IIT JAM Mathematics 2011 Past Year Paper | Past Year Papers of IIT JAM Mathematics PDF Download

 Page 1


 
A
 2011 MA
READ INSTRUCTIONS ON THE LEFT 
SIDE OF THIS PAGE CAREFULLY 
 
 REGISTRATION NUMBER 
        
 Name: 
 
 
 
 Test Centre: 
 
 
 
 
  
 
 
 
 Do not write your Registration Number 
or Name anywhere else in this 
question-cum-answer booklet. 
  
 
 
 
 I have read all the instructions and shall 
abide by them. 
 
 
 
 
 ……………………………………………... 
Signature of the Candidate 
 
  
        2011 MA 
Test Paper Code: MA 
 
        Time: 3 Hours             Maximum Marks: 300 
 
INSTRUCTIONS 
1. This question-cum-answer booklet has 40 pages 
and has 29 questions. Please ensure that the 
copy of the question-cum-answer booklet you 
have received contains all the questions.  
2. Write your Registration Number, Name and 
the name of the Test Centre in the appropriate 
space provided on the right side. 
3. Write the answers to the objective questions 
against each Question No. in the Answer Table 
for Objective Questions, provided on Page No. 
7. Do not write anything else on this page.  
4. Each objective question has 4 choices for its 
answer: (A), (B), (C) and (D). Only ONE of them 
is the correct answer. There will be negative 
marking for wrong answers to objective 
questions. The following marking scheme for 
objective questions shall be used: 
(a) For each correct answer, you will be 
awarded 6 (Six) marks.  
(b)  For each wrong answer, you will be 
awarded -2 (Negative two) marks. 
(c)  Multiple answers to a question will be 
treated as a wrong answer. 
(d)  For each un-attempted question, you will be 
awarded 0 (Zero) marks.  
(e) Negative marks for objective part will be 
carried over to total marks. 
5. Answer the subjective question only in the 
space provided after each question.  
6. Do not write more than one answer for the same 
question. In case you attempt a subjective 
question more than once, please cancel the 
answer(s) you consider wrong. Otherwise, the 
answer appearing last only will be evaluated. 
7. All answers must be written in blue/black/blue-
black ink only. Sketch pen, pencil or ink of any 
other colour should not be used. 
8. All rough work should be done in the space 
provided and scored out finally. 
9. No supplementary sheets will be provided to the 
candidates. 
10. Clip board, log tables, slide rule, calculator, 
cellular phone and electronic gadgets in any 
form are NOT allowed. 
11. The question-cum-answer booklet must be 
returned in its entirety to the Invigilator before 
leaving the examination hall. Do not remove any 
page from this booklet. 
12. Refer to special instructions/useful data on the 
reverse. 
 
 
 
 
 
I have verified the information filled by the 
Candidate above. 
 
 
 
 
 ……………………………………………... 
Signature of the Invigilator 
 
A 
MA-     / 40 i
Page 2


 
A
 2011 MA
READ INSTRUCTIONS ON THE LEFT 
SIDE OF THIS PAGE CAREFULLY 
 
 REGISTRATION NUMBER 
        
 Name: 
 
 
 
 Test Centre: 
 
 
 
 
  
 
 
 
 Do not write your Registration Number 
or Name anywhere else in this 
question-cum-answer booklet. 
  
 
 
 
 I have read all the instructions and shall 
abide by them. 
 
 
 
