Class 12 Mathematics (Maths) Previous Year Paper - 1 | Mathematics (Maths) Class 12 - JEE PDF Download

 Page 1


666  
65/1/1-11 Page 1 of 23 P.T.O.   
narjmWu àíZ-nÌ H$moS> >H$mo CÎma-nwpñVH$m Ho$ 
_wI-n¥ð >na Adí` {bIo§ & 
Candidates must write the Q.P. Code 
on the title page of the answer-book. 
 Series 
PQ1RS/1
  Set – 1  
  àíZ-nÌ H$moS>   
   
 
 
Q.P. Code  
AZwH«$_m§§H$
 
Roll No. 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 J{UV  
MATHEMATICS 
 
{ZYm©[aV g_` : 3 KÊQ>o   A{YH$V_ A§H$ : 80 
Time allowed : 3 hours  Maximum Marks : 80 
· H¥$n`m Om±M H$a b| {H$ Bg àíZ-nÌ _o§ _w{ÐV n¥ð> 23 h¢ & 
· H¥$n`m Om±M H$a b| {H$ Bg àíZ-nÌ _| >38 àíZ h¢ & 
· àíZ-nÌ _| Xm{hZo hmW H$s Amoa {XE JE àíZ-nÌ H$moS> H$mo narjmWu CÎma-nwpñVH$m Ho$  
_wI-n¥ð> na {bI| & 
· H¥$n`m àíZ H$m CÎma {bIZm ewê$ H$aZo go nhbo, CÎma-nwpñVH$m _| 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 23 printed pages. 
· Please check that this question paper contains 38 questions. 
· Q.P. Code 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 write down the serial number of the question in the answer-book 
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. 
65/1/1
 
* P Q 1 R S / 1 * 
Page 2


666  
65/1/1-11 Page 1 of 23 P.T.O.   
narjmWu àíZ-nÌ H$moS> >H$mo CÎma-nwpñVH$m Ho$ 
_wI-n¥ð >na Adí` {bIo§ & 
Candidates must write the Q.P. Code 
on the title page of the answer-book. 
 Series 
PQ1RS/1
  Set – 1  
  àíZ-nÌ H$moS>   
   
 
 
Q.P. Code  
AZwH«$_m§§H$
 
Roll No. 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 J{UV  
MATHEMATICS 
 
{ZYm©[aV g_` : 3 KÊQ>o   A{YH$V_ A§H$ : 80 
Time allowed : 3 hours  Maximum Marks : 80 
· H¥$n`m Om±M H$a b| {H$ Bg àíZ-nÌ _o§ _w{ÐV n¥ð> 23 h¢ & 
· H¥$n`m Om±M H$a b| {H$ Bg àíZ-nÌ _| >38 àíZ h¢ & 
· àíZ-nÌ _| Xm{hZo hmW H$s Amoa {XE JE àíZ-nÌ H$moS> H$mo narjmWu CÎma-nwpñVH$m Ho$  
_wI-n¥ð> na {bI| & 
· H¥$n`m àíZ H$m CÎma {bIZm ewê$ H$aZo go nhbo, CÎma-nwpñVH$m _| 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 23 printed pages. 
· Please check that this question paper contains 38 questions. 
· Q.P. Code 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 write down the serial number of the question in the answer-book 
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. 
65/1/1
 
* P Q 1 R S / 1 * 
666  
65/1/1-11 Page 3 of 23 P.T.O.   
General Instructions : 
Read the following instructions very carefully and strictly follow them : 
(i) This question paper contains 38 questions. All questions are compulsory.  
(ii) This question paper is divided into five Sections – A, B, C, D and E.  
(iii) In Section A, Questions no. 1 to 18 are multiple choice questions (MCQs) and 
questions number 19 and 20 are Assertion-Reason based questions of 1 mark 
each.  
(iv) In Section B, Questions no. 21 to 25 are very short answer (VSA) type 
questions, carrying 2 marks each.  
(v) In Section C, Questions no. 26 to 31 are short answer (SA) type questions, 
carrying 3 marks each. 
(vi) In Section D, Questions no. 32 to 35 are long answer (LA) type questions 
carrying 5 marks each.  
(vii) In Section E, Questions no. 36 to 38 are case study based questions carrying  
4 marks each.   
(viii) There is no overall choice. However, an internal choice has been provided in  
2 questions in Section B, 3 questions in Section C, 2 questions in Section D and 
2 questions in Section E.  
(ix) Use of calculators is not allowed.  
SECTION A 
This section comprises multiple choice questions (MCQs) of 1 mark each. 
  
