CBSE Class 12 Mathematics: Question Paper for 2020 (Term-II) | Toppers Answer Sheets for Class 12 PDF Download

 Page 1


  
 .65/1/2 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. 
  H$moS> Z§.  
   
 
 
Code No. 
amob Z§. 
Roll No. 
 
 
 
 
ZmoQ> 
 NOTE 
(I) 
H¥$n`m Om±M H$a b| {H$ Bg àíZ-nÌ _o§ _w{ÐV 
n¥ð> 15 h¢ & 
(I) Please check that this question 
paper contains 15 printed pages. 
(II) 
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| & 
(II) 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. 
(III) 
H¥$n`m Om±M H$a b| {H$ Bg àíZ-nÌ _|  
>36 àíZ h¢ & 
(III) Please check that this question 
paper contains 36 questions. 
(IV) 
H¥$n`m àíZ H$m CÎma {bIZm ewê$ H$aZo go 
nhbo, CÎma-nwpñVH$m _| àíZ H$m H«$_m§H$ 
Adí` {bI| &  
(IV) Please write down the Serial 
Number of the question in the 
answer-book before attempting it. 
(V) 
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 & 
(V) 15 minute 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. 
 J{UV 
MATHEMATICS 
{ZYm©[aV g_` : 3 KÊQ>o   A{YH$V_ A§H$ : 80 
Time allowed : 3 hours Maximum Marks : 80 
 
65/1/2
 
Page 2


  
 .65/1/2 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. 
  H$moS> Z§.  
   
 
 
Code No. 
amob Z§. 
Roll No. 
 
 
 
 
ZmoQ> 
 NOTE 
(I) 
H¥$n`m Om±M H$a b| {H$ Bg àíZ-nÌ _o§ _w{ÐV 
n¥ð> 15 h¢ & 
(I) Please check that this question 
paper contains 15 printed pages. 
(II) 
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| & 
(II) 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. 
(III) 
H¥$n`m Om±M H$a b| {H$ Bg àíZ-nÌ _|  
>36 àíZ h¢ & 
(III) Please check that this question 
paper contains 36 questions. 
(IV) 
H¥$n`m àíZ H$m CÎma {bIZm ewê$ H$aZo go 
nhbo, CÎma-nwpñVH$m _| àíZ H$m H«$_m§H$ 
Adí` {bI| &  
(IV) Please write down the Serial 
Number of the question in the 
answer-book before attempting it. 
(V) 
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 & 
(V) 15 minute 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. 
 J{UV 
MATHEMATICS 
{ZYm©[aV g_` : 3 KÊQ>o   A{YH$V_ A§H$ : 80 
Time allowed : 3 hours Maximum Marks : 80 
 
