CBSE Class 12 Mathematics: Question Paper for 2017 (Term-II) | Toppers Answer Sheets for Class 12 PDF Download

 Page 1


65/1 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 GBM 
 H$moS> Z§.  
   
 
 
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> 12 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 12 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 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. 
65/1
 
SET-1 
Page 2


65/1 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 GBM 
 H$moS> Z§.  
   
 
 
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> 12 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 12 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 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. 
65/1
 
SET-1 
65/1 2 
gm_mÝ` {ZX}e : 
(i) g^r àíZ A{Zdm`© h¢ & 
(ii) Bg àíZ nÌ _| 29 àíZ h¢ Omo Mma IÊS>m| _| {d^m{OV h¢ : A, ~, g VWm X &  IÊS> A _| 
4 àíZ h¢ {OZ_| go àË`oH$ EH$ A§H$ H$m h¡ & IÊS> ~ _| 8 àíZ h¢ {OZ_| go àË`oH$ Xmo A§H$ 
H$m h¡ & IÊS> g _| 11 àíZ h¢ {OZ_| go àË`oH$ Mma A§H$ H$m h¡ & IÊS> X _| 6 àíZ h¢ 
{OZ_| go àË`oH$ N>: A§H$ H$m h¡ &   
(iii) IÊS> A _| g^r àíZm| Ho$ CÎma EH$ eãX, EH$ dmŠ` AWdm àíZ H$s Amdí`H$VmZwgma {XE 
Om gH$Vo h¢ & 
(iv) nyU© àíZ nÌ _| {dH$ën Zht h¢ &  {\$a ^r Mma A§H$m| dmbo 3 àíZm| _| VWm N>… A§H$m| dmbo 
3 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 four sections A, B, 
C and D. Section A comprises of 4 questions of one mark each, Section B 
comprises of 8 questions of  two marks  each, Section C comprises of  
11 questions of four marks each and Section D comprises of 6 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  
3 questions of four marks each and 3 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


65/1 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 GBM 
 H$moS> Z§.  
   
 
 
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> 12 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 12 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 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. 
65/1
 
SET-1 
65/1 2 
gm_mÝ` {ZX}e : 
(i) g^r àíZ A{Zdm`© h¢ & 
(ii) Bg àíZ nÌ _| 29 àíZ h¢ Omo Mma IÊS>m| _| {d^m{OV h¢ : A, ~, g VWm X &  IÊS> A _| 
4 àíZ h¢ {OZ_| go àË`oH$ EH$ A§H$ H$m h¡ & IÊS> ~ _| 8 àíZ h¢ {OZ_| go àË`oH$ Xmo A§H$ 
H$m h¡ & IÊS> g _| 11 àíZ h¢ {OZ_| go àË`oH$ Mma A§H$ H$m h¡ & IÊS> X _| 6 àíZ h¢ 
{OZ_| go àË`oH$ N>: A§H$ H$m h¡ &   
(iii) IÊS> A _| g^r àíZm| Ho$ CÎma EH$ eãX, EH$ dmŠ` AWdm àíZ H$s Amdí`H$VmZwgma {XE 
Om gH$Vo h¢ & 
(iv) nyU© àíZ nÌ _| {dH$ën Zht h¢ &  {\$a ^r Mma A§H$m| dmbo 3 àíZm| _| VWm N>… A§H$m| dmbo 
3 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 four sections A, B, 
C and D. Section A comprises of 4 questions of one mark each, Section B 
comprises of 8 questions of  two marks  each, Section C comprises of  
11 questions of four marks each and Section D comprises of 6 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  
3 questions of four marks each and 3 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. 
65/1 3 P.T.O. 
IÊS> A 
SECTION A 
 
àíZ g§»`m 1 go 4 VH$ àË`oH$ àíZ 1 A§H$ H$m h¡ & 
Question numbers 1 to 4 carry 1 mark each. 
 
1. `{X {H$gr 2 ? 2 dJ© Amì`yh A Ho$ {bE, A(adj A) = 
?
?
?
?
?
?
?
?
8 0
0 8
 h¡, Vmo |A| H$m _mZ 
{b{IE &  
 
If for any 2 ? 2 square matrix A,  A(adj A) = 
?
?
?
?
?
?
?
?
8 0
0 8
, then write the value 
of |A|. 
 