1. A function f : R
+
 ® R (where R
+
 is the set of all non-negative real 
numbers) defined by f(x) = 4x + 3 is :  
(A) one-one but not onto  
(B) onto but not one-one 
(C) both one-one and onto  
(D) neither one-one nor onto 
2. If a matrix has 36 elements, the number of possible orders it can have,  
is : 
(A) 13  (B) 3 
(C) 5  (D) 9 
Page 3


666  
65/1/1-11 Page 1 of 23 P.T.O.   
narjmWu àíZ-nÌ H$moS> >H$mo CÎma-nwpñVH$m Ho$ 
_wI-n¥ð >na Adí` {bIo§ & 
Candidates must write the Q.P. Code 
on the title page of the answer-book. 
 Series 
PQ1RS/1
  Set – 1  
  àíZ-nÌ H$moS>   
   
 
 
Q.P. Code  
AZwH«$_m§§H$
 
Roll No. 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 J{UV  
MATHEMATICS 
 
{ZYm©[aV g_` : 3 KÊQ>o   A{YH$V_ A§H$ : 80 
Time allowed : 3 hours  Maximum Marks : 80 
· H¥$n`m Om±M H$a b| {H$ Bg àíZ-nÌ _o§ _w{ÐV n¥ð> 23 h¢ & 
· H¥$n`m Om±M H$a b| {H$ Bg àíZ-nÌ _| >38 àíZ h¢ & 
· àíZ-nÌ _| Xm{hZo hmW H$s Amoa {XE JE àíZ-nÌ H$moS> H$mo narjmWu CÎma-nwpñVH$m Ho$  
_wI-n¥ð> na {bI| & 
· H¥$n`m àíZ H$m CÎma {bIZm ewê$ H$aZo go nhbo, CÎma-nwpñVH$m _| 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 23 printed pages. 
· Please check that this question paper contains 38 questions. 
· Q.P. Code 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 write down the serial number of the question in the answer-book 
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. 
65/1/1
 
* P Q 1 R S / 1 * 
666  
65/1/1-11 Page 3 of 23 P.T.O.   
General Instructions : 
Read the following instructions very carefully and strictly follow them : 
(i) This question paper contains 38 questions. All questions are compulsory.  
(ii) This question paper is divided into five Sections – A, B, C, D and E.  
(iii) In Section A, Questions no. 1 to 18 are multiple choice questions (MCQs) and 
questions number 19 and 20 are Assertion-Reason based questions of 1 mark 
each.  
(iv) In Section B, Questions no. 21 to 25 are very short answer (VSA) type 
questions, carrying 2 marks each.  
(v) In Section C, Questions no. 26 to 31 are short answer (SA) type questions, 
carrying 3 marks each. 
(vi) In Section D, Questions no. 32 to 35 are long answer (LA) type questions 
carrying 5 marks each.  
(vii) In Section E, Questions no. 36 to 38 are case study based questions carrying  
4 marks each.   
(viii) There is no overall choice. However, an internal choice has been provided in  
2 questions in Section B, 3 questions in Section C, 2 questions in Section D and 
2 questions in Section E.  
(ix) Use of calculators is not allowed.  
SECTION A 
This section comprises multiple choice questions (MCQs) of 1 mark each. 
  
