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Exam Date: 07-Oct-2020
Exam Time: 09:00-12:00
Examination: 1. Course Code - M.A./M.Sc./M.C.A.
2. Field of Study - Mathematics (MATM)
SECTION 1 - PART I
Question No.1 (Question Id - 4)
Which of the following is a compact subset of ?
(A)
(Correct Answer)
(B)
(C)
(D)
Question No.2 (Question Id - 10)
Let G be an abelian group of order 16. Which of the following is true ?
(A)
There exists g G such that order of g is 8.
(B)
If there exists a subgroup H of G of order 8, then there exists g G with order 8.
(C)
If there exists g G with order 8, then G is cyclic.
(D)
There is a one-to-one group homomorphism : G Sm for some m 1. (Correct
Answer)
Question No.3 (Question Id - 8)
Consider the system of linear equations :
3x + y - z =
- x + 2y + 5z =
4x + z = 7
For which and does this system have a unique solution ?
(A)
For no , there is a unique solution.
(B)
is unique but can be arbitrary.
(C)
and are both unique.
(D)
For all , there is a unique solution. (Correct Answer)
Question No.4 (Question Id - 6)
(A)
(B)
(Correct Answer)
(C)
(D)
Question No.5 (Question Id - 1)
Let X, Y and Z be finite sets and let f : X Y and g : Y Z be maps.
Which of the following assertions is always true ?
(A)
If gof is a bijection, then both g and f are bijections.
(B)
If g is one to one, then gof is also one to one.
(C)
If f is onto, then gof is also onto.
(D)
If gof is onto, then |Z| |Y|, where |A| denotes the number of elements in any finite set
A.
(Correct Answer)
Question No.6 (Question Id - 9)
Consider the following subsets of 3 :
X = {(x, y, z) 3 : x 0, y 0, z 0}
Y = {(x, y, z) 3 : 3x + y = 2, y + z = 0}
Z = {(x, y, z) 3 : x2 + 2xy + y2 = 0}
W = {(x, y, z) 3 : x + y + z = 0, 4x + 3y - z = 0}
Which of the above are vector subspaces of 3 ?
(A)
X, Y, Z and W
(B)
Only W
(C)
Only Z and W (Correct Answer)
(D)
Only X and W
Question No.7 (Question Id - 7)
(A)
(Correct Answer)
(B)
(C)
(D)
Question No.8 (Question Id - 3)
(A)
lim sup an = 1 and lim inf an = -1. (Correct Answer)
(B)
lim sup an = 1 and lim inf an = 1.
(C)
lim sup an = -1 and lim inf an = -1.
(D)
lim sup an = - 1 and lim inf an = 1.
Question No.9 (Question Id - 2)
(A)
(B)
(C)
(Correct
Answer)
(D)
Question No.10 (Question Id - 5)
(A)
A, B and C only
(B)
A, B and D only (Correct Answer)
(C)
A and C only
(D)
B and C only
SECTION 2 - PART II
Question No.1 (Question Id - 12)
(A)
A only
(B)
A and B only (Correct Answer)
(C)
B and C only
(D)
D only
Question No.2 (Question Id - 23)
(A)
N is not a subgroup of G.
(B)
N is a subgroup of G, but N is not normal.
(C)
N is a subgroup of G and the number of cosets of N in G is finite.
(D)
N is a subgroup of G and there are infinitely many cosets of N in G. (Correct Answer)
Question No.3 (Question Id - 22)
Let G be a group in which every element other than identity has order 2. Then, which of the following
statements is necessarily true ?
(A)
G must be finite and abelian.
(B)
G can be infinite, but G must be abelian. (Correct Answer)
(C)
G is not necessarily abelian, but it must be finite.
(D)
G may be non-abelian as well as infinite.
Question No.4 (Question Id - 18)
(A)
(Correct Answer)
(B)
(C)
(D)
Question No.5 (Question Id - 21)
Let A M4x3(), B M3x4() and C M4x5(). Consider the following assertions :
A. The matrix ABC cannot have rank equal to 4.
B. AB can have rank 3 but BC cannot have rank 4.
C. ABC and BA can have ranks at most 3.
D. Rank of AB must be less than or equal to rank of BC.
Which of the above is/are correct statements ?
(A)
Only A
(B)
Only D
(C)
A, B and C only (Correct Answer)
(D)
B, C and D only
Question No.6 (Question Id - 16)
(A)
- log2
(B)
- 2 log2
(C)
1 - 2 log2 (Correct Answer)
(D)
1 - 3 log2
Question No.7 (Question Id - 17)
(A)
A only
(B)
A and C only (Correct Answer)
(C)
A, B and C only
(D)
A, B and D only
Question No.8 (Question Id - 13)
(A)
A and C only
(B)
B and D only (Correct Answer)
(C)
D only
(D)
B only
Question No.9 (Question Id - 19)
(A)
(B)
3R5
(C)
(D)
(Correct Answer)
Question No.10 (Question Id - 14)
(A)
B and D only (Correct Answer)
(B)
B, C and D only
(C)
A, C and D only
(D)
A and D only
Question No.11 (Question Id - 11)
Let X = {(x, y) 2 : x2 + y2 = 1}, Y = {(x, y) 2 : x = y} and
Z = {(x, y) 2 : y = - x}. Consider the following assertions :
A. X U Y U Z is an equivalence relation on .
B. X U Y is a reflexive relation on but not symmetric.
C. X U Y is an equivalence relation on .
D. Y U Z is an equivalence relation on .
Which of the above assertions are correct ?
(A)
A and B only
(B)
A, B and D only
(C)
A and D only (Correct Answer)
(D)
A, C and D only
Question No.12 (Question Id - 20)
Let V be a finite dimensional vector space over with dim V 2. Fix a non-zero vector v0 V.
Consider the following assertions :
A. There is a unique basis of V containing v0.
B. There exist infinitely many bases of V containing v0.
C. There is a unique injective linear map T : V V such that T(v0) = v0.
D. There exist infinitely many linear isomorphisms T : V V such that
T(v0) = v0.
Which of the above assertions is/are correct ?
(A)
A only
(B)
C only
(C)
A and C only
(D)
B and D only (Correct Answer)
Question No.13 (Question Id - 15)
(A)
one
(B)
two (Correct Answer)
(C)
more than 2 but finitely many
(D)
infinitely many
Question No.14 (Question Id - 24)
For which of the following n does n! have 2020 trailing zeros at the end ?
(A)
n = 8097 (Correct Answer)
(B)
n = 8085
(C)
n = 8080
(D)
n = 10100
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This post was last modified on 21 January 2021