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

This post was last modified on 27 December 2020

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This download link is referred from the post: DUET Last 10 Years 2011-2021 Question Papers With Answer Key || Delhi University Entrance Test conducted by the NTA

Topic:- MATHS MPHIL S2

Which of the following societies awards the Fields Medal?

[Question ID = 10202]
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1. International Mathematical Union
  [Option ID = 40802]
2. American Mathematical Society
  [Option ID = 40803]
3. European Mathematical Society
  [Option ID = 40804]
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4. International Mathematical Society for Outstanding Research
  [Option ID = 40806]
Correct Answer :-

• International Mathematical Union
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[Option ID = 40802]
Which of the following mathematicians won the famous Abel Prize for 2019 on the work of analysis, geometry and mathematical physics?

[Question ID = 10205]
1. Robert Langlands
  [Option ID = 40813]
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2. Karen Uhlenbeck
  [Option ID = 40814]
3. Louis Nirenberg
  [Option ID = 40816]
4. S.R. Srinivasa Varadhan
  [Option ID = 40817]
Correct Answer :-

• Karen Uhlenbeck

[Option ID = 40814]
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Which of the following Journals/Magazines/Newsletters is published in India

[Question ID = 10206]
1. The Mathematics Student
  [Option ID = 40818]
2. The College Mathematics Journal
  [Option ID = 40820]
3. Mathematics Magazine
  [Option ID = 40821]
4. Involve
  [Option ID = 40823]
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Correct Answer :-

• The Mathematics Student

[Option ID = 40818]

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Which of the following mathematicians is a recipient of the Padma Vibhushan, the third highest civilian honor in India?
1. M.S. Narasimhan
  [Option ID = 40830]
2. S.R. Srinivasa Vardhan
  [Option ID = 40832]
3. Akshay Venkatesh
  [Option ID = 40833]
Correct Answer :-

• Akshay Venkatesh

[Option ID = 40833]
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Which mathematician of the post-christian era wrote the mathematical treatise 'Ganita Sara Sangraha'?

[Question ID = 10210]
1. Mahavira
  [Option ID = 40834]
2. Sridharacharya
  [Option ID = 40835]
3. Brahmagupta
  [Option ID = 40836]
4. Varahamihira
  [Option ID = 40837]
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Correct Answer :-

• Mahavira

[Option ID = 40834]

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Let T be the set of square free positive integers i.e. the set of positive integers not divisible by any square larger than 1. Assuming that any positive integer can be written as n = m²k, k ∈ T, m ∈ N, the value of ∑_k∈T 1/k² is equal to

[Question ID = 10211]
1. π²/16
  [Option ID = 40838]
2. 2/π²
  [Option ID = 40839]
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3. 16/π²
  [Option ID = 40840]
4. 15/π²
  [Option ID = 40841]
Correct Answer :-

• 15/π²

[Option ID = 40841]
Which of the following statements is true?
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[Question ID = 10212]
1. f(x) = 1/(x-2) is uniformly continuous on (2,3].
  [Option ID = 40842]
2. f(x) = x² is uniformly continuous on [2,+∞).
3. f(x) = e^x is uniformly continuous on (-∞, +∞).
  [Option ID = 40845]
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Correct Answer :-

• f(x) = e^x is uniformly continuous on (-∞, +∞).

[Option ID = 40845]

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Applying term wise differentiation or otherwise, the value of ∑_n=1^∞ n²/n! is

[Question ID = 10213]
1. 6e²
  [Option ID = 40846]
2. 4e²
  [Option ID = 40847]
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3. e²
  [Option ID = 40848]
4. 6e
  [Option ID = 40849]
Correct Answer :-

• 6e²

[Option ID = 40846]
Which of the following statements is true?
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[Question ID = 10214]
1. If lim a_n/b_n = 0 and the series ∑_n=1^∞ b_n diverges, then the series ∑_n=1^∞ a_n diverges.
  [Option ID = 40850]
2. If ∑_n=1^∞ a_n = A and ∑_n=1^∞ |a_n| = B, A and B are finite then |A| = B.
  [Option ID = 40851]
3. If the series ∑_n=1^∞ a_n diverges and a_n > 0, then the series ∑_n=1^∞ a_n/(1+a_n) also diverges.
  [Option ID = 40852]
4. The series ∑_n=1^∞ (n+1)/(n+5) converges to 1/5.
  [Option ID = 40853]
Correct Answer :-
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• If the series ∑_n=1^∞ a_n diverges and a_n > 0, then the series ∑_n=1^∞ a_n/(1+a_n) also diverges.

