm| Ho$ CÎma EH$ eãX, EH$ dmŠ` AWdm àíZ H$s Amdí`H$Vm AZwgma 
{XE Om gH$Vo h¢ & 
(iv) nyU© àíZ nÌ _| {dH$ën Zht h¢ &  {\$a ^r Mma A§H$m| dmbo 4 àíZm| _| VWm N>… A§H$m| dmbo 
2 àíZm| _| AmÝV[aH$ {dH$ën h¡ &  Eogo g^r àíZm| _| go AmnH$mo EH$ hr {dH$ën hb H$aZm 
h¡ & 
(v) H¡$bHw$boQ>a Ho$ à`moJ H$s AZw_{V Zht h¡ & `{X Amdí`H$ hmo Vmo Amn bKwJUH$s` gma{U`m± 
_m±J gH$Vo h¢ & 
 
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 12 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 of four 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 ask for logarithmic tables, if 
required. 
Page 3


65/2 1 P.T.O. 
narjmWu H$moS >H$mo CÎma-nwpñVH$m Ho$ _wI-n¥ð 
>na Adí` {bIo§ & 
Candidates must write the Code on the 
title page of the answer-book. 
 Series OSR  
H$moS> Z§.     
65/2
 
 
Code No. 
amob Z§. 
Roll No. 
 
 
 
 
 
 
 
 
 
 
 
J{UV 
MATHEMATICS 
 
{ZYm©[aV g_` : 3 KÊQ>o   A{YH$V_ A§H$ : 100 
Time allowed : 3 hours Maximum Marks : 100 
? H¥$n`m Om±M H$a b| {H$ Bg àíZ-nÌ _o§ _w{ÐV n¥ð> 11 h¢ & 
? àíZ-nÌ _| Xm{hZo hmW H$s Amoa {XE JE H$moS >Zå~a H$mo N>mÌ CÎma-nwpñVH$m Ho$ _wI-n¥ð> na 
{bI| & 
? H¥$n`m Om±M H$a b| {H$ Bg àíZ-nÌ _| >29 àíZ h¢ & 
? H¥$n`m àíZ H$m CÎma {bIZm ewê$ H$aZo go nhbo, àíZ H$m H«$_m§H$ Adí` {bI| & 
? Bg  àíZ-nÌ  H$mo n‹T>Zo Ho$ {bE 15 {_ZQ >H$m g_` {X`m J`m h¡ &  àíZ-nÌ H$m {dVaU nydm©• 
_| 10.15 ~Oo {H$`m OmEJm &  10.15 ~Oo go 10.30 ~Oo VH$ N>mÌ Ho$db àíZ-nÌ H$mo n‹T>|Jo 
Am¡a Bg Ad{Y Ho$ Xm¡amZ do CÎma-nwpñVH$m na H$moB© CÎma Zht {bI|Jo & 
? Please check that this question paper contains 11 printed pages. 
? 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. 
? Please check that this question paper contains 29 questions. 
? Please write down the Serial Number of the question before 
attempting it. 
? 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. 
65/2 2 
gm_mÝ` {ZX}e : 
(i) g^r àíZ A{Zdm`© h¢ & 
(ii) Bg àíZ nÌ _| 29 àíZ h¢ Omo VrZ IÊS>m| _| {d^m{OV h¢ : A, ~ VWm g &  IÊS> A _| 
10 àíZ h¢ {OZ_| go àË`oH$ EH$ A§H$ H$m h¡ & IÊS> ~ _| 12 àíZ h¢ {OZ_| go àË`oH$ Mma 
A§H$ H$m h¡ & IÊS> g _| 7 àíZ h¢ {OZ_| go àË`oH$ N>… A§H$ H$m h¡ & 
(iii) IÊS> A _| g^r àíZm| Ho$ CÎma EH$ eãX, EH$ dmŠ` AWdm àíZ H$s Amdí`H$Vm AZwgma 
{XE Om gH$Vo h¢ & 
(iv) nyU© àíZ nÌ _| {dH$ën Zht h¢ &  {\$a ^r Mma A§H$m| dmbo 4 àíZm| _| VWm N>… A§H$m| dmbo 
2 àíZm| _| AmÝV[aH$ {dH$ën h¡ &  Eogo g^r àíZm| _| go AmnH$mo EH$ hr {dH$ën hb H$aZm 
h¡ & 
(v) H¡$bHw$boQ>a Ho$ à`moJ H$s AZw_{V Zht h¡ & `{X Amdí`H$ hmo Vmo Amn bKwJUH$s` gma{U`m± 
_m±J gH$Vo h¢ & 
 
