q

Delhi University Entrance Test (DUET) 2020 Previous Year Question Paper With Answer Key

DU MA MSc Mathematics
Topic: MATHS MA S2
1) Let
and
be sequences of real numbers such that
for all
, where is some positive integer.
Consider the following statements:
(a)
(b)
Which of the above statements is(are) correct?
[Question ID = 5742]
1. Neither (a) nor (b)
[Option ID = 22962]
2. Only (a)
[Option ID = 22963]
3. Only (b)
[Option ID = 22964]
4. Both (a) and (b)
[Option ID = 22965]
Correct Answer :
Both (a) and (b)
[Option ID = 22965]
2) Which of the sequences
and
of real numbers with
terms

,
has(have) convergent subsequences?
[Question ID = 5743]
1.
[Option ID = 22966]
2. Only
[Option ID = 22967]
3. Only
[Option ID = 22968]
4.
[Option ID = 22969]
Correct Answer :
[Option ID = 22969]
3) Consider the following series:
(a)
(b)
(c)
(d)
Which of the above series is (are) convergent?

---

[Question ID = 5744]
1. All of (a), (b), (c) and (d)
[Option ID = 22970]
2. Only (a), (c) and (d)
[Option ID = 22971]
3. Only (a) and (c)
[Option ID = 22972]
4. Only (c)
[Option ID = 22973]
Correct Answer :
Only (a) and (c)
[Option ID = 22972]
4) The union of infinitely many closed subsets of the real line is
[Question ID = 5745]
1. uncountable [Option ID = 22974]
2. finite [Option ID = 22975]
3. always closed [Option ID = 22976]
4. need not be closed [Option ID = 22977]
Correct Answer :
need not be closed [Option ID = 22977]
5)
[Question ID = 5746]
1. r/2 and 2r [Option ID = 22978]
2. r/3 and r [Option ID = 22979]
3. 2r/3 and 3r/2 [Option ID = 22980]
4. 0 and 1 [Option ID = 22981]
Correct Answer :
r/2 and 2r [Option ID = 22978]
6) Consider the following series:
(a)
(b)
(c)
Which of the above series converge uniformly on the indicated domain?
[Question ID = 5747]
1. Only (a) and (b)
[Option ID = 22982]
2. Only (b) and (c)
[Option ID = 22983]
3. Only (a) and (c)
[Option ID = 22984]
4. All of (a), (b) and (c)
[Option ID = 22985]
Correct Answer :
Only (a) and (c)
[Option ID = 22984]
7) Let
be a sequence of continuous functions on
converging uniformly to the function . Consider the following statements:
(a) is bounded on

---

(b)
(c) If each is differentiable, then the sequence
converges uniformly to on
, is the derivative of
Which of the following statements is(are) correct?
[Question ID = 5748]
1. Only (a) and (b)
[Option ID = 22986]
2. Only (a) and (c)
[Option ID = 22987]
3. Only (c)
[Option ID = 22988]
4. Only (b)
[Option ID = 22989]
Correct Answer :
Only (a) and (b)
[Option ID = 22986]
8)
[Question ID = 5749]
1.
[Option ID = 22990]
2.
[Option ID = 22991]
3.
[Option ID = 22992]
4.
= 0
[Option ID = 22993]
Correct Answer :
[Option ID = 22990]
9) Let
Consider the following statements:
a. is not continuous on
b. is continuous on but not differentiable at
c. is differentiable on but is not continuous on
d. is differentiable on and is continuous on
Which of the above statements is(are) correct?
[Question ID = 5750]
1. Only (a) and (d)
[Option ID = 22994]
2. Only (b) and (c)
[Option ID = 22995]
3. Only (c)
[Option ID = 22996]
4. Only (d)
[Option ID = 22997]
Correct Answer :
Only (d)
[Option ID = 22997]