65/1/2
 
  
 .65/1/2 2 
gm_mÝ` {ZX}e : 
{ZåZ{b{IV {ZX}em| H$mo ~hþV gmdYmZr go n{‹T>E Am¡a CZH$m g™Vr go nmbZ H$s{OE : 
(i) `h àíZ-nÌ Mma IÊS>m| _| {d^m{OV {H$`m J`m h¡ – H$, I, J Ed§ K & Bg àíZ-nÌ _| 36 àíZ 
h¢ & g^r  àíZ A{Zdm`© h¢ & 
(ii) IÊS> H$ _| àíZ g§»`m 1 go 20 VH$ 20 àíZ h¢ Ed§ àË`oH$ àíZ 1 A§H$ H$m h¡ &  
(iii) IÊS> I _| àíZ g§»`m 21 go 26 VH$ 6 àíZ h¢ Ed§ àË`oH$ àíZ 2 A§H$m| H$m h¡ & 
(iv) IÊS> J _| àíZ g§»`m 27 go 32 VH$ 6 àíZ h¢ Ed§ àË`oH$ àíZ 4 A§H$m| H$m h¡ &   
(v) IÊS> K _| àíZ g§»`m 33 go 36 VH$ 4 àíZ h¢ Ed§ àË`oH$ àíZ 6 A§H$m| H$m h¡ & 
(vi) àíZ-nÌ _| g_J« na H$moB© {dH$ën Zht h¡ & VWm{n EH$-EH$ A§H$ dmbo VrZ àíZm| _|, Xmo-Xmo A§H$m| 
dmbo Xmo àíZm| _|, Mma-Mma A§H$m| dmbo Xmo àíZm| _| Am¡a N :-N : A§H$m| dmbo Xmo àíZm| _| Am§V[aH$ 
{dH$ën {XE JE h¢ & Eogo àíZm| _| go Ho$db EH$ hr {dH$ën H$m CÎma {b{IE &  
(vii) BgHo$ A{V[aº$, Amdí`H$VmZwgma, àË`oH$ IÊS> Am¡a àíZ Ho$ gmW `Wmo{MV {ZX}e {XE JE h¢ & 
(viii) Ho$bHw$boQ>am| Ho$ à`moJ H$s AZw_{V Zht h¡ & 
IÊS> H$ 
àíZ g§»`m 1 go 20 VH$ àË`oH$ àíZ 1 A§H$ H$m h¡ &  
àíZ g§»`m 1 go 10 VH$ ~hþ{dH$ënr` àíZ h¢ & ghr {dH$ën Mw{ZE &  
1. `{X A EH$ 3 ? 3 Amì`yh h¡ Am¡a |A| = – 2 h¡, Vmo |A (adj A)| H$m _mZ hmoJm   >   
(A) – 2   
(B) 2  
(C) – 8   
(D) 8  
2. H$mo{Q> 2 dmbo AdH$b g_rH$aU Ho$ {deof hb _| ñdoÀN> AMam| H$s g§»`m hmoJr   
(A) 0  
(B) 1  
(C) 2  
(D) 3 
3. cos
–1
 (cos 
6
13 p
) H$m _w»` _mZ h¡  
(A) 
6
13 p
  
(B) 
2
p
  
(C) 
3
p
  
(D) 
6
p
 
Page 3


  
 .65/1/2 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. 
  H$moS> Z§.  
   
 
 
Code No. 
amob Z§. 
Roll No. 
 
 
 
 
ZmoQ> 
 NOTE 
(I) 
H¥$n`m Om±M H$a b| {H$ Bg àíZ-nÌ _o§ _w{ÐV 
n¥ð> 15 h¢ & 
(I) Please check that this question 
paper contains 15 printed pages. 
(II) 
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| & 
(II) 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. 
(III) 
H¥$n`m Om±M H$a b| {H$ Bg àíZ-nÌ _|  
>36 àíZ h¢ & 
(III) Please check that this question 
paper contains 36 questions. 
(IV) 
H¥$n`m àíZ H$m CÎma {bIZm ewê$ H$aZo go 
nhbo, CÎma-nwpñVH$m _| àíZ H$m H«$_m§H$ 
Adí` {bI| &  
(IV) Please write down the Serial 
Number of the question in the 
answer-book before attempting it. 
(V) 
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 & 
(V) 15 minute 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. 
 J{UV 
MATHEMATICS 
{ZYm©[aV g_` : 3 KÊQ>o   A{YH$V_ A§H$ : 80 
Time allowed : 3 hours Maximum Marks : 80 
 