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 12 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 of four 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 ask for logarithmic tables, if 
required. 
65/2 3 P.T.O. 
IÊS> A 
SECTION A 
àíZ g§»`m 1 go 10 VH$ àË`oH$ àíZ 1 A§H$ H$m h¡ & 
Question numbers 1 to 10 carry 1 mark each. 
1. `{X  
?
?
?
?
?
?
?
?
w y – x 2
z y – x
 = 
?
?
?
?
?
?
?
?
5 0
4 1 –
 h¡, Vmo x + y H$m _mZ kmV H$s{OE & 
If  
?
?
?
?
?
?
?
?
w y – x 2
z y – x
 = 
?
?
?
?
?
?
?
?
5 0
4 1 –
,  find the value of  x + y. 
2. `{X  
4 6
7 8
4 2 –
7 x 3
? h¡, Vmo x H$m _mZ kmV H$s{OE & 
If  
4 6
7 8
4 2 –
7 x 3
? ,  find the value of x. 
3. `{X  
?
?
x
0
dt t sin t ) x ( f  h¡, Vmo f ?(x) H$m _mZ kmV H$s{OE & 
If  
?
?
x
0
dt t sin t ) x ( f ,  then write the value of  f ?(x). 
4. `{X N na R = {(x, y) : x + 2y = 8}  EH$ g§~§Y h¡, Vmo R H$m n[aga {b{IE & 
If  R = {(x, y) : x + 2y = 8}  is a relation on N, write the range of R.  
5. `{X  tan
–1
 x + tan
–1
 y = 
4
?
,  xy < 1  h¡, Vmo x + y + xy  H$m _mZ {b{IE & 
If  tan
–1
 x + tan
–1
 y = 
4
?
,  xy < 1,  then write the value of  x + y + xy. 
Page 4


65/2 1 P.T.O. 
narjmWu H$moS >H$mo CÎma-nwpñVH$m Ho$ _wI-n¥ð 
>na Adí` {bIo§ & 
Candidates must write the Code on the 
title page of the answer-book. 
 Series OSR  
H$moS> Z§.     
65/2
 
 
Code No. 
amob Z§. 
Roll No. 
 
 
 
 
 
 
 
 
 
 
 
J{UV 
MATHEMATICS 
 
{ZYm©[aV g_` : 3 KÊQ>o   A{YH$V_ A§H$ : 100 
Time allowed : 3 hours Maximum Marks : 100 
? H¥$n`m Om±M H$a b| {H$ Bg àíZ-nÌ _o§ _w{ÐV n¥ð> 11 h¢ & 
? àíZ-nÌ _| Xm{hZo hmW H$s Amoa {XE JE H$moS >Zå~a H$mo N>mÌ CÎma-nwpñVH$m Ho$ _wI-n¥ð> na 
{bI| & 
? H¥$n`m Om±M H$a b| {H$ Bg àíZ-nÌ _| >29 àíZ h¢ & 
? H¥$n`m àíZ H$m CÎma {bIZm ewê$ H$aZo go nhbo, àíZ H$m H«$_m§H$ Adí` {bI| & 
? Bg  àíZ-nÌ  H$mo n‹T>Zo Ho$ {bE 15 {_ZQ >H$m g_` {X`m J`m h¡ &  àíZ-nÌ H$m {dVaU nydm©• 
_| 10.15 ~Oo {H$`m OmEJm &  10.15 ~Oo go 10.30 ~Oo VH$ N>mÌ Ho$db àíZ-nÌ H$mo n‹T>|Jo 
Am¡a Bg Ad{Y Ho$ Xm¡amZ do CÎma-nwpñVH$m na H$moB© CÎma Zht {bI|Jo & 
? Please check that this question paper contains 11 printed pages. 
? 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. 
? Please check that this question paper contains 29 questions. 
? Please write down the Serial Number of the question before 
attempting it. 
? 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. 
65/2 2 
gm_mÝ` {ZX}e : 
(i) g^r àíZ A{Zdm`© h¢ & 
(ii) Bg àíZ nÌ _| 29 àíZ h¢ Omo VrZ IÊS>m| _| {d^m{OV h¢ : A, ~ VWm g &  IÊS> A _| 
10 àíZ h¢ {OZ_| go àË`oH$ EH$ A§H$ H$m h¡ & IÊS> ~ _| 12 àíZ h¢ {OZ_| go àË`oH$ Mma 
A§H$ H$m h¡ & IÊS> g _| 7 àíZ h¢ {OZ_| go àË`oH$ N>… A§H$ H$m h¡ & 
(iii) IÊS> A _| g^r àíZm| Ho$ CÎma EH$ eãX, EH$ dmŠ` AWdm àíZ H$s Amdí`H$Vm AZwgma 
{XE Om gH$Vo h¢ & 
(iv) nyU© àíZ nÌ _| {dH$ën Zht h¢ &  {\$a ^r Mma A§H$m| dmbo 4 àíZm| _| VWm N>… A§H$m| dmbo 
2 àíZm| _| AmÝV[aH$ {dH$ën h¡ &  Eogo g^r àíZm| _| go AmnH$mo EH$ hr {dH$ën hb H$aZm 
h¡ & 
(v) H¡$bHw$boQ>a Ho$ à`moJ H$s AZw_{V Zht h¡ & `{X Amdí`H$ hmo Vmo Amn bKwJUH$s` gma{U`m± 
_m±J gH$Vo h¢ & 
 