 
 ……………………………………………... 
Signature of the Candidate 
 
  
        2011 MA 
Test Paper Code: MA 
 
        Time: 3 Hours             Maximum Marks: 300 
 
INSTRUCTIONS 
1. This question-cum-answer booklet has 40 pages 
and has 29 questions. Please ensure that the 
copy of the question-cum-answer booklet you 
have received contains all the questions.  
2. Write your Registration Number, Name and 
the name of the Test Centre in the appropriate 
space provided on the right side. 
3. Write the answers to the objective questions 
against each Question No. in the Answer Table 
for Objective Questions, provided on Page No. 
7. Do not write anything else on this page.  
4. Each objective question has 4 choices for its 
answer: (A), (B), (C) and (D). Only ONE of them 
is the correct answer. There will be negative 
marking for wrong answers to objective 
questions. The following marking scheme for 
objective questions shall be used: 
(a) For each correct answer, you will be 
awarded 6 (Six) marks.  
(b)  For each wrong answer, you will be 
awarded -2 (Negative two) marks. 
(c)  Multiple answers to a question will be 
treated as a wrong answer. 
(d)  For each un-attempted question, you will be 
awarded 0 (Zero) marks.  
(e) Negative marks for objective part will be 
carried over to total marks. 
5. Answer the subjective question only in the 
space provided after each question.  
6. Do not write more than one answer for the same 
question. In case you attempt a subjective 
question more than once, please cancel the 
answer(s) you consider wrong. Otherwise, the 
answer appearing last only will be evaluated. 
7. All answers must be written in blue/black/blue-
black ink only. Sketch pen, pencil or ink of any 
other colour should not be used. 
8. All rough work should be done in the space 
provided and scored out finally. 
9. No supplementary sheets will be provided to the 
candidates. 
10. Clip board, log tables, slide rule, calculator, 
cellular phone and electronic gadgets in any 
form are NOT allowed. 
11. The question-cum-answer booklet must be 
returned in its entirety to the Invigilator before 
leaving the examination hall. Do not remove any 
page from this booklet. 
12. Refer to special instructions/useful data on the 
reverse. 
 
 
 
 
 
I have verified the information filled by the 
Candidate above. 
 
 
 
 
 ……………………………………………... 
Signature of the Invigilator 
 
A 
MA-     / 40 i
 
A
 
Special Instructions/ Useful Data 
 
? :    The set of  all real numbers  
 
? :    The set of  all natural numbers, that is, the set of all positive  
          integers 1, 2, …. . 
 
? :    The set of  all integers 
 
 
MA-     / 40 ii
Page 3


 
A
 2011 MA
READ INSTRUCTIONS ON THE LEFT 
SIDE OF THIS PAGE CAREFULLY 
 
 REGISTRATION NUMBER 
        
 Name: 
 
 
 
 Test Centre: 
 
 
 
 
  
 
 
 
 Do not write your Registration Number 
or Name anywhere else in this 
question-cum-answer booklet. 
  
 
 
 
 I have read all the instructions and shall 
abide by them. 
 
 
 
 
 ……………………………………………... 
Signature of the Candidate 
 
  
        2011 MA 
Test Paper Code: MA 
 
        Time: 3 Hours             Maximum Marks: 300 
 
INSTRUCTIONS 
1. This question-cum-answer booklet has 40 pages 
and has 29 questions. Please ensure that the 
copy of the question-cum-answer booklet you 
have received contains all the questions.  
2. Write your Registration Number, Name and 
the name of the Test Centre in the appropriate 
space provided on the right side. 
3. Write the answers to the objective questions 
against each Question No. in the Answer Table 
for Objective Questions, provided on Page No. 
7. Do not write anything else on this page.  
4. Each objective question has 4 choices for its 
answer: (A), (B), (C) and (D). Only ONE of them 
is the correct answer. There will be negative 
marking for wrong answers to objective 
questions. The following marking scheme for 
objective questions shall be used: 
(a) For each correct answer, you will be 
awarded 6 (Six) marks.  
(b)  For each wrong answer, you will be 
awarded -2 (Negative two) marks. 
(c)  Multiple answers to a question will be 
treated as a wrong answer. 
(d)  For each un-attempted question, you will be 
awarded 0 (Zero) marks.  
(e) Negative marks for objective part will be 
carried over to total marks. 
5. Answer the subjective question only in the 
space provided after each question.  
6. Do not write more than one answer for the same 
question. In case you attempt a subjective 
question more than once, please cancel the 
answer(s) you consider wrong. Otherwise, the 
answer appearing last only will be evaluated. 
7. All answers must be written in blue/black/blue-
black ink only. Sketch pen, pencil or ink of any 
other colour should not be used. 
8. All rough work should be done in the space 
provided and scored out finally. 
9. No supplementary sheets will be provided to the 
candidates. 
10. Clip board, log tables, slide rule, calculator, 
cellular phone and electronic gadgets in any 
form are NOT allowed. 
11. The question-cum-answer booklet must be 
returned in its entirety to the Invigilator before 
leaving the examination hall. Do not remove any 
page from this booklet. 
12. Refer to special instructions/useful data on the 
reverse. 
 
 
 
 
 
I have verified the information filled by the 
Candidate above. 
 
