IIT JAM Mathematics 2015 Past Year Paper | Past Year Papers of IIT JAM Mathematics PDF Download

 Page 1


IIT JAM MATHEMATICS 2015  
          
 
 
SECTION – A 
 
MULTIPLE CHOICE QUESTIONS (MCQ) 
 Q.1 – Q.10 carry one mark each. 
1. Suppose N is a normal subgroup of a group G. Which one of the following is true? 
(A) If G is an infinite group then G/N is an infinite group 
(B) If G is a nonabelian group then G/N is a nonabelian group 
(C) If G is a cyclic group then G/N is an abelian group 
(D) If G is an abelian group then G/N is a cyclic group 
 
2. Let ?? (?? ) = ?? (?? ) ?????? ?? + ?? (?? ) ?????? ?? be a solution of the differential equation ?? " + ?? = ?????? ?? . Then 
?? (?? ) is 
(A) In ?????? ?? | + ??  
(B) -?? + ??   
(C) ?? + ??   
(D) In |?????? ?? | + ?? 
 
3. Let a,b,c,d be distinct non-zero real numbers with ?? + ?? = ?? + ?? . Then an eigenvalue of the matrix 
[
?? ?? ?? ?? ?? ?? ?? -?? ?? ] is  
(A) ?? + ??   
(B) ?? + ??   
(C) ?? - ??    
(D) ?? - ?? 
 
4. Let S be a nonempty subset of R. If S is a finite union of disjoint bounded intervals, thenwhich oneof the 
following is true? 
(A) If S is not compact, then sup ?? ? ?? and inf?? ? ?? 
(B) Even if sup ?? ? ?? and inf?? ? ?? , S need not be compact 
(C) If sup ?? ? ?? and inf?? ? ?? , then S is compact 
(D) Even if S is compact, it is not necessary that sup ?? ? ?? and inf?? ? ?? 
 
5.  Let {?? ?? } be a convergent sequence of real numbers. If ?? ?? > ?? + v?? and?? ?? +?? = ?? + v?? ?? - ?? for ?? = ?? , 
then which one of the following is the limit of this sequence? 
(A) ?? + 1   
(B) ?? + v2   
(C) ??    
(D) ?? + v?? 
 
6. The volume of the portion of the solid cylinder ?? ?? + ?? ?? = ?? bounded above by the surface?? = ?? ?? + ?? ?? 
and bounded below by the ???? -plane is 
(A) ??    
(B) 2??    
(C) ???    
(D) 4?? 
  
Page 2


IIT JAM MATHEMATICS 2015  
          
 
 
SECTION – A 
 
MULTIPLE CHOICE QUESTIONS (MCQ) 
 Q.1 – Q.10 carry one mark each. 
1. Suppose N is a normal subgroup of a group G. Which one of the following is true? 
(A) If G is an infinite group then G/N is an infinite group 
(B) If G is a nonabelian group then G/N is a nonabelian group 
(C) If G is a cyclic group then G/N is an abelian group 
(D) If G is an abelian group then G/N is a cyclic group 
 
2. Let ?? (?? ) = ?? (?? ) ?????? ?? + ?? (?? ) ?????? ?? be a solution of the differential equation ?? " + ?? = ?????? ?? . Then 
?? (?? ) is 
(A) In ?????? ?? | + ??  
(B) -?? + ??   
(C) ?? + ??   
(D) In |?????? ?? | + ?? 
 
3. Let a,b,c,d be distinct non-zero real numbers with ?? + ?? = ?? + ?? . Then an eigenvalue of the matrix 
[
?? ?? ?? ?? ?? ?? ?? -?? ?? ] is  
(A) ?? + ??   
(B) ?? + ??   
(C) ?? - ??    
(D) ?? - ?? 
 
4. Let S be a nonempty subset of R. If S is a finite union of disjoint bounded intervals, thenwhich oneof the 
following is true? 
(A) If S is not compact, then sup ?? ? ?? and inf?? ? ?? 
(B) Even if sup ?? ? ?? and inf?? ? ?? , S need not be compact 
(C) If sup ?? ? ?? and inf?? ? ?? , then S is compact 
(D) Even if S is compact, it is not necessary that sup ?? ? ?? and inf?? ? ?? 
 
5.  Let {?? ?? } be a convergent sequence of real numbers. If ?? ?? > ?? + v?? and?? ?? +?? = ?? + v?? ?? - ?? for ?? = ?? , 
then which one of the following is the limit of this sequence? 
(A) ?? + 1   
(B) ?? + v2   
(C) ??    
(D) ?? + v?? 
 