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 12 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 of four 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 ask for logarithmic tables, if 
required. 
65/2 3 P.T.O. 
IÊS> A 
SECTION A 
àíZ g§»`m 1 go 10 VH$ àË`oH$ àíZ 1 A§H$ H$m h¡ & 
Question numbers 1 to 10 carry 1 mark each. 
1. `{X  
?
?
?
?
?
?
?
?
w y – x 2
z y – x
 = 
?
?
?
?
?
?
?
?
5 0
4 1 –
 h¡, Vmo x + y H$m _mZ kmV H$s{OE & 
If  
?
?
?
?
?
?
?
?
w y – x 2
z y – x
 = 
?
?
?
?
?
?
?
?
5 0
4 1 –
,  find the value of  x + y. 
2. `{X  
4 6
7 8
4 2 –
7 x 3
? h¡, Vmo x H$m _mZ kmV H$s{OE & 
If  
4 6
7 8
4 2 –
7 x 3
? ,  find the value of x. 
3. `{X  
?
?
x
0
dt t sin t ) x ( f  h¡, Vmo f ?(x) H$m _mZ kmV H$s{OE & 
If  
?
?
x
0
dt t sin t ) x ( f ,  then write the value of  f ?(x). 
4. `{X N na R = {(x, y) : x + 2y = 8}  EH$ g§~§Y h¡, Vmo R H$m n[aga {b{IE & 
If  R = {(x, y) : x + 2y = 8}  is a relation on N, write the range of R.  
5. `{X  tan
–1
 x + tan
–1
 y = 
4
?
,  xy < 1  h¡, Vmo x + y + xy  H$m _mZ {b{IE & 
If  tan
–1
 x + tan
–1
 y = 
4
?
,  xy < 1,  then write the value of  x + y + xy. 
65/2 4 
6. `{X A EH$ Eogm dJ© Amì`yh h¡ {H$  A
2
 = A  h¡, Vmo  7A – (I + A)
3
  H$m _mZ {b{IE, 
Ohm± I EH$ VËg_H$ Amì`yh h¡ & 
If  A  is  a  square  matrix  such  that   A
2
 = A,   then  write  the  value of 
7A – (I + A)
3
, where I is an identity matrix. 
7. ‘p’ H$m dh _mZ kmV H$s{OE {OgHo$ {bE g{Xe 3
^
i + 2
^
j + 9
^
k VWm 
^
i – 2p
^
j + 3
^
k 
g_m§Va h¢ & 
Find the value of  ‘p’  for which the vectors  3
^
i + 2
^
j + 9
^
k  and  
^
i – 2p
^
j + 3
^
k are parallel. 
8. `{X EH$ aoIm Ho$ H$mVu` g_rH$aU 
4
6 – z 2
7
4 y
5
x – 3
?
?
? h¢, Vmo Cg aoIm H$m g{Xe 
g_rH$aU {b{IE & 
If the cartesian equations of a line are 
4
6 – z 2
7
4 y
5
x – 3
?
?
? , write the 
vector equation for the line. 
9. _mZ kmV H$s{OE : 
 
x log x
dx
2
e
e
?
 
Evaluate : 
 
x log x
dx
2
e
e
?
 
10. 5 2 n[a_mU H$m EH$ g{Xe 
?
a kmV H$s{OE, Omo x-Aj go 
4
?
 H$m H$moU, y-Aj go 
2
?
 