 
 
 
 ……………………………………………... 
Signature of the Invigilator 
 
A 
MA-     / 40 i
 
A
 
Special Instructions/ Useful Data 
 
? :    The set of  all real numbers  
 
? :    The set of  all natural numbers, that is, the set of all positive  
          integers 1, 2, …. . 
 
? :    The set of  all integers 
 
 
MA-     / 40 ii
 
A
 
IMPORTANT NOTE FOR CANDIDATES 
? Questions 1-15 (objective questions) carry six marks each and questions 16-29 (subjective 
questions) carry fifteen marks each. 
? Write the answers to the objective questions in the Answer Table for  Objective Questions 
provided on page 7 only. 
 
 
 
 Let   
2
1
,for .
n
n
k
n
a
nk
?
?
?
?
?n? Then the sequence ? ?
n
a is 
 
(A) Convergent (B) Bounded but not convergent 
Q.1  
(C) Diverges to  ? (D) Neither bounded nor diverges to ? 
 
 
The number of real roots of the equation 
3
10 xx ? ?? is 
 
Q.2  
(A)  0  (B)  1  (C)  2   (D)  3   
 
 
The value of  
2
1
1
lim
n
n
k nkn
??
? ?
?
  is  
 
Q.3  
(A)  2( 2 1) ? (B) 22 1 ? (C)  22 ? 
(D)  
1
(2 1)
2
? 
 
 
0, 2, 0, 0 xx y z
MA-     / 40 1
Let V be the  region bounded by the planes ? ?? ?and y 1 z ?? Q.4  . Then the 
value of the integral  is  
V
ydxdydz
???
 
(A)  
1
2
 (B)  
4
3
 
(C)  1  
(D)  
1
3
  
 
 
Q.5  
The solution    of the differential equation  () yx
2
2
44
dy dy
y
dx
dx
0 ? ??  satisfying the conditions 
(0) 4, y ?
dy
dx
 is  (0) 8 ?
 
(A) 
2
4
x
e (B) 
2
(16 4)
x
x e
?
? (C)   
2
41
x
ex
?
?6 (D)  
22
416
x x
ex
?
? e 
 
 
 
 
Page 4


 
A
 2011 MA
READ INSTRUCTIONS ON THE LEFT 
SIDE OF THIS PAGE CAREFULLY 
 
 REGISTRATION NUMBER 
        
 Name: 
 
 
 
 Test Centre: 
 
 
 
 
  
 
 
 
 Do not write your Registration Number 
or Name anywhere else in this 
question-cum-answer booklet. 
  
 
 
 
 I have read all the instructions and shall 
abide by them. 
 
 
 
 
 ……………………………………………... 
Signature of the Candidate 
 
  
        2011 MA 
Test Paper Code: MA 
 
        Time: 3 Hours             Maximum Marks: 300 
 
INSTRUCTIONS 
1. This question-cum-answer booklet has 40 pages 
and has 29 questions. Please ensure that the 
copy of the question-cum-answer booklet you 
have received contains all the questions.  
2. Write your Registration Number, Name and 
the name of the Test Centre in the appropriate 
space provided on the right side. 
3. Write the answers to the objective questions 
against each Question No. in the Answer Table 
for Objective Questions, provided on Page No. 
7. Do not write anything else on this page.  
4. Each objective question has 4 choices for its 
answer: (A), (B), (C) and (D). Only ONE of them 
is the correct answer. There will be negative 
marking for wrong answers to objective 
questions. The following marking scheme for 
objective questions shall be used: 
(a) For each correct answer, you will be 
awarded 6 (Six) marks.  
(b)  For each wrong answer, you will be 
awarded -2 (Negative two) marks. 
(c)  Multiple answers to a question will be 
treated as a wrong answer. 
(d)  For each un-attempted question, you will be 
awarded 0 (Zero) marks.  
(e) Negative marks for objective part will be 
carried over to total marks. 
5. Answer the subjective question only in the 
space provided after each question.  
6. Do not write more than one answer for the same 
question. In case you attempt a subjective 
question more than once, please cancel the 
answer(s) you consider wrong. Otherwise, the 
answer appearing last only will be evaluated. 
7. All answers must be written in blue/black/blue-
black ink only. Sketch pen, pencil or ink of any 
other colour should not be used. 
8. All rough work should be done in the space 
provided and scored out finally. 
9. No supplementary sheets will be provided to the 
candidates. 
10. Clip board, log tables, slide rule, calculator, 
cellular phone and electronic gadgets in any 
form are NOT allowed. 
11. The question-cum-answer booklet must be 
returned in its entirety to the Invigilator before 
leaving the examination hall. Do not remove any 
page from this booklet. 
12. Refer to special instructions/useful data on the 
reverse. 
 