[Option ID = 40852]
The value of ∫_C (2xy-x²)dx + (x + y²)dy, where C is the closed curve of the region bounded by y = x² and y² = x is

[Question ID = 10215]
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1. 1/30
  [Option ID = 40854]
2. 1/25
  [Option ID = 40855]
3. 2/35
  [Option ID = 40856]
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4. 3/17
  [Option ID = 40857]
Let A(2,5) be rotated by an angle of π/3 then the coordinates of the resulting point is
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[Question ID = 10216]
1. (1+5√3/2, 5/2 - √3)
  [Option ID = 40858]
2. (1-5√3/2, -5/2 - √3)
  [Option ID = 40859]
3. (1-5√3/2, 5/2 + √3)
  [Option ID = 40860]
4. (1+5√3/2, -5/2 + √3)
  [Option ID = 40861]
Correct Answer :-
--- Content provided by FirstRanker.com ---

• (1+5√3/2, -5/2 + √3)

[Option ID = 40861]
Let T: ℝ² → ℝ² be a length preserving linear transformation. Suppose that A = [a b] [c d] is the matrix corresponding to T. Which of the following relations is false?

[Question ID = 10217]
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1. a²+c² = 1
  [Option ID = 40862]
2. b²+d² = 1
  [Option ID = 40863]
3. ab + cd = 0
  [Option ID = 40864]
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4. ab + cd = 1
  [Option ID = 40865]
Correct Answer :-

• ab + cd = 1
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[Option ID = 40865]
Let T: V → W be a nonsingular linear transformation of a finite dimensional vector space V to any vector space W. Which of the following is true?

[Question ID = 10218]
1. dim V < Rank of T
  [Option ID = 40866]
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2. dim V = Rank of T
  [Option ID = 40867]
3. T maps a basis of V to a basis of W
  [Option ID = 40868]
4. dim V > Rank of T
  [Option ID = 40869]
Correct Answer :-

• dim V = Rank of T

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Let A be a n × n matrix over a field F of characteristic 0. Which of the following statements is true?

[Question ID = 10219]
1. If F = ℝ then A has nonzero eigenvalues.
  [Option ID = 40870]
2. If F = ℝ then A is diagonalizable
  [Option ID = 40871]
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3. If F = ℂ then A has real eigenvalues
  [Option ID = 40872]
4. If F = ℂ then A is diagonalizable
  [Option ID = 40873]
Correct Answer :-

• If F = ℂ then A is diagonalizable

[Option ID = 40873]
Let T: ℝ² → ℝ² be the linear map defined by
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T(x,y) = (2x-7y, 4x + 3y).

The matrix of T with respect to the ordered basis B = {(1,3), (2,5)} is

[Question ID = 10220]
1. [121 201] [-70 -116]
  [Option ID = 40874]
2. [-121 201] [70 -116]
  [Option ID = 40875]
3. [-121 201] [70 116]
  [Option ID = 40876]
4. [-121 -201] [70 116]
  [Option ID = 40877]
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Correct Answer :-

• [121 201] [-70 -116]

[Option ID = 40874]

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Let f be a bounded Lebesgue measurable function on [π,3π]. Then lim_n→∞ ∫_π^3π f(x) sin nx dx

[Question ID = 10221]
1. may not exist
  [Option ID = 40878]
2. exists and is equal to 1
  [Option ID = 40879]
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3. exists and is equal to 0
  [Option ID = 40880]
4. exists and is equal to 2
  [Option ID = 40881]
Correct Answer :-

• exists and is equal to 0

[Option ID = 40880]
Let A = {x ∈ (0,1): sin(1/x) = 0} and f:
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f(x) = sin(1/x), if x ∈ (0,1) ~ A, 0, if x ∈ A.
Which of the following statements is true?

[Question ID = 10222]
1. f is not Lebesgue measurable
  [Option ID = 40882]
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2. f is Lebesgue measurable but |f| is not Lebesgue measurable
  [Option ID = 40883]
3. f is Lebesgue measurable but not Lebesgue integrable on [0,1]
  [Option ID = 40884]
4. f is Lebesgue integrable on [0,1]
  [Option ID = 40885]
Correct Answer :-

• f is Lebesgue integrable on [0,1]

[Option ID = 40885]
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Let z = x + iy ∈ ℂ and

f(z) = (x³(1+i) - y³(1-i))/(x² + y²), if z ≠ 0, 0, otherwise.

