CBSE Class 12 Mathematics: Question Paper for 2018 (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 SGN 
 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 SGN 
 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 SGN 
 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$m h¡ & 
Question numbers 1 to 4 carry 1 mark each. 
1. tan
–1
3 –  cot
–1
(– 3 )  H$m _mZ kmV H$s{OE & 
Find the value of   tan
–1
3 –  cot
–1
(– 3 ). 
2. `{X Amì`yh 
?
?
?
?
?
?
?
?
?
?
?
0 1 b
1 – 0 2
3 – a 0
A {df_ g_{_V h¡, Vmo ‘a’ VWm ‘b’ Ho$ _mZ kmV  
H$s{OE & 
If the matrix  
?
?
?
?
?
?
?
?
?
?
?
0 1 b
1 – 0 2
3 – a 0
A is skew symmetric, find the values of ‘a’ 
and ‘b’. 
3. Xmo g{Xem| 
?
a VWm 
?
b , {OZHo$ n[a_mU g_mZ h¢, _| go àË`oH$ H$m n[a_mU kmV H$s{OE, 
O~{H$ CZHo$ ~rM H$m H$moU 60 ? h¡ VWm CZH$m A{Xe JwUZ\$b 
2
9
 h¡ & 
Find the magnitude of each of the two vectors 
?
a and 
?
b , having the 
same magnitude such that the angle between them is 60 ? and their scalar 
product is 
2
9
. 
4. `{X a 
*
 b, ‘a’ VWm ‘b’ _| go ~‹S>r g§»`m H$mo Xem©Vm h¡ VWm `{X a ? b = (a 
*
 b) + 3 h¡, 
Vmo (5) ? (10) H$m _mZ {b{IE, Ohm± 
*
 VWm ? {ÛAmYmar g§{H«$`mE± h¢ &  
If  a 
*
 b denotes the larger of ‘a’ and ‘b’ and if a ? b = (a 
*
 b) + 3, then 
write the value of (5) ? (10), where 
*
 and o are binary operations. 
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 SGN 
 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$m h¡ & 
Question numbers 1 to 4 carry 1 mark each. 
1. tan
–1
3 –  cot
–1
(– 3 )  H$m _mZ kmV H$s{OE & 
Find the value of   tan
–1
3 –  cot
–1
(– 3 ). 
2. `{X Amì`yh 
?
?
?
?
?
?
?
?
?
?
?
0 1 b
1 – 0 2
3 – a 0
A {df_ g_{_V h¡, Vmo ‘a’ VWm ‘b’ Ho$ _mZ kmV  
H$s{OE & 
If the matrix  
?
?
?
?
?
?
?
?
?
?
?
0 1 b
1 – 0 2
3 – a 0
A is skew symmetric, find the values of ‘a’ 
and ‘b’. 
3. Xmo g{Xem| 
?
a VWm 
?
b , {OZHo$ n[a_mU g_mZ h¢, _| go àË`oH$ H$m n[a_mU kmV H$s{OE, 
O~{H$ CZHo$ ~rM H$m H$moU 60 ? h¡ VWm CZH$m A{Xe JwUZ\$b 
2
9
 h¡ & 
Find the magnitude of each of the two vectors 
?
a and 
?
b , having the 
same magnitude such that the angle between them is 60 ? and their scalar 
product is 
2
9
. 
4. `{X a 
*
 b, ‘a’ VWm ‘b’ _| go ~‹S>r g§»`m H$mo Xem©Vm h¡ VWm `{X a ? b = (a 
*
 b) + 3 h¡, 
Vmo (5) ? (10) H$m _mZ {b{IE, Ohm± 
*
 VWm ? {ÛAmYmar g§{H«$`mE± h¢ &  
If  a 
*
 b denotes the larger of ‘a’ and ‘b’ and if a ? b = (a 
*
 b) + 3, then 
write the value of (5) ? (10), where 
*
 and o are binary operations. 
65/1 4 
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. {gÕ H$s{OE {H$ : 
 3 sin
–1
 x = sin
–1 
(3x – 4x
3
),  x ? 