 
 
 
 
I have verified the information filled by the 
Candidate above. 
 
 
 
 
 ……………………………………………... 
Signature of the Invigilator 
 
A 
MA-     / 40 i
 
A
 
Special Instructions/ Useful Data 
 
? :    The set of  all real numbers  
 
? :    The set of  all natural numbers, that is, the set of all positive  
          integers 1, 2, …. . 
 
? :    The set of  all integers 
 
 
MA-     / 40 ii
 
A
 
IMPORTANT NOTE FOR CANDIDATES 
? Questions 1-15 (objective questions) carry six marks each and questions 16-29 (subjective 
questions) carry fifteen marks each. 
? Write the answers to the objective questions in the Answer Table for  Objective Questions 
provided on page 7 only. 
 
 
 
 Let   
2
1
,for .
n
n
k
n
a
nk
?
?
?
?
?n? Then the sequence ? ?
n
a is 
 
(A) Convergent (B) Bounded but not convergent 
Q.1  
(C) Diverges to  ? (D) Neither bounded nor diverges to ? 
 
 
The number of real roots of the equation 
3
10 xx ? ?? is 
 
Q.2  
(A)  0  (B)  1  (C)  2   (D)  3   
 
 
The value of  
2
1
1
lim
n
n
k nkn
??
? ?
?
  is  
 
Q.3  
(A)  2( 2 1) ? (B) 22 1 ? (C)  22 ? 
(D)  
1
(2 1)
2
? 
 
 
0, 2, 0, 0 xx y z
MA-     / 40 1
Let V be the  region bounded by the planes ? ?? ?and y 1 z ?? Q.4  . Then the 
value of the integral  is  
V
ydxdydz
???
 
(A)  
1
2
 (B)  
4
3
 
(C)  1  
(D)  
1
3
  
 
 
Q.5  
The solution    of the differential equation  () yx
2
2
44
dy dy
y
dx
dx
0 ? ??  satisfying the conditions 
(0) 4, y ?
dy
dx
 is  (0) 8 ?
 
(A) 
2
4
x
e (B) 
2
(16 4)
x
x e
?
? (C)   
2
41
x
ex
?
?6 (D)  
22
416
x x
ex
?
? e 
 
 
 
 
 
A
 
Q.6  
MA-     / 40 2
If     is an integrating factor of the differential equation  
a
y
22
2(3 ) x y dx x y dy0 ? ?? ,   
then the value of  a  is 
 
(C) 1 ?  (D) 1   (A)  4 ? (B)  4
 
 
Let   and C be the positively oriented closed curve given by 
  If  
ˆ ˆˆ
Fayi zj xk ?? ?
?
22
1, 0. xy z ?? ?
C
Q.7  
Fd
?
r ? ??
?
? ?
,  then the value of a is  
 
(A)  1 ? (B) 0 
(C)  
1
2
 
(D)  1 
 
 
Consider the vector field where a is a constant. If 
  then the value of    is  
ˆ ˆˆ
() ( Fax y ai j x yk ??? ? ? ?
?
a
Q.8  
),
curl 0, FF ??
??
 
(C)  1 (A)  1 ? (B) 0 
(D)  
3
2
 
 
 
Let  denote the group of all  invertible matrices with entries from  Let   G 22 ? . ?
    
? ?
1
:det( ) 1 HAG A ?? ? and   ? ?
2
: is upper triangular HAGA ?? . 
 
Consider the following statements: 
 
Q.9  
      P :   
1
H is a normal subgroup of  G 
H is a normal subgroup of  .          Q :  G
2
 
Then 
 
(A) Both P and Q are true  (B) P is true and  is false Q
(C) P is false and  is true  Q (D)  Both P and Q are false  
 
 
For , let n ? ? ? ? : nnkk ?? ? .? ? Then the number of units of     and ,  
respectively,  are  
/11 ? /12 ??
 
Q.10  
(A) 11 , 12 (B) 10 , 11 (C)  10 , 4 (D)  10 , 8
 
 
 
 
Page 5


 
A
 2011 MA
READ INSTRUCTIONS ON THE LEFT 
SIDE OF THIS PAGE CAREFULLY 
 
 REGISTRATION NUMBER 
        
 Name: 
 
 
 
 Test Centre: 
 
 
 
 
  
 
 
 
 Do not write your Registration Number 
or Name anywhere else in this 
question-cum-answer booklet. 
  