If f(z) = u(x, y) + iv(x,y), then at origin

[Question ID = 10223]
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1. u and v do not satisfy Cauchy Riemann equations but f is differentiable
  [Option ID = 40886]
2. u and v do not satisfy Cauchy Riemann equations and f is not differentiable
  [Option ID = 40887]
3. u and v satisfy Cauchy Riemann equations but f is not differentiable
  [Option ID = 40888]
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4. u and v satisfy Cauchy Riemann equations and f is differentiable
  [Option ID = 40889]
Correct Answer :-

• u and v satisfy Cauchy Riemann equations but f is not differentiable
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[Option ID = 40888]
The value of the integral ∫_C dz/(z⁴ - e²z²) where C is the anticlockwise circle |z| = 3 is

[Question ID = 10224]
1. 2π(e⁴ + 2e⁴)i
  [Option ID = 40890]
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2. 2π(e⁴ – e²) i
  [Option ID = 40891]
3. 2π(e² – e⁴)i
  [Option ID = 40892]
4. 2π(e² – 2e⁴)i
  [Option ID = 40893]
Correct Answer :-

• 2π(e⁴ – e²)i

[Option ID = 40891]
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For n ∈ ℕ, let z_n = (-1)ⁿ and w_n = 0, if n is even, 1/n, if n is odd. functions such that f(1/n) = z_n and g(1/n) = w_n for all n ∈ ℕ. Then on ℂ

[Question ID = 10225]
1. f can be chosen to be analytic but g cannot be analytic
  [Option ID = 40894]
2. g can be chosen to be analytic but f cannot be analytic
  [Option ID = 40895]
3. Neither f nor g can be analytic
  [Option ID = 40896]
4. Both f and g can be chosen to be analytic
  [Option ID = 40897]
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Correct Answer :-

• Neither f nor g can be analytic

[Option ID = 40896]

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Let f(z) = a₀ + a₁z + ... + a₂₀₂₂z²⁰²², z∈ ℂ be such that |f(z)| ≤ 1, for |z| ≤ 1. Then for all n = 1,2,...,20,

[Question ID = 10226]
1. |a_n| ≤ 1 and f is a constant
  [Option ID = 40898]
2. |a_n| ≤ 1 but f need not be a constant
  [Option ID = 40899]
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3. 1<|a_n|[Option ID = 40900]
4. 1<|a_n|< n and f is a constant
  [Option ID = 40901]
Correct Answer :-
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• |a_n|≤1 but f need not be a constant

[Option ID = 40899]
Let S be compact subset of a metric space. Then,

[Question ID = 10227]
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1. S is complete and totally bounded
  [Option ID = 40902]
2. S is complete but need not be totally bounded
  [Option ID = 40903]
3. S is totally bounded but need not be complete
  [Option ID = 40904]
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4. S is totally bounded but S need not be totally bounded
  [Option ID = 40905]
Correct Answer :-

• S is complete and totally bounded
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[Option ID = 40902]
Which of the following statements is false?

[Question ID = 10228]
1. The connected subsets of ℝ, with the usual metric, are intervals.
  [Option ID = 40906]
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2. The real line ℝ and the Euclidean plane ℝ² are not homeomorphic
3. If A is a connected metric space then every continuous function f : A→ ℕ must be constant.
  
  [Option ID = 40908]
4. If B ⊆ ℝ is bounded and f: ℝ→ℝ is continuous on B then f(B) is bounded
  
  [Option ID = 40909]
Correct Answer :-

• If B ⊆ ℝ is bounded and f: ℝ→ℝ is continuous on B then f(B) is bounded
--- Content provided by FirstRanker.com ---

[Option ID = 40909]
Let {X_α: α ∈ Λ} be a family of topological spaces. Which of the following statements is false?