---

10) The zero of the function
defined on lie on the interval
[Question ID = 5751]
1. (1, 1)
[Option ID = 22998]
2. [3, 4]
[Option ID = 22999]
3. [2, 1]
[Option ID = 23000]
4. [5, 3]
[Option ID = 23001]
Correct Answer :
[2, 1]
[Option ID = 23000]
11) The Wronskian of
at
is
[Question ID = 5752]
1. 1
[Option ID = 23002]
2. 2
[Option ID = 23003]
3. 1
[Option ID = 23004]
4. 2
[Option ID = 23005]
Correct Answer :
2
[Option ID = 23003]
12) The solution of the initial value problem
is:
[Question ID = 5753]
1.
[Option ID = 23006]
2.
[Option ID = 23007]
3.
[Option ID = 23008]
4.
[Option ID = 23009]
Correct Answer :
[Option ID = 23007]
13) How many solution(s) does the initial value problem
have?
[Question ID = 5754]
1. No solution
[Option ID = 23010]
2. Unique solution
[Option ID = 23011]
3. Two solutions
[Option ID = 23012]
4. Infinitely many solutions
[Option ID = 23013]
Correct Answer :
Infinitely many solutions

---

[Option ID = 23013]
14)
[Question ID = 5755]
1.
[Option ID = 23014]
2.
[Option ID = 23015]
3.
[Option ID = 23016]
4.
[Option ID = 23017]
Correct Answer :
[Option ID = 23014]
15) The particular integral of the differential equation is
[Question ID = 5756]
1.
[Option ID = 23018]
2.
[Option ID = 23019]
3.
[Option ID = 23020]
4.
[Option ID = 23021]
Correct Answer :
[Option ID = 23021]
16) The complete integral of the partial differential equation
, where
is
(
are arbitrary constants)
[Question ID = 5757]
1.
[Option ID = 23022]
2.
[Option ID = 23023]
3.
[Option ID = 23024]
4.
[Option ID = 23025]
Correct Answer :
[Option ID = 23024]
17)
[Question ID = 5758]

---

1.
[Option ID = 23026]
2.
[Option ID = 23027]
3.
[Option ID = 23028]
4.
[Option ID = 23029]
Correct Answer :
[Option ID = 23027]
18) The partial differential equation
is
[Question ID = 5759]
1. Hyperbolic in
[Option ID = 23030]
2. Hyperbolic in
[Option ID = 23031]
3. Elliptic in
[Option ID = 23032]
4. Elliptic in
[Option ID = 23033]
Correct Answer :
Hyperbolic in
[Option ID = 23031]
19)
[Question ID = 5760]
1.
[Option ID = 23034]
2.
[Option ID = 23035]
3.
[Option ID = 23036]
4.
[Option ID = 23037]
Correct Answer :
[Option ID = 23035]
20) The general solution of
with
is
[Question ID = 5761]
1.
[Option ID = 23038]
2.
[Option ID = 23039]
3.
[Option ID = 23040]
4.

---

[Option ID = 23041]
Correct Answer :
[Option ID = 23039]
21) Let
be given by
Then,
[Question ID = 5762]
1.
[Option ID = 23042]
2.
[Option ID = 23043]
3.
[Option ID = 23044]
4.
[Option ID = 23045]
Correct Answer :
[Option ID = 23043]
22)
[Question ID = 5763]
1.
[Option ID = 23046]
2.
[Option ID = 23047]
3.
[Option ID = 23048]
4.
[Option ID = 23049]
Correct Answer :
[Option ID = 23049]
23) The unique polynomial of degree 2 passing through (1, 1), (3, 27) and (4, 64) obtained by Lagrange interpolation is
[Question ID = 5764]
1.
[Option ID = 23050]
2.
[Option ID = 23051]
3.
[Option ID = 23052]
4.
[Option ID = 23053]
Correct Answer :

---

[Option ID = 23053]
24) The approximate value of
by Simpson's 1/3rd rule, using the least number of equal subintervals, is
[Question ID = 5765]
1. 0.8512
[Option ID = 23054]
2. 0.8125
[Option ID = 23055]
3. 0.7625
[Option ID = 23056]
4. 0.6702
[Option ID = 23057]
Correct Answer :
0.8512
[Option ID = 23054]
25)
[Question ID = 5766]
1. 2.205 [Option ID = 23058]
2. 2.252 [Option ID = 23059]
3. 0.005 [Option ID = 23060]
4. 0.055 [Option ID = 23061]
Correct Answer :
0.005 [Option ID = 23060]
26) The approximate value of
obtained after two iterations of NewtonRaphson method starting with initial
approximation
is
[Question ID = 5767]
1. 2.7566
[Option ID = 23062]
2. 2.5826
[Option ID = 23063]
3. 2.6713
[Option ID = 23064]
4. 2.4566
[Option ID = 23065]
Correct Answer :
2.5826
[Option ID = 23063]
27) For an infinite discrete metric space
which of the following statements is correct?
[Question ID = 5768]
1.
is compact
[Option ID = 23066]
2.
[Option ID = 23067]
3.
is connected
[Option ID = 23068]
4.
is not totally bounded
[Option ID = 23069]
Correct Answer :
is not totally bounded