1. A function f : R
+
 ® R (where R
+
 is the set of all non-negative real 
numbers) defined by f(x) = 4x + 3 is :  
(A) one-one but not onto  
(B) onto but not one-one 
(C) both one-one and onto  
(D) neither one-one nor onto 
2. If a matrix has 36 elements, the number of possible orders it can have,  
is : 
(A) 13  (B) 3 
(C) 5  (D) 9 
666  
65/1/1-11 Page 5 of 23 P.T.O.   
3. Which of the following statements is true for the function  
f(x) = 
2
x 3, x 0
1 , x 0
ì
ï + ¹
í
=
ï
î
 ? 
(A) f(x) is continuous and differentiable ? x Î R 
(B) f(x) is continuous ? x Î R 
(C) f(x) is continuous and differentiable ? x Î R – {0} 
(D) f(x) is discontinuous at infinitely many points  
4. Let f(x) be a continuous function on [a, b] and differentiable on (a, b). 
Then, this function f(x) is strictly increasing in (a, b) if  
(A) f ¢(x) < 0, ? x Î (a, b)  
(B) f ¢(x) > 0, ? x Î (a, b) 
(C) f ¢(x) = 0, ? x Î (a, b)  
(D) f (x) > 0, ? x Î (a, b)  
5. If 
ú
û
ù
ê
ë
é
+
xy 5
2 y x
 = 
ú
û
ù
ê
ë
é
8 5
2 6
, then the value of 
÷
÷
ø
ö
ç
ç
è
æ
+
y
24
x
24
 is : 
(A) 7  (B) 6 
(C) 8  (D) 18 
6. 
ò
b
a
f(x) dx is equal to : 
(A) 
ò
b
a
f (a – x) dx  (B) 
ò
b
a
f (a + b – x) dx 
(C) 
ò
b
a
f (x – (a + b)) dx (D) 
ò
b
a
f ((a – x) + (b – x)) dx 
7. Let q be the angle between two unit vectors 
^
a and 
^
b such that sin q = 
5
3
. 
Then, 
^
a . 
^
b is equal to :  
(A) ± 
5
3
  (B) ± 
4
3
 
(C) ± 
5
4
  (D) ± 
3
4
 
Page 4


666  
65/1/1-11 Page 1 of 23 P.T.O.   
narjmWu àíZ-nÌ H$moS> >H$mo CÎma-nwpñVH$m Ho$ 
_wI-n¥ð >na Adí` {bIo§ & 
Candidates must write the Q.P. Code 
on the title page of the answer-book. 
 Series 
PQ1RS/1
  Set – 1  
  àíZ-nÌ H$moS>   
   
 
 
Q.P. Code  
AZwH«$_m§§H$
 
Roll No. 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 J{UV  
MATHEMATICS 
 
{ZYm©[aV g_` : 3 KÊQ>o   A{YH$V_ A§H$ : 80 
Time allowed : 3 hours  Maximum Marks : 80 
· H¥$n`m Om±M H$a b| {H$ Bg àíZ-nÌ _o§ _w{ÐV n¥ð> 23 h¢ & 
· H¥$n`m Om±M H$a b| {H$ Bg àíZ-nÌ _| >38 àíZ h¢ & 
· àíZ-nÌ _| Xm{hZo hmW H$s Amoa {XE JE àíZ-nÌ H$moS> H$mo narjmWu CÎma-nwpñVH$m Ho$  
_wI-n¥ð> na {bI| & 
· H¥$n`m àíZ H$m CÎma {bIZm ewê$ H$aZo go nhbo, CÎma-nwpñVH$m _| 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 23 printed pages. 
· Please check that this question paper contains 38 questions. 
· Q.P. Code 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 write down the serial number of the question in the answer-book 
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. 
65/1/1
 
* P Q 1 R S / 1 * 
666  
65/1/1-11 Page 3 of 23 P.T.O.   
General Instructions : 
Read the following instructions very carefully and strictly follow them : 
(i) This question paper contains 38 questions. All questions are compulsory.  
(ii) This question paper is divided into five Sections – A, B, C, D and E.  
(iii) In Section A, Questions no. 1 to 18 are multiple choice questions (MCQs) and 
questions number 19 and 20 are Assertion-Reason based questions of 1 mark 
each.  
(iv) In Section B, Questions no. 21 to 25 are very short answer (VSA) type 
questions, carrying 2 marks each.  
(v) In Section C, Questions no. 26 to 31 are short answer (SA) type questions, 
carrying 3 marks each. 
(vi) In Section D, Questions no. 32 to 35 are long answer (LA) type questions 
carrying 5 marks each.  
(vii) In Section E, Questions no. 36 to 38 are case study based questions carrying  
4 marks each.   
(viii) There is no overall choice. However, an internal choice has been provided in  
2 questions in Section B, 3 questions in Section C, 2 questions in Section D and 
2 questions in Section E.  
(ix) Use of calculators is not allowed.  
SECTION A 
This section comprises multiple choice questions (MCQs) of 1 mark each. 
  