?
?
?
?
?
?
2
1
,
2
1
– 
Prove that : 
 3 sin
–1
 x = sin
–1 
(3x – 4x
3
),  x ? 
?
?
?
?
?
?
2
1
,
2
1
– 
6. {X`m J`m h¡ {H$ 
?
?
?
?
?
?
?
?
?
7 4 –
3 – 2
A h¡, Vmo A
–1
  kmV H$s{OE VWm Xem©BE {H$  
2A
–1
 = 9I – A. 
Given 
?
?
?
?
?
?
?
?
?
7 4 –
3 – 2
A , compute A
–1
 and show that  2A
–1
 = 9I – A. 
7. tan
–1 
?
?
?
?
?
?
?
? ?
x sin
x cos 1
 H$m x Ho$ gmnoj AdH$bZ H$s{OE & 
Differentiate  tan
–1 
?
?
?
?
?
?
?
? ?
x sin
x cos 1
 with respect to x. 
8. {H$gr dñVw H$s x BH$mB`m| Ho$ CËnmXZ go gå~pÝYV Hw$b bmJV C(x),  
C(x) = 0·005x
3
 – 0·02x
2
 + 30x + 5000 go àXÎm h¡ & gr_m§V bmJV kmV H$s{OE 
O~{H$ 3 BH$mB© CËnm{XV H$s OmVr h¢, Ohm± gr_m§V bmJV (marginal cost) go A{^àm` h¡ 
CËnmXZ Ho$ {H$gr ñVa na g§nyU© bmJV _| VmËH$m{bH$ n[adV©Z H$s Xa & 
The total cost C(x) associated with the production of x units of an item is 
given by C(x) = 0·005x
3
 – 0·02x
2
 + 30x + 5000. Find the marginal cost 
when 3 units are produced, where by marginal cost we mean the 
instantaneous rate of change of total cost at any level of output. 
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 SGN 
 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$m h¡ & 
Question numbers 1 to 4 carry 1 mark each. 
1. tan
–1
3 –  cot
–1
(– 3 )  H$m _mZ kmV H$s{OE & 
Find the value of   tan
–1
3 –  cot
–1
(– 3 ). 
2. `{X Amì`yh 
?
?
?
?
?
?
?
?
?
?
?
0 1 b
1 – 0 2
3 – a 0
A {df_ g_{_V h¡, Vmo ‘a’ VWm ‘b’ Ho$ _mZ kmV  
H$s{OE & 
If the matrix  
?
?
?
?
?
?
?
?
?
?
?
0 1 b
1 – 0 2
3 – a 0
A is skew symmetric, find the values of ‘a’ 
and ‘b’. 
3. Xmo g{Xem| 
?
a VWm 
?
b , {OZHo$ n[a_mU g_mZ h¢, _| go àË`oH$ H$m n[a_mU kmV H$s{OE, 
O~{H$ CZHo$ ~rM H$m H$moU 60 ? h¡ VWm CZH$m A{Xe JwUZ\$b 
2
9
 h¡ & 
Find the magnitude of each of the two vectors 
?
a and 
?
b , having the 
same magnitude such that the angle between them is 60 ? and their scalar 
product is 
2
9
. 
4. `{X a 
*
 b, ‘a’ VWm ‘b’ _| go ~‹S>r g§»`m H$mo Xem©Vm h¡ VWm `{X a ? b = (a 
*
 b) + 3 h¡, 
Vmo (5) ? (10) H$m _mZ {b{IE, Ohm± 
*
 VWm ? {ÛAmYmar g§{H«$`mE± h¢ &  
If  a 
*
 b denotes the larger of ‘a’ and ‘b’ and if a ? b = (a 
*
 b) + 3, then 
write the value of (5) ? (10), where 
*
 and o are binary operations. 
65/1 4 
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. {gÕ H$s{OE {H$ : 
 3 sin
–1
 x = sin
–1 
(3x – 4x
3
),  x ? 