 
 
 
 I have read all the instructions and shall 
abide by them. 
 
 
 
 
 ……………………………………………... 
Signature of the Candidate 
 
  
        2011 MA 
Test Paper Code: MA 
 
        Time: 3 Hours             Maximum Marks: 300 
 
INSTRUCTIONS 
1. This question-cum-answer booklet has 40 pages 
and has 29 questions. Please ensure that the 
copy of the question-cum-answer booklet you 
have received contains all the questions.  
2. Write your Registration Number, Name and 
the name of the Test Centre in the appropriate 
space provided on the right side. 
3. Write the answers to the objective questions 
against each Question No. in the Answer Table 
for Objective Questions, provided on Page No. 
7. Do not write anything else on this page.  
4. Each objective question has 4 choices for its 
answer: (A), (B), (C) and (D). Only ONE of them 
is the correct answer. There will be negative 
marking for wrong answers to objective 
questions. The following marking scheme for 
objective questions shall be used: 
(a) For each correct answer, you will be 
awarded 6 (Six) marks.  
(b)  For each wrong answer, you will be 
awarded -2 (Negative two) marks. 
(c)  Multiple answers to a question will be 
treated as a wrong answer. 
(d)  For each un-attempted question, you will be 
awarded 0 (Zero) marks.  
(e) Negative marks for objective part will be 
carried over to total marks. 
5. Answer the subjective question only in the 
space provided after each question.  
6. Do not write more than one answer for the same 
question. In case you attempt a subjective 
question more than once, please cancel the 
answer(s) you consider wrong. Otherwise, the 
answer appearing last only will be evaluated. 
7. All answers must be written in blue/black/blue-
black ink only. Sketch pen, pencil or ink of any 
other colour should not be used. 
8. All rough work should be done in the space 
provided and scored out finally. 
9. No supplementary sheets will be provided to the 
candidates. 
10. Clip board, log tables, slide rule, calculator, 
cellular phone and electronic gadgets in any 
form are NOT allowed. 
11. The question-cum-answer booklet must be 
returned in its entirety to the Invigilator before 
leaving the examination hall. Do not remove any 
page from this booklet. 
12. Refer to special instructions/useful data on the 
reverse. 
 
 
 
 
 
I have verified the information filled by the 
Candidate above. 
 
 
 
 
 ……………………………………………... 
Signature of the Invigilator 
 
A 
MA-     / 40 i
 
A
 
Special Instructions/ Useful Data 
 
? :    The set of  all real numbers  
 
? :    The set of  all natural numbers, that is, the set of all positive  
          integers 1, 2, …. . 
 
? :    The set of  all integers 
 
 
MA-     / 40 ii
 
A
 
IMPORTANT NOTE FOR CANDIDATES 
? Questions 1-15 (objective questions) carry six marks each and questions 16-29 (subjective 
questions) carry fifteen marks each. 
? Write the answers to the objective questions in the Answer Table for  Objective Questions 
provided on page 7 only. 
 
 
 
 Let   
2
1
,for .
n
n
k
n
a
nk
?
?
?
?
?n? Then the sequence ? ?
n
a is 
 
(A) Convergent (B) Bounded but not convergent 
Q.1  
(C) Diverges to  ? (D) Neither bounded nor diverges to ? 
 
 
The number of real roots of the equation 
3
10 xx ? ?? is 
 
Q.2  
(A)  0  (B)  1  (C)  2   (D)  3   
 
 
The value of  
2
1
1
lim
n
n
k nkn
??
? ?
?
  is  
 
Q.3  
(A)  2( 2 1) ? (B) 22 1 ? (C)  22 ? 
(D)  
1
(2 1)
2
? 
 
 
0, 2, 0, 0 xx y z
MA-     / 40 1
Let V be the  region bounded by the planes ? ?? ?and y 1 z ?? Q.4  . Then the 
value of the integral  is  
V
ydxdydz
???
 
(A)  
1
2
 (B)  
4
3
 
(C)  1  
(D)  
1
3
  
 
 
Q.5  
The solution    of the differential equation  () yx
2
2
44
dy dy
y
dx
dx
0 ? ??  satisfying the conditions 
(0) 4, y ?
dy
dx
 is  (0) 8 ?
 