1. A function f : R
+
 ® R (where R
+
 is the set of all non-negative real 
numbers) defined by f(x) = 4x + 3 is :  
(A) one-one but not onto  
(B) onto but not one-one 
(C) both one-one and onto  
(D) neither one-one nor onto 
2. If a matrix has 36 elements, the number of possible orders it can have,  
is : 
(A) 13  (B) 3 
(C) 5  (D) 9 
666  
65/1/1-11 Page 5 of 23 P.T.O.   
3. Which of the following statements is true for the function  
f(x) = 
2
x 3, x 0
1 , x 0
ì
ï + ¹
í
=
ï
î
 ? 
(A) f(x) is continuous and differentiable ? x Î R 
(B) f(x) is continuous ? x Î R 
(C) f(x) is continuous and differentiable ? x Î R – {0} 
(D) f(x) is discontinuous at infinitely many points  
4. Let f(x) be a continuous function on [a, b] and differentiable on (a, b). 
Then, this function f(x) is strictly increasing in (a, b) if  
(A) f ¢(x) < 0, ? x Î (a, b)  
(B) f ¢(x) > 0, ? x Î (a, b) 
(C) f ¢(x) = 0, ? x Î (a, b)  
(D) f (x) > 0, ? x Î (a, b)  
5. If 
ú
û
ù
ê
ë
é
+
xy 5
2 y x
 = 
ú
û
ù
ê
ë
é
8 5
2 6
, then the value of 
÷
÷
ø
ö
ç
ç
è
æ
+
y
24
x
24
 is : 
(A) 7  (B) 6 
(C) 8  (D) 18 
6. 
ò
b
a
f(x) dx is equal to : 
(A) 
ò
b
a
f (a – x) dx  (B) 
ò
b
a
f (a + b – x) dx 
(C) 
ò
b
a
f (x – (a + b)) dx (D) 
ò
b
a
f ((a – x) + (b – x)) dx 
7. Let q be the angle between two unit vectors 
^
a and 
^
b such that sin q = 
5
3
. 
Then, 
^
a . 
^
b is equal to :  
(A) ± 
5
3
  (B) ± 
4
3
 
(C) ± 
5
4
  (D) ± 
3
4
 
666  
65/1/1-11 Page 7 of 23 P.T.O.   
8. The integrating factor of the differential equation (1 – x
2
) 
dx
dy
 + xy = ax,  
– 1 < x < 1, is :  
(A) 
1 – x
1
2
 (B) 
1 – x
1
2
 
(C) 
2
x – 1
1
 (D) 
2
x – 1
1
 
9. If the direction cosines of a line are 3 k, 3 k, 3 k, then the value of k 
is : 
(A) ± 1  (B) ± 3 
(C) ± 3  (D) ± 
3
1
 
10. A linear programming problem deals with the optimization of a/an :  
(A) logarithmic function  (B) linear function  
(C) quadratic function  (D) exponential function  
11. If P(A|B) = P(A ¢|B), then which of the following statements is true ? 
(A) P(A) = P(A ¢) (B) P(A) = 2 P(B) 
(C) P(A 3 B) = 
2
1
 P(B) (D) P(A 3 B) = 2 P(B) 
12. 
2 2
x 1 x – 1
x x 1 x – x 1
+
+ + +
 is equal to : 
(A) 2x
3
  (B) 2 
(C) 0  (D) 2x
3
 – 2 
13. The derivative of sin (x
2
) w.r.t. x, at x = p is : 
(A) 1  (B) – 1 
(C) – 2 p (D) 2 p 
Page 5