65/1/2
 
  
 .65/1/2 2 
gm_mÝ` {ZX}e : 
{ZåZ{b{IV {ZX}em| H$mo ~hþV gmdYmZr go n{‹T>E Am¡a CZH$m g™Vr go nmbZ H$s{OE : 
(i) `h àíZ-nÌ Mma IÊS>m| _| {d^m{OV {H$`m J`m h¡ – H$, I, J Ed§ K & Bg àíZ-nÌ _| 36 àíZ 
h¢ & g^r  àíZ A{Zdm`© h¢ & 
(ii) IÊS> H$ _| àíZ g§»`m 1 go 20 VH$ 20 àíZ h¢ Ed§ àË`oH$ àíZ 1 A§H$ H$m h¡ &  
(iii) IÊS> I _| àíZ g§»`m 21 go 26 VH$ 6 àíZ h¢ Ed§ àË`oH$ àíZ 2 A§H$m| H$m h¡ & 
(iv) IÊS> J _| àíZ g§»`m 27 go 32 VH$ 6 àíZ h¢ Ed§ àË`oH$ àíZ 4 A§H$m| H$m h¡ &   
(v) IÊS> K _| àíZ g§»`m 33 go 36 VH$ 4 àíZ h¢ Ed§ àË`oH$ àíZ 6 A§H$m| H$m h¡ & 
(vi) àíZ-nÌ _| g_J« na H$moB© {dH$ën Zht h¡ & VWm{n EH$-EH$ A§H$ dmbo VrZ àíZm| _|, Xmo-Xmo A§H$m| 
dmbo Xmo àíZm| _|, Mma-Mma A§H$m| dmbo Xmo àíZm| _| Am¡a N :-N : A§H$m| dmbo Xmo àíZm| _| Am§V[aH$ 
{dH$ën {XE JE h¢ & Eogo àíZm| _| go Ho$db EH$ hr {dH$ën H$m CÎma {b{IE &  
(vii) BgHo$ A{V[aº$, Amdí`H$VmZwgma, àË`oH$ IÊS> Am¡a àíZ Ho$ gmW `Wmo{MV {ZX}e {XE JE h¢ & 
(viii) Ho$bHw$boQ>am| Ho$ à`moJ H$s AZw_{V Zht h¡ & 
IÊS> H$ 
àíZ g§»`m 1 go 20 VH$ àË`oH$ àíZ 1 A§H$ H$m h¡ &  
àíZ g§»`m 1 go 10 VH$ ~hþ{dH$ënr` àíZ h¢ & ghr {dH$ën Mw{ZE &  
1. `{X A EH$ 3 ? 3 Amì`yh h¡ Am¡a |A| = – 2 h¡, Vmo |A (adj A)| H$m _mZ hmoJm   >   
(A) – 2   
(B) 2  
(C) – 8   
(D) 8  
2. H$mo{Q> 2 dmbo AdH$b g_rH$aU Ho$ {deof hb _| ñdoÀN> AMam| H$s g§»`m hmoJr   
(A) 0  
(B) 1  
(C) 2  
(D) 3 
3. cos
–1
 (cos 
6
13 p
) H$m _w»` _mZ h¡  
(A) 
6
13 p
  
(B) 
2
p
  
(C) 
3
p
  
(D) 
6
p
 
  
 .65/1/2 3 P.T.O. 
General Instructions : 
Read the following instructions very carefully and strictly follow them : 
(i) This question paper comprises four Sections A, B, C and D. This question 
paper carries 36 questions. All questions are compulsory.  
(ii) Section A – Questions no. 1 to 20 comprises of 20 questions of 1 mark each. 
(iii) Section B – Questions no. 21 to 26 comprises of 6 questions of 2 marks each. 
(iv) Section C – Questions no. 27 to 32 comprises of 6 questions of 4 marks each. 
(v) Section D – Questions no. 33 to 36 comprises of 4 questions of 6 marks each. 
(vi) There is no overall choice in the question paper. However, an internal choice 
has been provided in 3 questions of one mark, 2 questions of two marks,  
2 questions of four marks and 2 questions of six marks. Only one of the choices 
in such questions have to be attempted.  
(vii) In addition to this, separate instructions are given with each section and 
question, wherever necessary.  
(viii) Use of calculators is not permitted.  
SECTION A 
Question numbers 1 to 20 carry 1 mark each.  
Question numbers 1 to 10 are multiple choice type questions. Select the correct option. 
1. If  A is a 3 ? 3 matrix and |A| = – 2, then value of |A (adj A)| is   
(A) – 2   
(B) 2  
(C) – 8   
(D) 8 
2. The number of arbitrary constants in the particular solution of a 
differential equation of second order is (are)     
(A) 0  
(B) 1  
(C) 2  
(D) 3 
3. The principal value of  cos
–1
 (cos 
6
13 p
) is  
(A) 
6
13 p
  
(B) 
2
p
  
(C) 
3
p
  
(D) 
6
p
 
Page 4


  
 .65/1/2 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. 
  H$moS> Z§.  
   