2. ‘k’ H$m _mZ kmV H$s{OE {OgHo$ {bE {ZåZ{b{IV \$bZ x = 3 na g§VV h¡ : 
 f(x) = 
?
?
?
?
?
?
?
?
?
?
? ?
3 x , k
3 x ,
3 x
36 ) 3 x (
2
 
Determine the value of ‘k’ for which the following function is continuous 
at x = 3 : 
 f(x) = 
?
?
?
?
?
?
?
?
?
?
? ?
3 x , k
3 x ,
3 x
36 ) 3 x (
2
 
Page 4


65/1 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 GBM 
 H$moS> Z§.  
   
 
 
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> 12 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 12 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 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. 
65/1
 
SET-1 
65/1 2 
gm_mÝ` {ZX}e : 
(i) g^r àíZ A{Zdm`© h¢ & 
(ii) Bg àíZ nÌ _| 29 àíZ h¢ Omo Mma IÊS>m| _| {d^m{OV h¢ : A, ~, g VWm X &  IÊS> A _| 
4 àíZ h¢ {OZ_| go àË`oH$ EH$ A§H$ H$m h¡ & IÊS> ~ _| 8 àíZ h¢ {OZ_| go àË`oH$ Xmo A§H$ 
H$m h¡ & IÊS> g _| 11 àíZ h¢ {OZ_| go àË`oH$ Mma A§H$ H$m h¡ & IÊS> X _| 6 àíZ h¢ 
{OZ_| go àË`oH$ N>: A§H$ H$m h¡ &   
(iii) IÊS> A _| g^r àíZm| Ho$ CÎma EH$ eãX, EH$ dmŠ` AWdm àíZ H$s Amdí`H$VmZwgma {XE 
Om gH$Vo h¢ & 
(iv) nyU© àíZ nÌ _| {dH$ën Zht h¢ &  {\$a ^r Mma A§H$m| dmbo 3 àíZm| _| VWm N>… A§H$m| dmbo 
3 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 four sections A, B, 
C and D. Section A comprises of 4 questions of one mark each, Section B 
comprises of 8 questions of  two marks  each, Section C comprises of  
11 questions of four marks each and Section D comprises of 6 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  
3 questions of four marks each and 3 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. 
65/1 3 P.T.O. 
IÊS> A 
SECTION A 
 
àíZ g§»`m 1 go 4 VH$ àË`oH$ àíZ 1 A§H$ H$m h¡ & 
Question numbers 1 to 4 carry 1 mark each. 
 
1. `{X {H$gr 2 ? 2 dJ© Amì`yh A Ho$ {bE, A(adj A) = 
?
?
?
?
?
?
?
?
8 0
0 8
 h¡, Vmo |A| H$m _mZ 
{b{IE &  
 
If for any 2 ? 2 square matrix A,  A(adj A) = 
?
?
?
?
?
?
?
?
8 0
0 8
, then write the value 
of |A|. 
 
2. ‘k’ H$m _mZ kmV H$s{OE {OgHo$ {bE {ZåZ{b{IV \$bZ x = 3 na g§VV h¡ : 
 f(x) = 
?
?
?
?
?
?
?
?
?
?
? ?
3 x , k
3 x ,
3 x
36 ) 3 x (
2
 
Determine the value of ‘k’ for which the following function is continuous 
at x = 3 : 
 f(x) = 
?
?
?
?
?
?
?
?
?
?
? ?
3 x , k
3 x ,
3 x
36 ) 3 x (
2
 
65/1 4 
3. kmV H$s{OE : 
 
?
?
dx
x cos x sin
x cos x sin
2 2
 
Find :  
 
?
?
dx
x cos x sin
x cos x sin
2 2
  
4. g_Vbm| 2x – y + 2z = 5  VWm  5x – 2
.
5y + 5z = 20  Ho$ ~rM H$s Xÿar kmV H$s{OE & 
Find the distance between the planes 2x  –  y  +  2z  =  5 and  
5x – 2
.
5y + 5z = 20. 
 