666  
65/1/1-11 Page 1 of 23 P.T.O.   
narjmWu àíZ-nÌ H$moS> >H$mo CÎma-nwpñVH$m Ho$ 
_wI-n¥ð >na Adí` {bIo§ & 
Candidates must write the Q.P. Code 
on the title page of the answer-book. 
 Series 
PQ1RS/1
  Set – 1  
  àíZ-nÌ H$moS>   
   
 
 
Q.P. Code  
AZwH«$_m§§H$
 
Roll No. 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 J{UV  
MATHEMATICS 
 
{ZYm©[aV g_` : 3 KÊQ>o   A{YH$V_ A§H$ : 80 
Time allowed : 3 hours  Maximum Marks : 80 
· H¥$n`m Om±M H$a b| {H$ Bg àíZ-nÌ _o§ _w{ÐV n¥ð> 23 h¢ & 
· H¥$n`m Om±M H$a b| {H$ Bg àíZ-nÌ _| >38 àíZ h¢ & 
· àíZ-nÌ _| Xm{hZo hmW H$s Amoa {XE JE àíZ-nÌ H$moS> H$mo narjmWu CÎma-nwpñVH$m Ho$  
_wI-n¥ð> na {bI| & 
· H¥$n`m àíZ H$m CÎma {bIZm ewê$ H$aZo go nhbo, CÎma-nwpñVH$m _| 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 23 printed pages. 
· Please check that this question paper contains 38 questions. 
· Q.P. Code 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 write down the serial number of the question in the answer-book 
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. 
65/1/1
 
* P Q 1 R S / 1 * 
666  
65/1/1-11 Page 3 of 23 P.T.O.   
General Instructions : 
Read the following instructions very carefully and strictly follow them : 
(i) This question paper contains 38 questions. All questions are compulsory.  
(ii) This question paper is divided into five Sections – A, B, C, D and E.  
(iii) In Section A, Questions no. 1 to 18 are multiple choice questions (MCQs) and 
questions number 19 and 20 are Assertion-Reason based questions of 1 mark 
each.  
(iv) In Section B, Questions no. 21 to 25 are very short answer (VSA) type 
questions, carrying 2 marks each.  
(v) In Section C, Questions no. 26 to 31 are short answer (SA) type questions, 
carrying 3 marks each. 
(vi) In Section D, Questions no. 32 to 35 are long answer (LA) type questions 
carrying 5 marks each.  
(vii) In Section E, Questions no. 36 to 38 are case study based questions carrying  
4 marks each.   
(viii) There is no overall choice. However, an internal choice has been provided in  
2 questions in Section B, 3 questions in Section C, 2 questions in Section D and 
2 questions in Section E.  
(ix) Use of calculators is not allowed.  
SECTION A 
This section comprises multiple choice questions (MCQs) of 1 mark each. 
  
1. A function f : R
+
 ® R (where R
+
 is the set of all non-negative real 
numbers) defined by f(x) = 4x + 3 is :  
(A) one-one but not onto  
(B) onto but not one-one 
(C) both one-one and onto  
(D) neither one-one nor onto 
2. If a matrix has 36 elements, the number of possible orders it can have,  
is : 
(A) 13  (B) 3 
(C) 5  (D) 9 
666  
65/1/1-11 Page 5 of 23 P.T.O.   
3. Which of the following statements is true for the function  
f(x) = 
2
x 3, x 0
1 , x 0
ì
ï + ¹
í
=
ï
î
 ? 
(A) f(x) is continuous and differentiable ? x Î R 
(B) f(x) is continuous ? x Î R 
(C) f(x) is continuous and differentiable ? x Î R – {0} 
(D) f(x) is discontinuous at infinitely many points  
4. Let f(x) be a continuous function on [a, b] and differentiable on (a, b). 
Then, this function f(x) is strictly increasing in (a, b) if  
(A) f ¢(x) < 0, ? x Î (a, b)  
(B) f ¢(x) > 0, ? x Î (a, b) 
(C) f ¢(x) = 0, ? x Î (a, b)  
(D) f (x) > 0, ? x Î (a, b)  
5. If 
ú
û
ù
ê
ë
é
+
xy 5
2 y x
 = 
ú
û
ù
ê
ë
é
8 5
2 6
, then the value of 
÷
÷
ø
ö
ç
ç
è
æ
+
y
24
x
24
 is : 
(A) 7  (B) 6 
(C) 8  (D) 18 
6. 
ò
b
a
f(x) dx is equal to : 
(A) 
ò
b
a
f (a – x) dx  (B) 
ò
b
a
f (a + b – x) dx 
(C) 
ò
b
a
f (x – (a + b)) dx (D) 
ò
b
a
f ((a – x) + (b – x)) dx 
7. Let q be the angle between two unit vectors 
^
a and 
^
b such that sin q = 
5
3
. 
Then, 
^
a . 
^
b is equal to :  
(A) ± 
5
3
  (B) ± 
4
3
 