 
 
Code No. 
amob Z§. 
Roll No. 
 
 
 
 
ZmoQ> 
 NOTE 
(I) 
H¥$n`m Om±M H$a b| {H$ Bg àíZ-nÌ _o§ _w{ÐV 
n¥ð> 15 h¢ & 
(I) Please check that this question 
paper contains 15 printed pages. 
(II) 
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| & 
(II) 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. 
(III) 
H¥$n`m Om±M H$a b| {H$ Bg àíZ-nÌ _|  
>36 àíZ h¢ & 
(III) Please check that this question 
paper contains 36 questions. 
(IV) 
H¥$n`m àíZ H$m CÎma {bIZm ewê$ H$aZo go 
nhbo, CÎma-nwpñVH$m _| àíZ H$m H«$_m§H$ 
Adí` {bI| &  
(IV) Please write down the Serial 
Number of the question in the 
answer-book before attempting it. 
(V) 
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 & 
(V) 15 minute 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. 
 J{UV 
MATHEMATICS 
{ZYm©[aV g_` : 3 KÊQ>o   A{YH$V_ A§H$ : 80 
Time allowed : 3 hours Maximum Marks : 80 
 
65/1/2
 
  
 .65/1/2 2 
gm_mÝ` {ZX}e : 
{ZåZ{b{IV {ZX}em| H$mo ~hþV gmdYmZr go n{‹T>E Am¡a CZH$m g™Vr go nmbZ H$s{OE : 
(i) `h àíZ-nÌ Mma IÊS>m| _| {d^m{OV {H$`m J`m h¡ – H$, I, J Ed§ K & Bg àíZ-nÌ _| 36 àíZ 
h¢ & g^r  àíZ A{Zdm`© h¢ & 
(ii) IÊS> H$ _| àíZ g§»`m 1 go 20 VH$ 20 àíZ h¢ Ed§ àË`oH$ àíZ 1 A§H$ H$m h¡ &  
(iii) IÊS> I _| àíZ g§»`m 21 go 26 VH$ 6 àíZ h¢ Ed§ àË`oH$ àíZ 2 A§H$m| H$m h¡ & 
(iv) IÊS> J _| àíZ g§»`m 27 go 32 VH$ 6 àíZ h¢ Ed§ àË`oH$ àíZ 4 A§H$m| H$m h¡ &   
(v) IÊS> K _| àíZ g§»`m 33 go 36 VH$ 4 àíZ h¢ Ed§ àË`oH$ àíZ 6 A§H$m| H$m h¡ & 
(vi) àíZ-nÌ _| g_J« na H$moB© {dH$ën Zht h¡ & VWm{n EH$-EH$ A§H$ dmbo VrZ àíZm| _|, Xmo-Xmo A§H$m| 
dmbo Xmo àíZm| _|, Mma-Mma A§H$m| dmbo Xmo àíZm| _| Am¡a N :-N : A§H$m| dmbo Xmo àíZm| _| Am§V[aH$ 
{dH$ën {XE JE h¢ & Eogo àíZm| _| go Ho$db EH$ hr {dH$ën H$m CÎma {b{IE &  
(vii) BgHo$ A{V[aº$, Amdí`H$VmZwgma, àË`oH$ IÊS> Am¡a àíZ Ho$ gmW `Wmo{MV {ZX}e {XE JE h¢ & 
(viii) Ho$bHw$boQ>am| Ho$ à`moJ H$s AZw_{V Zht h¡ & 
IÊS> H$ 
àíZ g§»`m 1 go 20 VH$ àË`oH$ àíZ 1 A§H$ H$m h¡ &  
àíZ g§»`m 1 go 10 VH$ ~hþ{dH$ënr` àíZ h¢ & ghr {dH$ën Mw{ZE &  
1. `{X A EH$ 3 ? 3 Amì`yh h¡ Am¡a |A| = – 2 h¡, Vmo |A (adj A)| H$m _mZ hmoJm   >   
(A) – 2   
(B) 2  
(C) – 8   
(D) 8  
2. H$mo{Q> 2 dmbo AdH$b g_rH$aU Ho$ {deof hb _| ñdoÀN> AMam| H$s g§»`m hmoJr   
(A) 0  
(B) 1  
(C) 2  
(D) 3 
3. cos
–1
 (cos 
6
13 p
) H$m _w»` _mZ h¡  
(A) 
6
13 p
  