6. The volume of the portion of the solid cylinder ?? ?? + ?? ?? = ?? bounded above by the surface?? = ?? ?? + ?? ?? 
and bounded below by the ???? -plane is 
(A) ??    
(B) 2??    
(C) ???    
(D) 4?? 
  
 
          
 
 
7. Let ?? :R ? R be differentiable function with ?? (?? ) = ?? . If for all ?? ?? R,?? < ?? '
(?? ) < 2, then which one of the 
following statements is true on (?? , 8)? 
 (A) ?? is unbounded      
(B) ?? is increasing and bounded  
 (C) ?? has at least one zero     
(D) ?? is periodic  
 
8. If integral curve of the differential (?? - ?? )
????
????
= ?? passes through (0,0) and (?? ,?? ), then ?? is equal to  
 (A) 2- ?? -1
   
(B) 1 - ?? -1
   
(C) ?? -1
    
(D) 1 + e 
 
9. An integrating factor of the differential equation 
????
????
=
?? ????
?? +?? ?? -?? ?? ?? is 
 (A) 
1
??    
(B) 
1
?? 2
    
(C) y    
(D) ?? 2
 
 
10. Let A be a nonempty subset of R. Let ?? (?? ) denote the set of interior points of A. Then ?? (?? ) can be  
 (A) empy 
 (B) singleton 
 (C) a finite set containing more than one element 
 (D) countable but not finite  
 
Q.11-Q.30 carry two marks each.  
 
11. Let ?? ?? be the group f permutations of three distinct symbols. The direct sum ?? ?? ? ?? ?? has an element of 
order  
 (A) 4    
(B) 6    
(C) 9    
(D) 18 
 
12. The orthogonal trajectories of the family of curves ?? = ?? ?? ?? ?? are  
 (A) 2?? 2
+ 3?? 2
= ?? 2
     
(B) 3?? 2
+ ?? 2
= ?? 2
 
 (C) 3?? 2
+ 2?? 2
= ?? 2
     
(D) ?? 2
+ 3?? 2
= ?? 2
 
 
13. Let G be a nonabelian group. Let ?? ?? ?? have order 4 and let ?? ?? ??  have order 3. Then the order of the 
element ???? in G  
 (A) is 6       
(B) is 12 
 (C) is of the form 12k for ?? = 2   
(D) need not be finite  
 
  
Page 3


IIT JAM MATHEMATICS 2015  
          
 
 
SECTION – A 
 
MULTIPLE CHOICE QUESTIONS (MCQ) 
 Q.1 – Q.10 carry one mark each. 
1. Suppose N is a normal subgroup of a group G. Which one of the following is true? 
(A) If G is an infinite group then G/N is an infinite group 
(B) If G is a nonabelian group then G/N is a nonabelian group 
(C) If G is a cyclic group then G/N is an abelian group 
(D) If G is an abelian group then G/N is a cyclic group 
 
2. Let ?? (?? ) = ?? (?? ) ?????? ?? + ?? (?? ) ?????? ?? be a solution of the differential equation ?? " + ?? = ?????? ?? . Then 
?? (?? ) is 
(A) In ?????? ?? | + ??  
(B) -?? + ??   
(C) ?? + ??   
(D) In |?????? ?? | + ?? 
 
3. Let a,b,c,d be distinct non-zero real numbers with ?? + ?? = ?? + ?? . Then an eigenvalue of the matrix 
[
?? ?? ?? ?? ?? ?? ?? -?? ?? ] is  
(A) ?? + ??   
(B) ?? + ??   
(C) ?? - ??    
(D) ?? - ?? 
 
4. Let S be a nonempty subset of R. If S is a finite union of disjoint bounded intervals, thenwhich oneof the 
following is true? 
(A) If S is not compact, then sup ?? ? ?? and inf?? ? ?? 
(B) Even if sup ?? ? ?? and inf?? ? ?? , S need not be compact 
(C) If sup ?? ? ?? and inf?? ? ?? , then S is compact 
(D) Even if S is compact, it is not necessary that sup ?? ? ?? and inf?? ? ?? 
 
5.  Let {?? ?? } be a convergent sequence of real numbers. If ?? ?? > ?? + v?? and?? ?? +?? = ?? + v?? ?? - ?? for ?? = ?? , 
then which one of the following is the limit of this sequence? 
(A) ?? + 1   
(B) ?? + v2   
(C) ??    
(D) ?? + v?? 
 