?
?
?
?
?
?
2
1
,
2
1
– 
Prove that : 
 3 sin
–1
 x = sin
–1 
(3x – 4x
3
),  x ? 
?
?
?
?
?
?
2
1
,
2
1
– 
6. {X`m J`m h¡ {H$ 
?
?
?
?
?
?
?
?
?
7 4 –
3 – 2
A h¡, Vmo A
–1
  kmV H$s{OE VWm Xem©BE {H$  
2A
–1
 = 9I – A. 
Given 
?
?
?
?
?
?
?
?
?
7 4 –
3 – 2
A , compute A
–1
 and show that  2A
–1
 = 9I – A. 
7. tan
–1 
?
?
?
?
?
?
?
? ?
x sin
x cos 1
 H$m x Ho$ gmnoj AdH$bZ H$s{OE & 
Differentiate  tan
–1 
?
?
?
?
?
?
?
? ?
x sin
x cos 1
 with respect to x. 
8. {H$gr dñVw H$s x BH$mB`m| Ho$ CËnmXZ go gå~pÝYV Hw$b bmJV C(x),  
C(x) = 0·005x
3
 – 0·02x
2
 + 30x + 5000 go àXÎm h¡ & gr_m§V bmJV kmV H$s{OE 
O~{H$ 3 BH$mB© CËnm{XV H$s OmVr h¢, Ohm± gr_m§V bmJV (marginal cost) go A{^àm` h¡ 
CËnmXZ Ho$ {H$gr ñVa na g§nyU© bmJV _| VmËH$m{bH$ n[adV©Z H$s Xa & 
The total cost C(x) associated with the production of x units of an item is 
given by C(x) = 0·005x
3
 – 0·02x
2
 + 30x + 5000. Find the marginal cost 
when 3 units are produced, where by marginal cost we mean the 
instantaneous rate of change of total cost at any level of output. 
65/1 5 P.T.O. 
9. _yë`m§H$Z H$s{OE : 
 dx
x cos
x sin 2 x 2 cos
2
2
?
?
 
Evaluate : 
 dx
x cos
x sin 2 x 2 cos
2
2
?
?
 
10. dH«$ Hw$b  y = a e
bx+5
 H$mo {Zê${nV H$aZo dmbm EH$ AdH$b g_rH$aU kmV H$s{OE, Ohm± 
a VWm b ñdoÀN> AMa h¢ & 
Find the differential equation representing the family of curves  
y = a e
bx+5
, where a and b are arbitrary constants. 
11. `{X Xmo g{Xem| 
^
i – 2
^
j + 3
^
k VWm 3
^
i – 2
^
j + 
^
k Ho$ ~rM H$m H$moU ? h¡, Vmo sin ? 
kmV H$s{OE & 
If  ?  is the angle between two vectors 
^
i – 2
^
j + 3
^
k and 3
^
i – 2
^
j + 
^
k , 
find sin ?. 
12. EH$ H$mbm VWm EH$ bmb nmgm EH$ gmW CN>mbo OmVo h¢ & nmgm| na AmZo dmbr g§»`mAm§o 
H$m `moJ\$b 8 AmZo H$s gà{V~§Y àm{`H$Vm kmV H$s{OE, {X`m J`m h¡ {H$ bmb nmgo na 
AmZo dmbr g§»`m 4 go H$_ h¡ & 
A black and a red die are rolled together. Find the conditional probability 
of obtaining the sum 8, given that the red die resulted in a number less 
than 4. 
IÊS> g 
SECTION C 
àíZ g§»`m 13 go 23 VH$ àË`oH$ àíZ Ho$ 4 A§H$ h¢ & 
Question numbers 13 to 23 carry 4 marks each. 
13. gma{UH$m| Ho$ JwUY_mªo H$m à`moJ H$aHo$ {gÕ H$s{OE {H$  
 ) zx yz xy xyz 3 ( 9
1 z 3 1 1
1 1 y 3 1
x 3 1 1 1
? ? ? ?
?
?
?