H$m H$moU VWm z-Aj go Ý`yZ H$moU ? ~ZmVm h¡ & 
Find a vector 
?
a of magnitude 5 2 , making an angle of 
4
?
 with x-axis,  
2
?
 with y-axis and an acute angle ? with z-axis. 
 
Page 5


65/2 1 P.T.O. 
narjmWu H$moS >H$mo CÎma-nwpñVH$m Ho$ _wI-n¥ð 
>na Adí` {bIo§ & 
Candidates must write the Code on the 
title page of the answer-book. 
 Series OSR  
H$moS> Z§.     
65/2
 
 
Code No. 
amob Z§. 
Roll No. 
 
 
 
 
 
 
 
 
 
 
 
J{UV 
MATHEMATICS 
 
{ZYm©[aV g_` : 3 KÊQ>o   A{YH$V_ A§H$ : 100 
Time allowed : 3 hours Maximum Marks : 100 
? H¥$n`m Om±M H$a b| {H$ Bg àíZ-nÌ _o§ _w{ÐV n¥ð> 11 h¢ & 
? àíZ-nÌ _| Xm{hZo hmW H$s Amoa {XE JE H$moS >Zå~a H$mo N>mÌ CÎma-nwpñVH$m Ho$ _wI-n¥ð> na 
{bI| & 
? H¥$n`m Om±M H$a b| {H$ Bg àíZ-nÌ _| >29 àíZ h¢ & 
? H¥$n`m àíZ H$m CÎma {bIZm ewê$ H$aZo go nhbo, àíZ H$m H«$_m§H$ Adí` {bI| & 
? Bg  àíZ-nÌ  H$mo n‹T>Zo Ho$ {bE 15 {_ZQ >H$m g_` {X`m J`m h¡ &  àíZ-nÌ H$m {dVaU nydm©• 
_| 10.15 ~Oo {H$`m OmEJm &  10.15 ~Oo go 10.30 ~Oo VH$ N>mÌ Ho$db àíZ-nÌ H$mo n‹T>|Jo 
Am¡a Bg Ad{Y Ho$ Xm¡amZ do CÎma-nwpñVH$m na H$moB© CÎma Zht {bI|Jo & 
? Please check that this question paper contains 11 printed pages. 
? 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. 
? Please check that this question paper contains 29 questions. 
? Please write down the Serial Number of the question before 
attempting it. 
? 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. 
65/2 2 
gm_mÝ` {ZX}e : 
(i) g^r àíZ A{Zdm`© h¢ & 
(ii) Bg àíZ nÌ _| 29 àíZ h¢ Omo VrZ IÊS>m| _| {d^m{OV h¢ : A, ~ VWm g &  IÊS> A _| 
10 àíZ h¢ {OZ_| go àË`oH$ EH$ A§H$ H$m h¡ & IÊS> ~ _| 12 àíZ h¢ {OZ_| go àË`oH$ Mma 
A§H$ H$m h¡ & IÊS> g _| 7 àíZ h¢ {OZ_| go àË`oH$ N>… A§H$ H$m h¡ & 
(iii) IÊS> A _| g^r àíZm| Ho$ CÎma EH$ eãX, EH$ dmŠ` AWdm àíZ H$s Amdí`H$Vm AZwgma 
{XE Om gH$Vo h¢ & 
(iv) nyU© àíZ nÌ _| {dH$ën Zht h¢ &  {\$a ^r Mma A§H$m| dmbo 4 àíZm| _| VWm N>… A§H$m| dmbo 
2 àíZm| _| AmÝV[aH$ {dH$ën h¡ &  Eogo g^r àíZm| _| go AmnH$mo EH$ hr {dH$ën hb H$aZm 
h¡ & 
(v) H¡$bHw$boQ>a Ho$ à`moJ H$s AZw_{V Zht h¡ & `{X Amdí`H$ hmo Vmo Amn bKwJUH$s` gma{U`m± 
_m±J gH$Vo h¢ & 
 