(B) 
2
p
  
(C) 
3
p
  
(D) 
6
p
 
  
 .65/1/2 3 P.T.O. 
General Instructions : 
Read the following instructions very carefully and strictly follow them : 
(i) This question paper comprises four Sections A, B, C and D. This question 
paper carries 36 questions. All questions are compulsory.  
(ii) Section A – Questions no. 1 to 20 comprises of 20 questions of 1 mark each. 
(iii) Section B – Questions no. 21 to 26 comprises of 6 questions of 2 marks each. 
(iv) Section C – Questions no. 27 to 32 comprises of 6 questions of 4 marks each. 
(v) Section D – Questions no. 33 to 36 comprises of 4 questions of 6 marks each. 
(vi) There is no overall choice in the question paper. However, an internal choice 
has been provided in 3 questions of one mark, 2 questions of two marks,  
2 questions of four marks and 2 questions of six marks. Only one of the choices 
in such questions have to be attempted.  
(vii) In addition to this, separate instructions are given with each section and 
question, wherever necessary.  
(viii) Use of calculators is not permitted.  
SECTION A 
Question numbers 1 to 20 carry 1 mark each.  
Question numbers 1 to 10 are multiple choice type questions. Select the correct option. 
1. If  A is a 3 ? 3 matrix and |A| = – 2, then value of |A (adj A)| is   
(A) – 2   
(B) 2  
(C) – 8   
(D) 8 
2. The number of arbitrary constants in the particular solution of a 
differential equation of second order is (are)     
(A) 0  
(B) 1  
(C) 2  
(D) 3 
3. The principal value of  cos
–1
 (cos 
6
13 p
) is  
(A) 
6
13 p
  
(B) 
2
p
  
(C) 
3
p
  
(D) 
6
p
 
  
 .65/1/2 4 
4. a¡{IH$ Ag{_H$mAm| Ho$ {ZH$m` go {Z`V gwg§JV joÌ Ho$ H$moZr` q~Xþ (0, 0), (4, 0), (2, 4) 
VWm (0, 5) h¢ & `{X z = ax + by, Ohm± a, b > 0 H$m A{YH$V_ _mZ {~ÝXþAmo§ (2, 4) VWm 
(4, 0) XmoZm| na hmo, Vmo    
(A) a = 2b  
(B) 2a = b  
(C) a = b  
(D) 3a = b  
5. `{X A Am¡a B Xmo ñdV§Ì KQ>ZmE± h¢, Ohm± P(A) = 
3
1
 d P(B) = 
4
1
 h¡, Vmo P(B
?
|A) 
~am~a h¡   
(A) 
4
1
  
(B) 
3
1
  
(C) 
4
3
  
(D) 1 
6. `{X A EH$ dJ© Amì`yh h¡ VWm A
2
 = A hmo, Vmo (I – A)
3
 + A ~am~a h¡     
(A) I  
(B) 0  
(C) I – A  
(D) I + A  
7. , dx
x
1
sin
x
1
2
2
–
2
?
?
?
?
?
?
?
p
p
 Ohm± x ? 0, ~am~a h¡   
(A) – 2  
(B) 0  
(C) 1  
(D) ? 
Page 5


  
 .65/1/2 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. 
  H$moS> Z§.  
   
 
 
Code No. 
amob Z§. 
Roll No. 
 