(C) ± 
5
4
  (D) ± 
3
4
 
666  
65/1/1-11 Page 7 of 23 P.T.O.   
8. The integrating factor of the differential equation (1 – x
2
) 
dx
dy
 + xy = ax,  
– 1 < x < 1, is :  
(A) 
1 – x
1
2
 (B) 
1 – x
1
2
 
(C) 
2
x – 1
1
 (D) 
2
x – 1
1
 
9. If the direction cosines of a line are 3 k, 3 k, 3 k, then the value of k 
is : 
(A) ± 1  (B) ± 3 
(C) ± 3  (D) ± 
3
1
 
10. A linear programming problem deals with the optimization of a/an :  
(A) logarithmic function  (B) linear function  
(C) quadratic function  (D) exponential function  
11. If P(A|B) = P(A ¢|B), then which of the following statements is true ? 
(A) P(A) = P(A ¢) (B) P(A) = 2 P(B) 
(C) P(A 3 B) = 
2
1
 P(B) (D) P(A 3 B) = 2 P(B) 
12. 
2 2
x 1 x – 1
x x 1 x – x 1
+
+ + +
 is equal to : 
(A) 2x
3
  (B) 2 
(C) 0  (D) 2x
3
 – 2 
13. The derivative of sin (x
2
) w.r.t. x, at x = p is : 
(A) 1  (B) – 1 
(C) – 2 p (D) 2 p 
666  
65/1/1-11 Page 9 of 23 P.T.O.   
14. The order and degree of the differential equation 
3
2
dx
dy
1
ú
ú
û
ù
ê
ê
ë
é
÷
ø
ö
ç
è
æ
+ = 
2
2
dx
y d
 
respectively are : 
(A) 1, 2  (B) 2, 3 
(C) 2, 1  (D) 2, 6 
15. The vector with terminal point A (2, – 3, 5) and initial point B (3, – 4, 7) 
is : 
(A) 
^
i – 
^
j + 2
^
k (B) 
^
i + 
^
j + 2
^
k 
(C) – 
^
i – 
^
j – 2
^
k (D) – 
^
i + 
^
j –  2
^
k 
16. The distance of point P(a, b, c) from y-axis is :  
(A) b  (B) b
2
 
(C) 
2 2
c a + (D) a
2
 + c
2
 
17. The number of corner points of the feasible region determined by 
constraints x ³ 0, y ³ 0, x + y ³ 4 is : 
(A) 0  (B) 1 
(C) 2  (D) 3 
18. If A and B are two non-zero square matrices of same order such that  
(A + B)
2
 = A
2
 + B
2
, then : 
(A) AB = O (B) AB = – BA 
(C) BA = O (D) AB = BA 
Questions number 19 and 20 are Assertion and Reason based questions. Two 
statements are given, one labelled Assertion (A) and the other labelled Reason 
(R). Select the correct answer from the codes (A), (B), (C) and (D) as given below.  
(A) Both Assertion (A) and Reason (R) are true and Reason (R) is the 
correct explanation of the Assertion (A). 
(B) Both Assertion (A) and Reason (R) are true, but Reason (R) is not 
the correct explanation of the Assertion (A). 
(C) Assertion (A) is true, but Reason (R) is false.  
(D) Assertion (A) is false, but Reason (R) is true.