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 12 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 of four 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 ask for logarithmic tables, if 
required. 
65/2 3 P.T.O. 
IÊS> A 
SECTION A 
àíZ g§»`m 1 go 10 VH$ àË`oH$ àíZ 1 A§H$ H$m h¡ & 
Question numbers 1 to 10 carry 1 mark each. 
1. `{X  
?
?
?
?
?
?
?
?
w y – x 2
z y – x
 = 
?
?
?
?
?
?
?
?
5 0
4 1 –
 h¡, Vmo x + y H$m _mZ kmV H$s{OE & 
If  
?
?
?
?
?
?
?
?
w y – x 2
z y – x
 = 
?
?
?
?
?
?
?
?
5 0
4 1 –
,  find the value of  x + y. 
2. `{X  
4 6
7 8
4 2 –
7 x 3
? h¡, Vmo x H$m _mZ kmV H$s{OE & 
If  
4 6
7 8
4 2 –
7 x 3
? ,  find the value of x. 
3. `{X  
?
?
x
0
dt t sin t ) x ( f  h¡, Vmo f ?(x) H$m _mZ kmV H$s{OE & 
If  
?
?
x
0
dt t sin t ) x ( f ,  then write the value of  f ?(x). 
4. `{X N na R = {(x, y) : x + 2y = 8}  EH$ g§~§Y h¡, Vmo R H$m n[aga {b{IE & 
If  R = {(x, y) : x + 2y = 8}  is a relation on N, write the range of R.  
5. `{X  tan
–1
 x + tan
–1
 y = 
4
?
,  xy < 1  h¡, Vmo x + y + xy  H$m _mZ {b{IE & 
If  tan
–1
 x + tan
–1
 y = 
4
?
,  xy < 1,  then write the value of  x + y + xy. 
65/2 4 
6. `{X A EH$ Eogm dJ© Amì`yh h¡ {H$  A
2
 = A  h¡, Vmo  7A – (I + A)
3
  H$m _mZ {b{IE, 
Ohm± I EH$ VËg_H$ Amì`yh h¡ & 
If  A  is  a  square  matrix  such  that   A
2
 = A,   then  write  the  value of 
7A – (I + A)
3
, where I is an identity matrix. 
7. ‘p’ H$m dh _mZ kmV H$s{OE {OgHo$ {bE g{Xe 3
^
i + 2
^
j + 9
^
k VWm 
^
i – 2p
^
j + 3
^
k 
g_m§Va h¢ & 
Find the value of  ‘p’  for which the vectors  3
^
i + 2
^
j + 9
^
k  and  
^
i – 2p
^
j + 3
^
k are parallel. 
8. `{X EH$ aoIm Ho$ H$mVu` g_rH$aU 
4
6 – z 2
7
4 y
5
x – 3
?
?
? h¢, Vmo Cg aoIm H$m g{Xe 
g_rH$aU {b{IE & 
If the cartesian equations of a line are 
4
6 – z 2
7
4 y
5
x – 3
?
?
? , write the 
vector equation for the line. 
9. _mZ kmV H$s{OE : 
 
x log x
dx
2
e
e
?
 
Evaluate : 
 
x log x
dx
2
e
e
?
 
10. 5 2 n[a_mU H$m EH$ g{Xe 
?
a kmV H$s{OE, Omo x-Aj go 
4
?
 H$m H$moU, y-Aj go 
2
?
 
H$m H$moU VWm z-Aj go Ý`yZ H$moU ? ~ZmVm h¡ & 
Find a vector 
?
a of magnitude 5 2 , making an angle of 
4
?
 with x-axis,  
2
?
 with y-axis and an acute angle ? with z-axis. 
 
65/2 5 P.T.O. 
IÊS> ~ 
SECTION B 
àíZ g§»`m 11 go 22 VH$ àË`oH$ àíZ 4 A§H$ H$m h¡ & 
Question numbers 11 to 22 carry 4 marks each. 
11. x Ho$ dh _mZ kmV H$s{OE {OgHo$ {bE  y = [x (x – 2)]
2
  EH$ dY©_mZ \$bZ h¡ & 
AWdm 
 dH«$ 1
b
y
–
a
x
2
2
2
2
? Ho$ {~ÝXþ ( 2 a, b) na ñne© aoIm VWm A{^b§~ Ho$ g_rH$aU kmV 
H$s{OE & 
Find the value(s) of x for which  y = [x (x – 2)]
2  
is an increasing function. 
OR 
Find the equations of the tangent and normal to the curve 1
b
y
–
a
x
2
2
2
2
? 
at the point ( 2 a, b). 
 
12. _mZ kmV H$s{OE : 
dx
x cos 1
x sin x 4
0
2
?
?
?
 
      AWdm       
_mZ kmV H$s{OE : 
dx
6 x 5 x
2 x
2
?
? ?
?
 
Evaluate : 
dx
x cos 1
x sin x 4
0
2
?
?
?
 
          OR 
Evaluate : 
dx
6 x 5 x
2 x
2
?
? ?
?