---

[Option ID = 23069]
28) Consider the metric space
of square summable sequences in with the Euclidean metric. Let
where is the sequence of 1 at the
place and 0 elsewhere. Then,
[Question ID = 5769]
1.
is not compact and has no limit point
[Option ID = 23070]
2.
[Option ID = 23071]
3.
is not compact and has a limit point
[Option ID = 23072]
4.
is compact and has no limit point
[Option ID = 23073]
Correct Answer :
is not compact and has no limit point
[Option ID = 23070]
29) Let
be the set of real valued continuous functions on [0, 1] with supmetric. Let
and
be the subspaces of
. Then,
[Question ID = 5770]
1.
[Option ID = 23074]
2.
[Option ID = 23075]
3.
[Option ID = 23076]
4.
[Option ID = 23077]
Correct Answer :
[Option ID = 23075]
30) Let
and
be the metric spaces with the discrete metric space and usual metric respectively.
Let
and
be the functions given by
Then,
[Question ID = 5771]
1.
[Option ID = 23078]
2.
[Option ID = 23079]
3.
[Option ID = 23080]
4.
[Option ID = 23081]
Correct Answer :
[Option ID = 23080]
31)

---

[Question ID = 5772]
1.
is connected
[Option ID = 23082]
2.
is connected
[Option ID = 23083]
3.
is disconnected
[Option ID = 23084]
4.
[Option ID = 23085]
Correct Answer :
is connected
[Option ID = 23082]
32) Let be the set of all realvalued Riemann integrable functions on and let be the function given by
Which of the following statements is correct?
[Question ID = 5773]
1.
[Option ID = 23086]
2.
[Option ID = 23087]
3.
[Option ID = 23088]
4.
[Option ID = 23089]
Correct Answer :
[Option ID = 23089]
33) The improper integral
[Question ID = 5774]
1. Converges to
[Option ID = 23090]
2. Converges to
[Option ID = 23091]
3. Converges to 0
[Option ID = 23092]
4. Diverges
[Option ID = 23093]
Correct Answer :
Converges to
[Option ID = 23090]
34) Consider the functions
and
Then
[Question ID = 5775]
1.

---

[Option ID = 23094]
2.
[Option ID = 23095]
3.
[Option ID = 23096]
4.
[Option ID = 23097]
Correct Answer :
[Option ID = 23096]
35) What is the length of the interval on which the function
is decreasing?
[Question ID = 5776]
1. 8
[Option ID = 23098]
2. 6
[Option ID = 23099]
3. 4
[Option ID = 23100]
4. 2
[Option ID = 23101]
Correct Answer :
6
[Option ID = 23099]
36) Let
be a monotonic function. Consider the following statements:
a. The function obeys the maximum principle
b. The function is Riemann integrable on [a, b]
Which of the above statement(s) is(are) true?
[Question ID = 5777]
1. Only (a)
[Option ID = 23102]
2. Only (b)
[Option ID = 23103]
3. Both (a) and (b)
[Option ID = 23104]
4. Neither (a) nor (b)
[Option ID = 23105]
Correct Answer :
Both (a) and (b)
[Option ID = 23104]
37) Consider the following:
a.
b.
where
,
are polynomials over
Which of the above is(are) an inner product?
[Question ID = 5778]
1. Neither (a) nor (b)
[Option ID = 23106]
2. Both (a) and (b)
[Option ID = 23107]
3. Only (a)
[Option ID = 23108]
4. Only (b)