6. The volume of the portion of the solid cylinder ?? ?? + ?? ?? = ?? bounded above by the surface?? = ?? ?? + ?? ?? 
and bounded below by the ???? -plane is 
(A) ??    
(B) 2??    
(C) ???    
(D) 4?? 
  
 
          
 
 
7. Let ?? :R ? R be differentiable function with ?? (?? ) = ?? . If for all ?? ?? R,?? < ?? '
(?? ) < 2, then which one of the 
following statements is true on (?? , 8)? 
 (A) ?? is unbounded      
(B) ?? is increasing and bounded  
 (C) ?? has at least one zero     
(D) ?? is periodic  
 
8. If integral curve of the differential (?? - ?? )
????
????
= ?? passes through (0,0) and (?? ,?? ), then ?? is equal to  
 (A) 2- ?? -1
   
(B) 1 - ?? -1
   
(C) ?? -1
    
(D) 1 + e 
 
9. An integrating factor of the differential equation 
????
????
=
?? ????
?? +?? ?? -?? ?? ?? is 
 (A) 
1
??    
(B) 
1
?? 2
    
(C) y    
(D) ?? 2
 
 
10. Let A be a nonempty subset of R. Let ?? (?? ) denote the set of interior points of A. Then ?? (?? ) can be  
 (A) empy 
 (B) singleton 
 (C) a finite set containing more than one element 
 (D) countable but not finite  
 
Q.11-Q.30 carry two marks each.  
 
11. Let ?? ?? be the group f permutations of three distinct symbols. The direct sum ?? ?? ? ?? ?? has an element of 
order  
 (A) 4    
(B) 6    
(C) 9    
(D) 18 
 
12. The orthogonal trajectories of the family of curves ?? = ?? ?? ?? ?? are  
 (A) 2?? 2
+ 3?? 2
= ?? 2
     
(B) 3?? 2
+ ?? 2
= ?? 2
 
 (C) 3?? 2
+ 2?? 2
= ?? 2
     
(D) ?? 2
+ 3?? 2
= ?? 2
 
 
13. Let G be a nonabelian group. Let ?? ?? ?? have order 4 and let ?? ?? ??  have order 3. Then the order of the 
element ???? in G  
 (A) is 6       
(B) is 12 
 (C) is of the form 12k for ?? = 2   
(D) need not be finite  
 
  
 
          
 
 
14. Let S be the bounded surface of the cylinder ?? ?? + ?? ?? = ?? cut by the planes ?? = ?? and ?? = ?? + ?? . Then 
the value of the surface integral ?
?? ?? ?? ?? ???? is equal to 
 (A) ?
0
2?? (1 + cos?? )
3
????     
(B) ?
0
2?? sin?? cos?? (1 + cos?? )
2
???? 
(C) ?
0
2?? (1 + 2cos?? )
3
????    
(D) ?
0
2?? sin?? cos?? (1 + 2cos?? )
2
???? 
 
15. Suppose that the dependent variables ?? and ?? are functions of the independent variables?? and?? , defined 
by the equations ?? (?? ,?? ,?? ,?? ) = ?? and ?? (?? ,?? ,?? ,?? ) = ?? , where?? ?? ?? ?? - ?? ?? ?? ?? = ?? .Which one of the 
following is correct? 
(A) ?? ?? = ?? ?? ?? ?? - ?? ?? ?? ??     
(B) ?? ?? = ?? ?? ?? ?? - ?? ?? ?? ?? 
(C) ?? ?? = ?? ?? ?? ?? - ?? ?? ?? ??     
(D) ?? ?? = ?? ?? ?? ?? - ?? ?? ?? ?? 
 
16. Let ?? = [
?? ?? - ?? -?? - ?? ?? ]and ?? = ?? ?? ?? ¯
. Then  
(A) an eigenvalue of B is purely imaginary 
(B) an eigenvalue of A is zero 
(C) all eigenvalues of B are real 
(D) A has a non-zero real eigenvalue 
 