IÊS> ~ 
SECTION B 
 
àíZ g§»`m 5 go 12 VH$ àË`oH$ àíZ Ho$ 2 A§H$ h¢ & 
Question numbers 5 to 12 carry 2 marks each. 
5. `{X A H$mo{Q> 3 H$m EH$ {df_-g_{_V Amì`yh h¡, Vmo {gÕ H$s{OE {H$ det A = 0.  
If A is a skew-symmetric matrix of order 3, then prove that det A = 0. 
6. \$bZ f(x) = x
3
 – 3x,  [– 3 , 0] Ho$ {bE amobo Ho$ à_o` Ho$ à`moJ go c H$m _mZ kmV 
H$s{OE & 
Find the value of c in Rolle’s theorem for the function f(x) = x
3
 – 3x in  
[– 3 , 0]. 
7. EH$ KZ H$m Am`VZ 9 KZ go_r/go. H$s Xa go ~‹T> ahm h¡ & O~ KZ H$s ^wOm 10 go_r h¡, 
Vmo CgHo$ n¥ð>r` joÌ\$b _| ~‹T>moVar H$s Xa kmV H$s{OE & 
The volume of a cube is increasing at the rate of 9 cm
3
/s. How fast is its 
surface area increasing when the length of an edge is 10 cm ?  
Page 5


65/1 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 GBM 
 H$moS> Z§.  
   
 
 
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> 12 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 12 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 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. 
65/1
 
SET-1 
65/1 2 
gm_mÝ` {ZX}e : 
(i) g^r àíZ A{Zdm`© h¢ & 
(ii) Bg àíZ nÌ _| 29 àíZ h¢ Omo Mma IÊS>m| _| {d^m{OV h¢ : A, ~, g VWm X &  IÊS> A _| 
4 àíZ h¢ {OZ_| go àË`oH$ EH$ A§H$ H$m h¡ & IÊS> ~ _| 8 àíZ h¢ {OZ_| go àË`oH$ Xmo A§H$ 
H$m h¡ & IÊS> g _| 11 àíZ h¢ {OZ_| go àË`oH$ Mma A§H$ H$m h¡ & IÊS> X _| 6 àíZ h¢ 
{OZ_| go àË`oH$ N>: A§H$ H$m h¡ &   
(iii) IÊS> A _| g^r àíZm| Ho$ CÎma EH$ eãX, EH$ dmŠ` AWdm àíZ H$s Amdí`H$VmZwgma {XE 
Om gH$Vo h¢ & 
(iv) nyU© àíZ nÌ _| {dH$ën Zht h¢ &  {\$a ^r Mma A§H$m| dmbo 3 àíZm| _| VWm N>… A§H$m| dmbo 
3 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 four sections A, B, 
C and D. Section A comprises of 4 questions of one mark each, Section B 
comprises of 8 questions of  two marks  each, Section C comprises of  
11 questions of four marks each and Section D comprises of 6 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  
3 questions of four marks each and 3 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. 
65/1 3 P.T.O. 
IÊS> A 
SECTION A 
 
àíZ g§»`m 1 go 4 VH$ àË`oH$ àíZ 1 A§H$ H$m h¡ & 
Question numbers 1 to 4 carry 1 mark each. 
 
1. `{X {H$gr 2 ? 2 dJ© Amì`yh A Ho$ {bE, A(adj A) = 
?
?
?
?
?
?
?
?
8 0
0 8
 h¡, Vmo |A| H$m _mZ 
{b{IE &  
 
If for any 2 ? 2 square matrix A,  A(adj A) = 
?
?
?
?
?
?
?
?
8 0
0 8
, then write the value 
of |A|. 
 
2. ‘k’ H$m _mZ kmV H$s{OE {OgHo$ {bE {ZåZ{b{IV \$bZ x = 3 na g§VV h¡ : 
 f(x) = 
?
?
?
?
?
?
?
?
?
?
? ?
3 x , k
3 x ,
3 x
36 ) 3 x (
2
 
Determine the value of ‘k’ for which the following function is continuous 
at x = 3 : 
 f(x) = 
?
?
?
?
?
?
?
?
?
?
? ?
3 x , k
3 x ,
3 x
36 ) 3 x (
2
 
65/1 4 
3. kmV H$s{OE : 
 
?
?
dx
x cos x sin
x cos x sin
2 2
 
Find :  
 
?
?
dx
x cos x sin
x cos x sin
2 2
  
4. g_Vbm| 2x – y + 2z = 5  VWm  5x – 2
.
5y + 5z = 20  Ho$ ~rM H$s Xÿar kmV H$s{OE & 
Find the distance between the planes 2x  –  y  +  2z  =  5 and  
5x – 2
.
5y + 5z = 20. 
 