---

[Option ID = 23109]
Correct Answer :
Neither (a) nor (b)
[Option ID = 23106]
38) Let
. Then
is equal to
[Question ID = 5779]
1.
[Option ID = 23110]
2.
[Option ID = 23111]
3.
[Option ID = 23112]
4.
[Option ID = 23113]
Correct Answer :
[Option ID = 23111]
39) Let be an infinite dimensional vector space over a field .
Consider the following statements:
a. Any oneone linear transformation from to itself is onto
b. Any onto linear transformation from to itself must be oneone
Which of the above statements is (are) correct?
[Question ID = 5780]
1. Both (a) and (b)
[Option ID = 23114]
2. Only (a)
[Option ID = 23115]
3. Only (b)
[Option ID = 23116]
4. Neither (a) nor (b)
[Option ID = 23117]
Correct Answer :
Neither (a) nor (b)
[Option ID = 23117]
40) Let
be the set of all polynomials over of degree at most . Let
be given by
. Then
[Question ID = 5781]
1.
is oneone and onto linear transformation
[Option ID = 23118]
2.
is an onto function but neither a linear transformation nor oneone
[Option ID = 23119]
3.
is not onto but a oneone linear transformation
[Option ID = 23120]
4.
is oneone but neither a linear transformation nor onto
[Option ID = 23121]
Correct Answer :
is not onto but a oneone linear transformation
[Option ID = 23120]

---

41)
[Question ID = 5782]
1.
[Option ID = 23122]
2.
[Option ID = 23123]
3.
[Option ID = 23124]
4.
[Option ID = 23125]
Correct Answer :
[Option ID = 23124]
42)
[Question ID = 5783]
1.
[Option ID = 23126]
2. Order of is even
[Option ID = 23127]
3.
is abelian
[Option ID = 23128]
4.
[Option ID = 23129]
Correct Answer :
Order of is even
[Option ID = 23127]
43) Let and be finite groups such that
and
. Suppose does not have a normal subgroup of order
3. Let be the set of all group homomorphism from to . Then the number of elements in is
[Question ID = 5784]
1. 1
[Option ID = 23130]
2. 3
[Option ID = 23131]
3. 5
[Option ID = 23132]
4. 7
[Option ID = 23133]
Correct Answer :
1
[Option ID = 23130]
44) Let be a finite group of
has exactly two conjugates. Suppose that
and
.
Which of the following statements is incorrect?
[Question ID = 5785]
1. The number of elements in
is a prime number
[Option ID = 23134]
2.
is a simple group
[Option ID = 23135]

---

3.
[Option ID = 23136]
4.
[Option ID = 23137]
Correct Answer :
is a simple group
[Option ID = 23135]
45)
[Question ID = 5786]
1.
[Option ID = 23138]
2.
[Option ID = 23139]
3.
[Option ID = 23140]
4.
[Option ID = 23141]
Correct Answer :
[Option ID = 23140]
46) The remainder when 2020
is divided by 12 is
2020
[Question ID = 5787]
1. 0 [Option ID = 23142]
2. 2 [Option ID = 23143]
3. 4 [Option ID = 23144]
4. 8 [Option ID = 23145]
Correct Answer :
4 [Option ID = 23144]
47) The smallest integer
such that

,
and
is
[Question ID = 5788]
1. 14
[Option ID = 23146]
2. 56
[Option ID = 23147]
3. 122
[Option ID = 23148]
4. 62
[Option ID = 23149]
Correct Answer :
62
[Option ID = 23149]
48) Let
be a ring and
be given by
. Which of the following statements is
incorrect?
[Question ID = 5789]
1.
is a ring homomorphism
[Option ID = 23150]
2.
is a prime ideal but not maximal
[Option ID = 23151]
3.
is maximal ideal

---

[Option ID = 23152]
4.
is surjective
[Option ID = 23153]
Correct Answer :
is maximal ideal
[Option ID = 23152]
49) Consider the following statements
a. A polynomial is irreducible over a field if it has no zeros in
b. Let
If
is reducible over , then it is reducible over
c. For any prime , the polynomial
is irreducible over
Which of the above statements is (are) correct?
[Question ID = 5790]
1. Only (a) and (b)
[Option ID = 23154]
2. Only (a) and (c)
[Option ID = 23155]
3. Only (b) and (c)
[Option ID = 23156]
4. All of (a), (b) and (c)
[Option ID = 23157]
Correct Answer :
Only (b) and (c)
[Option ID = 23156]
50) Which of the following is a Euclidean domain?
[Question ID = 5791]
1.
[Option ID = 23158]
2.
[Option ID = 23159]
3.
[Option ID = 23160]
4. None of these
[Option ID = 23161]
Correct Answer :
[Option ID = 23158]

---