17. The limit ?????? ?? ??? +
?? ?????? ?? ?? ??? ?? ?? ??????
-?? ?? ???? is equal to  
 (A) 0     
(B) 
1
8
    
(C) 
1
4
    
(D) 
3
8
 
 
18. Let ?? ?? (R) be the vector space of polynomials in x of degree at most 2 with real coefficients. Let ?? ?? (R) be 
the vector space of 2 x 2 real matrices. If a linear transformation ?? : ?? ?? (R) ?? ?? (R)is defined as ?? (?? ) =
[
?? (?? )- ?? (?? ) ?? ?? ?? (?? )
]. Then 
(A) T is one-one but not onto 
(B) T is onto but not one-one 
(C) Range (T) = span {[
0 0
0 1
] [
-2 0
0 1
]} 
(D) Null (T) = span {?? 2
- 2?? ,1 - ?? } 
 
19. Let ?? ?? = {(?? ,?? ),(?? ,-?? )} and ?? ?? = {(?? ,?? ),(?? ,?? )} be ordered bases of R
?? . If ?? : R
?? ? R
?? isa linear 
transformation such that [?? ]
?? ?? ,?? ?? , the matrix of T with respect to ?? ?? and ?? ?? , is[
?? ?? ?? -?? ]then T(5,5) is 
equal to 
(A) (-9,8)   
(B) (9,8)   
(C) (-15, -2)    
(D) (15, 2) 
 
  
Page 4


IIT JAM MATHEMATICS 2015  
          
 
 
SECTION – A 
 
MULTIPLE CHOICE QUESTIONS (MCQ) 
 Q.1 – Q.10 carry one mark each. 
1. Suppose N is a normal subgroup of a group G. Which one of the following is true? 
(A) If G is an infinite group then G/N is an infinite group 
(B) If G is a nonabelian group then G/N is a nonabelian group 
(C) If G is a cyclic group then G/N is an abelian group 
(D) If G is an abelian group then G/N is a cyclic group 
 
2. Let ?? (?? ) = ?? (?? ) ?????? ?? + ?? (?? ) ?????? ?? be a solution of the differential equation ?? " + ?? = ?????? ?? . Then 
?? (?? ) is 
(A) In ?????? ?? | + ??  
(B) -?? + ??   
(C) ?? + ??   
(D) In |?????? ?? | + ?? 
 
3. Let a,b,c,d be distinct non-zero real numbers with ?? + ?? = ?? + ?? . Then an eigenvalue of the matrix 
[
?? ?? ?? ?? ?? ?? ?? -?? ?? ] is  
(A) ?? + ??   
(B) ?? + ??   
(C) ?? - ??    
(D) ?? - ?? 
 
4. Let S be a nonempty subset of R. If S is a finite union of disjoint bounded intervals, thenwhich oneof the 
following is true? 
(A) If S is not compact, then sup ?? ? ?? and inf?? ? ?? 
(B) Even if sup ?? ? ?? and inf?? ? ?? , S need not be compact 
(C) If sup ?? ? ?? and inf?? ? ?? , then S is compact 
(D) Even if S is compact, it is not necessary that sup ?? ? ?? and inf?? ? ?? 
 
5.  Let {?? ?? } be a convergent sequence of real numbers. If ?? ?? > ?? + v?? and?? ?? +?? = ?? + v?? ?? - ?? for ?? = ?? , 
then which one of the following is the limit of this sequence? 
(A) ?? + 1   
(B) ?? + v2   
(C) ??    
(D) ?? + v?? 
 
6. The volume of the portion of the solid cylinder ?? ?? + ?? ?? = ?? bounded above by the surface?? = ?? ?? + ?? ?? 
and bounded below by the ???? -plane is 
(A) ??    
(B) 2??    
(C) ???    
(D) 4?? 
  
 
          
 
 
7. Let ?? :R ? R be differentiable function with ?? (?? ) = ?? . If for all ?? ?? R,?? < ?? '
(?? ) < 2, then which one of the 
following statements is true on (?? , 8)? 
 (A) ?? is unbounded      
(B) ?? is increasing and bounded  
 (C) ?? has at least one zero     
(D) ?? is periodic  
 
8. If integral curve of the differential (?? - ?? )
????
????
= ?? passes through (0,0) and (?? ,?? ), then ?? is equal to  
 (A) 2- ?? -1
   
(B) 1 - ?? -1
   
(C) ?? -1
    
(D) 1 + e 
 
9. An integrating factor of the differential equation 
????
????
=
?? ????
?? +?? ?? -?? ?? ?? is 
 (A) 
1
??    
(B) 
1
?? 2
    
(C) y    
(D) ?? 2
 
 
10. Let A be a nonempty subset of R. Let ?? (?? ) denote the set of interior points of A. Then ?? (?? ) can be  
 (A) empy 
 (B) singleton 
 (C) a finite set containing more than one element 
 (D) countable but not finite  
 
Q.11-Q.30 carry two marks each.  
 