 
 
 
ZmoQ> 
 NOTE 
(I) 
H¥$n`m Om±M H$a b| {H$ Bg àíZ-nÌ _o§ _w{ÐV 
n¥ð> 15 h¢ & 
(I) Please check that this question 
paper contains 15 printed pages. 
(II) 
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| & 
(II) 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. 
(III) 
H¥$n`m Om±M H$a b| {H$ Bg àíZ-nÌ _|  
>36 àíZ h¢ & 
(III) Please check that this question 
paper contains 36 questions. 
(IV) 
H¥$n`m àíZ H$m CÎma {bIZm ewê$ H$aZo go 
nhbo, CÎma-nwpñVH$m _| àíZ H$m H«$_m§H$ 
Adí` {bI| &  
(IV) Please write down the Serial 
Number of the question in the 
answer-book before attempting it. 
(V) 
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 & 
(V) 15 minute 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. 
 J{UV 
MATHEMATICS 
{ZYm©[aV g_` : 3 KÊQ>o   A{YH$V_ A§H$ : 80 
Time allowed : 3 hours Maximum Marks : 80 
 
65/1/2
 
  
 .65/1/2 2 
gm_mÝ` {ZX}e : 
{ZåZ{b{IV {ZX}em| H$mo ~hþV gmdYmZr go n{‹T>E Am¡a CZH$m g™Vr go nmbZ H$s{OE : 
(i) `h àíZ-nÌ Mma IÊS>m| _| {d^m{OV {H$`m J`m h¡ – H$, I, J Ed§ K & Bg àíZ-nÌ _| 36 àíZ 
h¢ & g^r  àíZ A{Zdm`© h¢ & 
(ii) IÊS> H$ _| àíZ g§»`m 1 go 20 VH$ 20 àíZ h¢ Ed§ àË`oH$ àíZ 1 A§H$ H$m h¡ &  
(iii) IÊS> I _| àíZ g§»`m 21 go 26 VH$ 6 àíZ h¢ Ed§ àË`oH$ àíZ 2 A§H$m| H$m h¡ & 
(iv) IÊS> J _| àíZ g§»`m 27 go 32 VH$ 6 àíZ h¢ Ed§ àË`oH$ àíZ 4 A§H$m| H$m h¡ &   
(v) IÊS> K _| àíZ g§»`m 33 go 36 VH$ 4 àíZ h¢ Ed§ àË`oH$ àíZ 6 A§H$m| H$m h¡ & 
(vi) àíZ-nÌ _| g_J« na H$moB© {dH$ën Zht h¡ & VWm{n EH$-EH$ A§H$ dmbo VrZ àíZm| _|, Xmo-Xmo A§H$m| 
dmbo Xmo àíZm| _|, Mma-Mma A§H$m| dmbo Xmo àíZm| _| Am¡a N :-N : A§H$m| dmbo Xmo àíZm| _| Am§V[aH$ 
{dH$ën {XE JE h¢ & Eogo àíZm| _| go Ho$db EH$ hr {dH$ën H$m CÎma {b{IE &  
(vii) BgHo$ A{V[aº$, Amdí`H$VmZwgma, àË`oH$ IÊS> Am¡a àíZ Ho$ gmW `Wmo{MV {ZX}e {XE JE h¢ & 
(viii) Ho$bHw$boQ>am| Ho$ à`moJ H$s AZw_{V Zht h¡ & 
IÊS> H$ 
àíZ g§»`m 1 go 20 VH$ àË`oH$ àíZ 1 A§H$ H$m h¡ &  
àíZ g§»`m 1 go 10 VH$ ~hþ{dH$ënr` àíZ h¢ & ghr {dH$ën Mw{ZE &  
1. `{X A EH$ 3 ? 3 Amì`yh h¡ Am¡a |A| = – 2 h¡, Vmo |A (adj A)| H$m _mZ hmoJm   >   
(A) – 2   
(B) 2  
(C) – 8   
(D) 8  
2. H$mo{Q> 2 dmbo AdH$b g_rH$aU Ho$ {deof hb _| ñdoÀN> AMam| H$s g§»`m hmoJr   
(A) 0  
(B) 1  
(C) 2  
(D) 3 
3. cos
–1
 (cos 
6
13 p
) H$m _w»` _mZ h¡  
(A) 
6
13 p
  