[Question ID = 10229]
1. Product topology on ∏_α∈ΛX_α is finer than the box topology on ∏_α∈ΛX_α
  [Option ID = 40910]
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2. If ∏_α∈ΛX_α has product topology and E_α ⊆ X_α then ∏_α∈ΛE_α = ∏_α∈ΛE_α
  [Option ID = 40911]
3. If each X_α is completely regular then ∏_α∈ΛX_α under product topology is completely regular
  [Option ID = 40912]
4. If B_α is a basis for X_α then the family of sets ∏_α∈ΛB_α, B_α∈ B_α is a basis for ∏_α∈ΛX_α under box topology
  [Option ID = 40913]
Correct Answer :-

• Product topology on ∏_α∈ΛX_α is finer than the box topology on ∏_α∈ΛX_α

[Option ID = 40910]
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Which of the following statements is false?

[Question ID = 10230]
1. The open continuous image of a first countable space is first countable.
  [Option ID = 40914]
2. The space ℝ_l, (ℝ with lower limit topology) is separable but not second countable.
  [Option ID = 40915]
3. Let X and Y be topological spaces, f: X → Y be such that x_n → x in X implies f(x_n) → f(x) in Y, then f is continuous.
  [Option ID = 40916]
4. Both axioms of first countability and second countability are hereditary.
  [Option ID = 40917]
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Correct Answer :-

• Let X and Y be topological spaces, f: X → Y be such that x_n → x in X implies f(x_n) → f(x) in Y, then f is continuous.

[Option ID = 40916]

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Which of the following statements is false?

[Question ID = 10231]
1. All the cubes, spheres and discs are compact in ℝⁿ.
  [Option ID = 40918]
2. For a metric space (X, d) and x∈X, {y ∈ X: d(x,y) [Option ID = 40919]
3. If (A_n) _n∈ℕ is a sequence of nowhere dense sets in a complete metric space X then X ≠ ∪_n∈ℕA_n
  [Option ID = 40920]
4. Any continuous function from [a, b] to ℝ is limit sequence of polynomials.
  [Option ID = 40921]

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For a metric space (X, d) and x∈X, {y ∈ X: d(x,y)
[Option ID = 40919]
Which of the following statements is false?

[Question ID = 10232]
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1. The space of all real valued continuous functions on [a, b] under supremum norm is separable.
  [Option ID = 40922]
2. Every subset of the Euclidean space ℝⁿ is separable.
  [Option ID = 40923]
3. Every compact metric space is separable.
  [Option ID = 40924]
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4. A closed subspace of a separable topological space is separable.
  [Option ID = 40925]
Correct Answer :-

• A closed subspace of a separable topological space is separable.
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[Option ID = 40925]
Let (X, ||. ||_X) and (Y, ||. ||_Y) be two normed spaces. For (x, y) ∈ X×Y define ||(x,y) ||₁ = ||x||_X + ||y||_Y; ||(x,y)||₂ = (||x||_X^1/2 + ||y||_Y^1/2)² ¡|| (x,y)||₃ = (||x||_X³ + ||y||_Y³)^1/3.

Consider the following statements:

a. ||. ||₁ defines a norm on X × Y.
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b. ||. ||₂ defines a norm on X × Y.

c. ||. ||₃ defines a norm on X × Y.

Which of the following options is correct?

[Question ID = 10233]
1. Only a) and b) are correct.
  [Option ID = 40926]
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2. Only a) and c) are correct.
  [Option ID = 40927]
3. Only b) and c) are correct.
  [Option ID = 40928]
4. None of a), b) and c) is correct.
  [Option ID = 40929]
Correct Answer :-

• Only a) and c) are correct.

[Option ID = 40927]
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For a normed space X consider the following statements:

a. For a sequence (x_n) _n∈ℕ in X if ∑_m=1^∞||x_n|| < ∞, then the series ∑_m=1^∞x_n converges in X.

b. If X is complete and for a sequence (x_n) _n∈ℕ in X if ∑_m=1^∞||x_n|| < ∞, then the series ∑_m=1^∞x_n converges in X.

c. If ∑_m=1^∞||x_n|| < ∞ implies the series ∑_m=1^∞x_n converges for any sequence (x_n) _n∈ℕ in X, then X is complete.
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d. For a sequence (x_n) _n∈ℕ in X if the series ∑_m=1^∞x_n converges in X, then ∑_m=1^∞||x_n|| < ∞.

Which of the following options is correct?

[Question ID = 10234]
1. Only a)
  This download link is referred from the post: DUET Last 10 Years 2011-2021 Question Papers With Answer Key || Delhi University Entrance Test conducted by the NTA
  --- Content provided by FirstRanker.com ---

This post was last modified on 27 December 2020

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