11. Let ?? ?? be the group f permutations of three distinct symbols. The direct sum ?? ?? ? ?? ?? has an element of 
order  
 (A) 4    
(B) 6    
(C) 9    
(D) 18 
 
12. The orthogonal trajectories of the family of curves ?? = ?? ?? ?? ?? are  
 (A) 2?? 2
+ 3?? 2
= ?? 2
     
(B) 3?? 2
+ ?? 2
= ?? 2
 
 (C) 3?? 2
+ 2?? 2
= ?? 2
     
(D) ?? 2
+ 3?? 2
= ?? 2
 
 
13. Let G be a nonabelian group. Let ?? ?? ?? have order 4 and let ?? ?? ??  have order 3. Then the order of the 
element ???? in G  
 (A) is 6       
(B) is 12 
 (C) is of the form 12k for ?? = 2   
(D) need not be finite  
 
  
 
          
 
 
14. Let S be the bounded surface of the cylinder ?? ?? + ?? ?? = ?? cut by the planes ?? = ?? and ?? = ?? + ?? . Then 
the value of the surface integral ?
?? ?? ?? ?? ???? is equal to 
 (A) ?
0
2?? (1 + cos?? )
3
????     
(B) ?
0
2?? sin?? cos?? (1 + cos?? )
2
???? 
(C) ?
0
2?? (1 + 2cos?? )
3
????    
(D) ?
0
2?? sin?? cos?? (1 + 2cos?? )
2
???? 
 
15. Suppose that the dependent variables ?? and ?? are functions of the independent variables?? and?? , defined 
by the equations ?? (?? ,?? ,?? ,?? ) = ?? and ?? (?? ,?? ,?? ,?? ) = ?? , where?? ?? ?? ?? - ?? ?? ?? ?? = ?? .Which one of the 
following is correct? 
(A) ?? ?? = ?? ?? ?? ?? - ?? ?? ?? ??     
(B) ?? ?? = ?? ?? ?? ?? - ?? ?? ?? ?? 
(C) ?? ?? = ?? ?? ?? ?? - ?? ?? ?? ??     
(D) ?? ?? = ?? ?? ?? ?? - ?? ?? ?? ?? 
 
16. Let ?? = [
?? ?? - ?? -?? - ?? ?? ]and ?? = ?? ?? ?? ¯
. Then  
(A) an eigenvalue of B is purely imaginary 
(B) an eigenvalue of A is zero 
(C) all eigenvalues of B are real 
(D) A has a non-zero real eigenvalue 
 
17. The limit ?????? ?? ??? +
?? ?????? ?? ?? ??? ?? ?? ??????
-?? ?? ???? is equal to  
 (A) 0     
(B) 
1
8
    
(C) 
1
4
    
(D) 
3
8
 
 
18. Let ?? ?? (R) be the vector space of polynomials in x of degree at most 2 with real coefficients. Let ?? ?? (R) be 
the vector space of 2 x 2 real matrices. If a linear transformation ?? : ?? ?? (R) ?? ?? (R)is defined as ?? (?? ) =
[
?? (?? )- ?? (?? ) ?? ?? ?? (?? )
]. Then 
(A) T is one-one but not onto 
(B) T is onto but not one-one 
(C) Range (T) = span {[
0 0
0 1
] [
-2 0
0 1
]} 
(D) Null (T) = span {?? 2
- 2?? ,1 - ?? } 
 
19. Let ?? ?? = {(?? ,?? ),(?? ,-?? )} and ?? ?? = {(?? ,?? ),(?? ,?? )} be ordered bases of R
?? . If ?? : R
?? ? R
?? isa linear 
transformation such that [?? ]
?? ?? ,?? ?? , the matrix of T with respect to ?? ?? and ?? ?? , is[
?? ?? ?? -?? ]then T(5,5) is 
equal to 
(A) (-9,8)   
(B) (9,8)   
(C) (-15, -2)    
(D) (15, 2) 
 
  
 
          
 
 
20. Let?? =n
?? =?? 8
([?? ,
?? ?? ?? +?? ] ? [
?? ?? ?? ,?? ]). Which one of the following statements is FALSE? 
(A) There exist sequences {?? ?? } and {?? ?? } in [0,1] such that ?? = [0,1]\?
?? =1
8
(?? ?? ,?? ?? ) 
(B) [0, 1]\S is an open set 
(C) If A is an infinite subset of S, then A has a limit point 
(D) There exists an infinite subset of Shaving no limit points 
 