IÊS> ~ 
SECTION B 
 
àíZ g§»`m 5 go 12 VH$ àË`oH$ àíZ Ho$ 2 A§H$ h¢ & 
Question numbers 5 to 12 carry 2 marks each. 
5. `{X A H$mo{Q> 3 H$m EH$ {df_-g_{_V Amì`yh h¡, Vmo {gÕ H$s{OE {H$ det A = 0.  
If A is a skew-symmetric matrix of order 3, then prove that det A = 0. 
6. \$bZ f(x) = x
3
 – 3x,  [– 3 , 0] Ho$ {bE amobo Ho$ à_o` Ho$ à`moJ go c H$m _mZ kmV 
H$s{OE & 
Find the value of c in Rolle’s theorem for the function f(x) = x
3
 – 3x in  
[– 3 , 0]. 
7. EH$ KZ H$m Am`VZ 9 KZ go_r/go. H$s Xa go ~‹T> ahm h¡ & O~ KZ H$s ^wOm 10 go_r h¡, 
Vmo CgHo$ n¥ð>r` joÌ\$b _| ~‹T>moVar H$s Xa kmV H$s{OE & 
The volume of a cube is increasing at the rate of 9 cm
3
/s. How fast is its 
surface area increasing when the length of an edge is 10 cm ?  
65/1 5 P.T.O. 
8. Xem©BE {H$ \$bZ f(x) = x
3
 – 3x
2
 + 6x – 100,     na dY©_mZ h¡ &  
Show that the function  f(x) = x
3
 – 3x
2
 + 6x – 100  is increasing on    . 
9. q~XþAm| P(2, 2, 1) VWm Q(5, 1, – 2) H$mo {_bmZo dmbr aoIm na pñWV EH$ q~Xþ H$m  
x-{ZX}em§H$ 4 h¡ & CgH$m z-{ZX}em§H$$kmV H$s{OE & 
The x-coordinate of a point on the line joining the points P(2, 2, 1) and 
Q(5, 1, – 2) is 4. Find its z-coordinate. 
10. EH$ nmgm, {OgHo$ \$bH$m| na A§H$ 1, 2, 3 bmb a§J _| {bIo h¢ VWm 4, 5, 6 hao a§J _| 
{bIo h¢, H$mo CN>mbm J`m & _mZm KQ>Zm A h¡ : ‘‘àmßV g§»`m g_ h¡’’ VWm KQ>Zm B h¡ : 
‘‘àmßV g§»`m bmb h¡’’ & kmV H$s{OE {H$ Š`m A VWm B ñdV§Ì KQ>ZmE± h¢ & 
A die, whose faces are marked 1, 2, 3 in red and 4, 5, 6 in green, is tossed. 
Let A be the event ‘‘number obtained is even’’ and B be the event 
‘‘number obtained is red’’. Find if A and B are independent events. 
11. Xmo XOu, A VWm B, à{V{XZ H«$_e: < 300 VWm < 400 H$_mVo h¢ &  A EH$ {XZ _|  
6 H$_rµO| VWm 4 n¢Q>| {gb gH$Vm h¡ O~{H$ B à{V{XZ 10 H$_rµOo§ VWm 4 n¢Q>o§ {gb gH$Vm  
h¡ & `h kmV H$aZo Ho$ {bE {H$ H$_-go-H$_ 60 H$_rµO§o VWm 32 n¢Q>| {gbZo Ho$ {bE àË`oH$ 
{H$VZo {XZ H$m`© H$ao {H$ l_ bmJV H$_-go-H$_ hmo, a¡{IH$ àmoJ«m_Z g_ñ`m Ho$ ê$n _| 
gyÌ~Õ H$s{OE & 
Two tailors, A and B, earn < 300 and < 400 per day respectively. A can 
stitch 6 shirts and 4 pairs of trousers while B can stitch 10 shirts and  
4 pairs of trousers per day. To find how many days should each of them 
work and if it is desired to produce at least 60 shirts and 32 pairs of 
trousers at a minimum labour cost, formulate this as an LPP. 
12. kmV H$s{OE : 
 
? ? ?
2
x x 8 5
dx
 
Find : 
 
? ? ?
2
x x 8 5
dx