(B) 
2
p
  
(C) 
3
p
  
(D) 
6
p
 
  
 .65/1/2 3 P.T.O. 
General Instructions : 
Read the following instructions very carefully and strictly follow them : 
(i) This question paper comprises four Sections A, B, C and D. This question 
paper carries 36 questions. All questions are compulsory.  
(ii) Section A – Questions no. 1 to 20 comprises of 20 questions of 1 mark each. 
(iii) Section B – Questions no. 21 to 26 comprises of 6 questions of 2 marks each. 
(iv) Section C – Questions no. 27 to 32 comprises of 6 questions of 4 marks each. 
(v) Section D – Questions no. 33 to 36 comprises of 4 questions of 6 marks each. 
(vi) There is no overall choice in the question paper. However, an internal choice 
has been provided in 3 questions of one mark, 2 questions of two marks,  
2 questions of four marks and 2 questions of six marks. Only one of the choices 
in such questions have to be attempted.  
(vii) In addition to this, separate instructions are given with each section and 
question, wherever necessary.  
(viii) Use of calculators is not permitted.  
SECTION A 
Question numbers 1 to 20 carry 1 mark each.  
Question numbers 1 to 10 are multiple choice type questions. Select the correct option. 
1. If  A is a 3 ? 3 matrix and |A| = – 2, then value of |A (adj A)| is   
(A) – 2   
(B) 2  
(C) – 8   
(D) 8 
2. The number of arbitrary constants in the particular solution of a 
differential equation of second order is (are)     
(A) 0  
(B) 1  
(C) 2  
(D) 3 
3. The principal value of  cos
–1
 (cos 
6
13 p
) is  
(A) 
6
13 p
  
(B) 
2
p
  
(C) 
3
p
  
(D) 
6
p
 
  
 .65/1/2 4 
4. a¡{IH$ Ag{_H$mAm| Ho$ {ZH$m` go {Z`V gwg§JV joÌ Ho$ H$moZr` q~Xþ (0, 0), (4, 0), (2, 4) 
VWm (0, 5) h¢ & `{X z = ax + by, Ohm± a, b > 0 H$m A{YH$V_ _mZ {~ÝXþAmo§ (2, 4) VWm 
(4, 0) XmoZm| na hmo, Vmo    
(A) a = 2b  
(B) 2a = b  
(C) a = b  
(D) 3a = b  
5. `{X A Am¡a B Xmo ñdV§Ì KQ>ZmE± h¢, Ohm± P(A) = 
3
1
 d P(B) = 
4
1
 h¡, Vmo P(B
?
|A) 
~am~a h¡   
(A) 
4
1
  
(B) 
3
1
  
(C) 
4
3
  
(D) 1 
6. `{X A EH$ dJ© Amì`yh h¡ VWm A
2
 = A hmo, Vmo (I – A)
3
 + A ~am~a h¡     
(A) I  
(B) 0  
(C) I – A  
(D) I + A  
7. , dx
x
1
sin
x
1
2
2
–
2
?
?
?
?
?
?
?
p
p
 Ohm± x ? 0, ~am~a h¡   
(A) – 2  
(B) 0  
(C) 1  
(D) ? 
  
 .65/1/2 5 P.T.O. 
4. The corner points of the feasible region determined by the system of 
linear inequalities are (0, 0), (4, 0), (2, 4) and (0, 5). If the maximum value 
of z = ax + by, where a, b > 0 occurs at both (2, 4) and (4, 0), then    
(A) a = 2b  
(B) 2a = b  
(C) a = b  
(D) 3a = b 
5. If A and B are two independent events with P(A) = 
3
1
 and P(B) = 
4
1
, then 
P(B
?
 | A) is equal to  
(A) 
4
1
  
(B) 
3
1
  
(C) 
4
3
  
(D) 1 
6. If A is a square matrix such that  A
2
 = A,  then  (I – A)
3
 + A  is equal to  
(A) I  
(B) 0  
(C) I – A  
(D) I + A 
7. , dx
x
1
sin
x
1
2
2
–
2
?
?
?
?
?
?
?
p
p
 where x ? 0, is equal to   
(A) – 2  
(B) 0  
(C) 1  
(D) ?