21. Let?? :R - R be a strictly increasing continuous function. If {?? ?? } is a sequence in [0,1],then the sequence 
{?? ({?? ?? })} is 
(A) increasing 
(B) bounded 
(C) convergent 
(D) not necessarily bounded 
 
22. Which one of the following statements is true for the series? (-?? )
?? (?? ?? )!
?? ?? ?? 8
?? =?? ? 
(A) The series converges conditionally but not absolutely 
(B) The series converges absolutely 
(C) The sequence of partial sums of the series is bounded but not convergent 
(D) The sequence of partial sums of the series is unbounded 
 
23. The sequence {?????? (
?? ?? ?????? -?? (-
?? ?? )
?? )} is  
(A) monotone and convergent 
(B) monotone but not convergent 
(C) convergent but not monotone 
(D) neither monotone nor convergent 
 
24. If y(t) is a solution of the differential equation ?? " + ?? ?? = ?? ?? ?? , then ?????? ?? ? 8
?? -?? ?? (?? ) is equal to  
 (A) 
2
3
         
(B) 
2
5
     
(C) 
2
7
         
(D) 
2
9
 
 
25. For what real values of x and y, does the integral ?
?? ?? (?? - ?? - ?? ?? )???? attain its maximum? 
(A) x = -3, y = 2         
(B) x = 2,  y = 3 
(C) x = -2, y = 2        
(D) x = -3, y = 4 
 
26. The area of the planar region bounded by the curves ?? = ?? ?? ?? – ?? and ?? = ?? ?? ?? is 
(A) 
v2
3
         
(B) 
2v2
3
     
(C) 
4v2
3
         
(D) v2 
 
  
Page 5


IIT JAM MATHEMATICS 2015  
          
 
 
SECTION – A 
 
MULTIPLE CHOICE QUESTIONS (MCQ) 
 Q.1 – Q.10 carry one mark each. 
1. Suppose N is a normal subgroup of a group G. Which one of the following is true? 
(A) If G is an infinite group then G/N is an infinite group 
(B) If G is a nonabelian group then G/N is a nonabelian group 
(C) If G is a cyclic group then G/N is an abelian group 
(D) If G is an abelian group then G/N is a cyclic group 
 
2. Let ?? (?? ) = ?? (?? ) ?????? ?? + ?? (?? ) ?????? ?? be a solution of the differential equation ?? " + ?? = ?????? ?? . Then 
?? (?? ) is 
(A) In ?????? ?? | + ??  
(B) -?? + ??   
(C) ?? + ??   
(D) In |?????? ?? | + ?? 
 
3. Let a,b,c,d be distinct non-zero real numbers with ?? + ?? = ?? + ?? . Then an eigenvalue of the matrix 
[
?? ?? ?? ?? ?? ?? ?? -?? ?? ] is  
(A) ?? + ??   
(B) ?? + ??   
(C) ?? - ??    
(D) ?? - ?? 
 
4. Let S be a nonempty subset of R. If S is a finite union of disjoint bounded intervals, thenwhich oneof the 
following is true? 
(A) If S is not compact, then sup ?? ? ?? and inf?? ? ?? 
(B) Even if sup ?? ? ?? and inf?? ? ?? , S need not be compact 
(C) If sup ?? ? ?? and inf?? ? ?? , then S is compact 
(D) Even if S is compact, it is not necessary that sup ?? ? ?? and inf?? ? ?? 
 
5.  Let {?? ?? } be a convergent sequence of real numbers. If ?? ?? > ?? + v?? and?? ?? +?? = ?? + v?? ?? - ?? for ?? = ?? , 
then which one of the following is the limit of this sequence? 
(A) ?? + 1   
(B) ?? + v2   
(C) ??    
(D) ?? + v?? 
 
6. The volume of the portion of the solid cylinder ?? ?? + ?? ?? = ?? bounded above by the surface?? = ?? ?? + ?? ?? 
and bounded below by the ???? -plane is 
(A) ??    
(B) 2??    
(C) ???    
(D) 4?? 
  