(A) 
2
4
x
e (B) 
2
(16 4)
x
x e
?
? (C)   
2
41
x
ex
?
?6 (D)  
22
416
x x
ex
?
? e 
 
 
 
 
 
A
 
Q.6  
MA-     / 40 2
If     is an integrating factor of the differential equation  
a
y
22
2(3 ) x y dx x y dy0 ? ?? ,   
then the value of  a  is 
 
(C) 1 ?  (D) 1   (A)  4 ? (B)  4
 
 
Let   and C be the positively oriented closed curve given by 
  If  
ˆ ˆˆ
Fayi zj xk ?? ?
?
22
1, 0. xy z ?? ?
C
Q.7  
Fd
?
r ? ??
?
? ?
,  then the value of a is  
 
(A)  1 ? (B) 0 
(C)  
1
2
 
(D)  1 
 
 
Consider the vector field where a is a constant. If 
  then the value of    is  
ˆ ˆˆ
() ( Fax y ai j x yk ??? ? ? ?
?
a
Q.8  
),
curl 0, FF ??
??
 
(C)  1 (A)  1 ? (B) 0 
(D)  
3
2
 
 
 
Let  denote the group of all  invertible matrices with entries from  Let   G 22 ? . ?
    
? ?
1
:det( ) 1 HAG A ?? ? and   ? ?
2
: is upper triangular HAGA ?? . 
 
Consider the following statements: 
 
Q.9  
      P :   
1
H is a normal subgroup of  G 
H is a normal subgroup of  .          Q :  G
2
 
Then 
 
(A) Both P and Q are true  (B) P is true and  is false Q
(C) P is false and  is true  Q (D)  Both P and Q are false  
 
 
For , let n ? ? ? ? : nnkk ?? ? .? ? Then the number of units of     and ,  
respectively,  are  
/11 ? /12 ??
 
Q.10  
(A) 11 , 12 (B) 10 , 11 (C)  10 , 4 (D)  10 , 8
 
 
 
 
 
A
 Let   be a  3 matrix with   = 3 and det  =  2.  If   1   is an eigenvalue  of  , 
then the eigenvalues  of  the matrix  
A 3 ? trace ( ) A
2
A
( ) A A
2
Q.11  
I ?  are  
 
(A) 1,                                       (B)  2( 1 ), 2( 1 ) ii ?? ? 1, 2( 1 ), 2( 1 ) ii ? ?? 
(C)  1,                                       (D) 2( 1 ), 2( 1 ) ii ?? ? 1, 2( 1 ), 2( 1 ) ii ? ?? ? 
 
 
Q.12  
Let  be a linear transformation, where   For :
n
T ? ??
n
2. n ? , kn ?  let   
       
??
12
, ,...,
n
k
Evv v ?? ? and ? ?
12
, ,..., .
k
FTvTv Tv ? 
Then 
 
(A)  If is linearly independent, then  is linearly independent  E F
(B)  If  is linearly independent, then  is linearly independent F E
(C)  If   is linearly independent, then  is linearly dependent E F
(D)  If  is linearly independent, then  is linearly dependent F E
 
 
Q.13  
For   let   be linear transformations such that  is 
bijective. Then  
12
: and :
nm m
TT ? ?? ? ?
n
? , nm ?
12
TT
 
(A) rank ( ) =  and rank ( ) =  
1
T n
2
T m (B) rank ( ) =  and rank ( ) =  
1
T
2
T m n
(C) rank ( ) =  and rank ( ) = n 
1
T n
2
T (D) rank ( ) =  and rank ( ) =  
1
T m
2
T m
 
 
MA-     / 40 3
The set of all 
 
Q.14  
x at which the power series   
3
2
1
(2)
(2 1)
n
n
n
x
n
?
?
?
?
?
  converges is  
(A) ? ? 1, 1 ? (B) ? ? 1,1 ? (C)  ? ? 1, 3 (D)  ? ? 1, 3 
 
 
Consider the following subsets of ? : Q.15  
1
:, :0
11
n
En F x
nx
?? ?
?? ? ?
?? ?
??
?? ?
? 1.
?
?
?
?
 
Then  
 
(A)  Both  and  are closed  E F (B)  is closed and E F is NOT closed 
(C)  is NOT closed and  is closed E F (D)  Neither  nor E F is  closed