 
          
 
 
7. Let ?? :R ? R be differentiable function with ?? (?? ) = ?? . If for all ?? ?? R,?? < ?? '
(?? ) < 2, then which one of the 
following statements is true on (?? , 8)? 
 (A) ?? is unbounded      
(B) ?? is increasing and bounded  
 (C) ?? has at least one zero     
(D) ?? is periodic  
 
8. If integral curve of the differential (?? - ?? )
????
????
= ?? passes through (0,0) and (?? ,?? ), then ?? is equal to  
 (A) 2- ?? -1
   
(B) 1 - ?? -1
   
(C) ?? -1
    
(D) 1 + e 
 
9. An integrating factor of the differential equation 
????
????
=
?? ????
?? +?? ?? -?? ?? ?? is 
 (A) 
1
??    
(B) 
1
?? 2
    
(C) y    
(D) ?? 2
 
 
10. Let A be a nonempty subset of R. Let ?? (?? ) denote the set of interior points of A. Then ?? (?? ) can be  
 (A) empy 
 (B) singleton 
 (C) a finite set containing more than one element 
 (D) countable but not finite  
 
Q.11-Q.30 carry two marks each.  
 
11. Let ?? ?? be the group f permutations of three distinct symbols. The direct sum ?? ?? ? ?? ?? has an element of 
order  
 (A) 4    
(B) 6    
(C) 9    
(D) 18 
 
12. The orthogonal trajectories of the family of curves ?? = ?? ?? ?? ?? are  
 (A) 2?? 2
+ 3?? 2
= ?? 2
     
(B) 3?? 2
+ ?? 2
= ?? 2
 
 (C) 3?? 2
+ 2?? 2
= ?? 2
     
(D) ?? 2
+ 3?? 2
= ?? 2
 
 
13. Let G be a nonabelian group. Let ?? ?? ?? have order 4 and let ?? ?? ??  have order 3. Then the order of the 
element ???? in G  
 (A) is 6       
(B) is 12 
 (C) is of the form 12k for ?? = 2   
(D) need not be finite  
 
  
 
          
 
 
14. Let S be the bounded surface of the cylinder ?? ?? + ?? ?? = ?? cut by the planes ?? = ?? and ?? = ?? + ?? . Then 
the value of the surface integral ?
?? ?? ?? ?? ???? is equal to 
 (A) ?
0
2?? (1 + cos?? )
3
????     
(B) ?
0
2?? sin?? cos?? (1 + cos?? )
2
???? 
(C) ?
0
2?? (1 + 2cos?? )
3
????    
(D) ?
0
2?? sin?? cos?? (1 + 2cos?? )
2
???? 
 
15. Suppose that the dependent variables ?? and ?? are functions of the independent variables?? and?? , defined 
by the equations ?? (?? ,?? ,?? ,?? ) = ?? and ?? (?? ,?? ,?? ,?? ) = ?? , where?? ?? ?? ?? - ?? ?? ?? ?? = ?? .Which one of the 
following is correct? 
(A) ?? ?? = ?? ?? ?? ?? - ?? ?? ?? ??     
(B) ?? ?? = ?? ?? ?? ?? - ?? ?? ?? ?? 
(C) ?? ?? = ?? ?? ?? ?? - ?? ?? ?? ??     
(D) ?? ?? = ?? ?? ?? ?? - ?? ?? ?? ?? 
 
16. Let ?? = [
?? ?? - ?? -?? - ?? ?? ]and ?? = ?? ?? ?? ¯
. Then  
(A) an eigenvalue of B is purely imaginary 
(B) an eigenvalue of A is zero 
(C) all eigenvalues of B are real 
(D) A has a non-zero real eigenvalue 
 
17. The limit ?????? ?? ??? +
?? ?????? ?? ?? ??? ?? ?? ??????
-?? ?? ???? is equal to  
 (A) 0     
(B) 
1
8
    
(C) 
1
4
    
(D) 
3
8
 
 
18. Let ?? ?? (R) be the vector space of polynomials in x of degree at most 2 with real coefficients. Let ?? ?? (R) be 
the vector space of 2 x 2 real matrices. If a linear transformation ?? : ?? ?? (R) ?? ?? (R)is defined as ?? (?? ) =
[
?? (?? )- ?? (?? ) ?? ?? ?? (?? )
]. Then 
(A) T is one-one but not onto 
(B) T is onto but not one-one 
(C) Range (T) = span {[
0 0
0 1
] [
-2 0
0 1
]} 
(D) Null (T) = span {?? 2
- 2?? ,1 - ?? } 
 
19. Let ?? ?? = {(?? ,?? ),(?? ,-?? )} and ?? ?? = {(?? ,?? ),(?? ,?? )} be ordered bases of R
?? . If ?? : R
?? ? R
?? isa linear 
transformation such that [?? ]
?? ?? ,?? ?? , the matrix of T with respect to ?? ?? and ?? ?? , is[
?? ?? ?? -?? ]then T(5,5) is 
equal to 
(A) (-9,8)   
(B) (9,8)   
(C) (-15, -2)    
(D) (15, 2) 
 
  
 
          
 
 
20. Let?? =n
?? =?? 8
([?? ,
?? ?? ?? +?? ] ? [
?? ?? ?? ,?? ]). Which one of the following statements is FALSE? 
(A) There exist sequences {?? ?? } and {?? ?? } in [0,1] such that ?? = [0,1]\?
?? =1
8
(?? ?? ,?? ?? ) 
(B) [0, 1]\S is an open set 
(C) If A is an infinite subset of S, then A has a limit point 
(D) There exists an infinite subset of Shaving no limit points 
 
21. Let?? :R - R be a strictly increasing continuous function. If {?? ?? } is a sequence in [0,1],then the sequence 
{?? ({?? ?? })} is 
(A) increasing 
(B) bounded 
(C) convergent 
(D) not necessarily bounded 
 
22. Which one of the following statements is true for the series? (-?? )
?? (?? ?? )!
?? ?? ?? 8
?? =?? ? 
(A) The series converges conditionally but not absolutely 
(B) The series converges absolutely 
(C) The sequence of partial sums of the series is bounded but not convergent 
(D) The sequence of partial sums of the series is unbounded 
 
23. The sequence {?????? (
?? ?? ?????? -?? (-
?? ?? )
?? )} is  
(A) monotone and convergent 
(B) monotone but not convergent 
(C) convergent but not monotone 
(D) neither monotone nor convergent 
 
24. If y(t) is a solution of the differential equation ?? " + ?? ?? = ?? ?? ?? , then ?????? ?? ? 8
?? -?? ?? (?? ) is equal to  
 (A) 
2
3
         
(B) 
2
5
     
(C) 
2
7
         
(D) 
2
9
 
 
25. For what real values of x and y, does the integral ?
?? ?? (?? - ?? - ?? ?? )???? attain its maximum? 
(A) x = -3, y = 2         
(B) x = 2,  y = 3 
(C) x = -2, y = 2        
(D) x = -3, y = 4 
 
26. The area of the planar region bounded by the curves ?? = ?? ?? ?? – ?? and ?? = ?? ?? ?? is 
(A) 
v2
3
         
(B) 
2v2
3
     
(C) 
4v2
3
         
(D) v2 
 
  
 
          
 
 
27. For ?? = ?? , let ?? ?? : R ? R be given by ?? ?? (?? ) = ?? ?? ?????? ?? . Then at ?? = ?? , ?? ?? has a 
(A) local maximum if n is even 
(B) local maximum if n is odd 
(C) local minimum if n is even 
(D) local minimum if n is odd 
 
28. For ?? ,?? ?? N define ?? ?? ,?? (?? ) = {
?? ?? ?????? (
?? ?? ?? ), ?? ? ?? ?? ?? = ?? 
(A) differentiable for each pair m,n with ?? > ?? 
(B) differentiable for each pair m,n with ?? < ?? 
(C) not differentiable for each pair m, n with ?? > ?? 
(D) not differentiable for each pair m, n with ?? < ?? 
 
29. Let G and H be nonempty subsets of R, where G is connected and ?? ? ?? is not connected.Which one of 
the following statements is true for all such G and H ? 
(A) If ?? n ?? = Ø, then H is connected 
(B) If ?? n ?? = Ø, then H is not connected 
(C) If ?? n ?? ? Ø, then H is connected 
(D) If ?? n ?? ? Ø, then H is not connected 
 
30. Let ?? :{(?? ,?? ) ?? R
?? :?? > 0,?? > 0} ? R be given by ?? (?? ,?? ) = ?? ?? ?? ?? -
?? ?? ?????? -?? (
?? ?? ) +
?? v?? ?? +?? ?? .Then the value of  
 ?? (?? ,?? ) =
?? ?? ?? (?? ,?? )+?? ?? ?? (?? ,?? )
?? (?? ,?? )
 
 (A) changes with ?? but not with ?? 
 (B) changes with ?? but not with ?? 
 (C) changes with ?? and also with ?? 
 (D) neither changes with ?? nor with ??