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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

DUET Last 10 Years 2011-2021 Question Papers With Answer Key || Delhi University Entrance Test conducted by the NTA


Topic:- MATHS MPHIL S2

  1. Which of the following societies awards the Fields Medal?

    [Question ID = 10202]

    --- Content provided by‍ FirstRanker.com ---

    1. International Mathematical Union

      [Option ID = 40802]

    2. American Mathematical Society

      [Option ID = 40803]

    3. European Mathematical Society

      [Option ID = 40804]

      --- Content provided by‌ FirstRanker.com ---

    4. International Mathematical Society for Outstanding Research

      [Option ID = 40806]

    Correct Answer :-

    • International Mathematical Union

    --- Content provided by‌ FirstRanker.com ---

    [Option ID = 40802]

  2. 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]

      --- Content provided by​ FirstRanker.com ---

    2. Karen Uhlenbeck

      [Option ID = 40814]

    3. Louis Nirenberg

      [Option ID = 40816]

    4. --- Content provided by​ FirstRanker.com ---

    5. S.R. Srinivasa Varadhan

      [Option ID = 40817]

    Correct Answer :-

    • Karen Uhlenbeck

    [Option ID = 40814]

    --- Content provided by​ FirstRanker.com ---

  3. Which of the following Journals/Magazines/Newsletters is published in India

    [Question ID = 10206]

    1. The Mathematics Student

      [Option ID = 40818]

    2. --- Content provided by FirstRanker.com ---

    3. The College Mathematics Journal

      [Option ID = 40820]

    4. Mathematics Magazine

      [Option ID = 40821]

    5. Involve

      [Option ID = 40823]

      --- Content provided by⁠ FirstRanker.com ---

    Correct Answer :-

    • The Mathematics Student

    [Option ID = 40818]

  4. --- Content provided by FirstRanker.com ---

  5. 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. --- Content provided by⁠ FirstRanker.com ---

    4. Akshay Venkatesh

      [Option ID = 40833]

    Correct Answer :-

    • Akshay Venkatesh

    [Option ID = 40833]

    --- Content provided by FirstRanker.com ---

  6. Which mathematician of the post-christian era wrote the mathematical treatise 'Ganita Sara Sangraha'?

    [Question ID = 10210]

    1. Mahavira

      [Option ID = 40834]

    2. --- Content provided by‌ FirstRanker.com ---

    3. Sridharacharya

      [Option ID = 40835]

    4. Brahmagupta

      [Option ID = 40836]

    5. Varahamihira

      [Option ID = 40837]

      --- Content provided by⁠ FirstRanker.com ---

    Correct Answer :-

    • Mahavira

    [Option ID = 40834]

  7. --- Content provided by​ FirstRanker.com ---

  8. 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. p2/16

      [Option ID = 40838]

    2. 2/p2

      [Option ID = 40839]

      --- Content provided by⁠ FirstRanker.com ---

    3. 16/p2

      [Option ID = 40840]

    4. 15/p2

      [Option ID = 40841]

    5. --- Content provided by⁠ FirstRanker.com ---

    Correct Answer :-

    • 15/p2

    [Option ID = 40841]

  9. Which of the following statements is true?

    --- Content provided by‍ FirstRanker.com ---

    [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,+8).
    3. f(x) = ex is uniformly continuous on (-8, +8).

      [Option ID = 40845]

      --- Content provided by FirstRanker.com ---

    Correct Answer :-

    • f(x) = ex is uniformly continuous on (-8, +8).

    [Option ID = 40845]

  10. --- Content provided by‍ FirstRanker.com ---

  11. Applying term wise differentiation or otherwise, the value of ?n=18 n2/n! is

    [Question ID = 10213]

    1. 6e2

      [Option ID = 40846]

    2. 4e2

      [Option ID = 40847]

      --- Content provided by‌ FirstRanker.com ---

    3. e2

      [Option ID = 40848]

    4. 6e

      [Option ID = 40849]

    5. --- Content provided by‍ FirstRanker.com ---

    Correct Answer :-

    • 6e2

    [Option ID = 40846]

  12. Which of the following statements is true?

    --- Content provided by FirstRanker.com ---

    [Question ID = 10214]

    1. If lim an/bn = 0 and the series ?n=18 bn diverges, then the series ?n=18 an diverges.

      [Option ID = 40850]

    2. If ?n=18 an = A and ?n=18 |an| = B, A and B are finite then |A| = B.

      [Option ID = 40851]

    3. --- Content provided by​ FirstRanker.com ---

    4. If the series ?n=18 an diverges and an > 0, then the series ?n=18 an/(1+an) also diverges.

      [Option ID = 40852]

    5. The series ?n=18 (n+1)/(n+5) converges to 1/5.

      [Option ID = 40853]

    Correct Answer :-

    --- Content provided by‌ FirstRanker.com ---

    • If the series ?n=18 an diverges and an > 0, then the series ?n=18 an/(1+an) also diverges.

    [Option ID = 40852]

  13. 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]

    --- Content provided by⁠ FirstRanker.com ---

    1. 1/30

      [Option ID = 40854]

    2. 1/25

      [Option ID = 40855]

    3. 2/35

      [Option ID = 40856]

      --- Content provided by​ FirstRanker.com ---

    4. 3/17

      [Option ID = 40857]

  14. Let A(2,5) be rotated by an angle of p/3 then the coordinates of the resulting point is

    --- Content provided by FirstRanker.com ---

    [Question ID = 10216]

    1. (1+5v3/2, 5/2 - v3)

      [Option ID = 40858]

    2. (1-5v3/2, -5/2 - v3)

      [Option ID = 40859]

    3. --- Content provided by‍ FirstRanker.com ---

    4. (1-5v3/2, 5/2 + v3)

      [Option ID = 40860]

    5. (1+5v3/2, -5/2 + v3)

      [Option ID = 40861]

    Correct Answer :-

    --- Content provided by‍ FirstRanker.com ---

    • (1+5v3/2, -5/2 + v3)

    [Option ID = 40861]

  15. 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]

    --- Content provided by‌ FirstRanker.com ---

    1. a2+c2 = 1

      [Option ID = 40862]

    2. b2+d2 = 1

      [Option ID = 40863]

    3. ab + cd = 0

      [Option ID = 40864]

      --- Content provided by‍ FirstRanker.com ---

    4. ab + cd = 1

      [Option ID = 40865]

    Correct Answer :-

    • ab + cd = 1

    --- Content provided by‌ FirstRanker.com ---

    [Option ID = 40865]

  16. 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]

      --- Content provided by‍ FirstRanker.com ---

    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. --- Content provided by FirstRanker.com ---

    5. dim V > Rank of T

      [Option ID = 40869]

    Correct Answer :-

    • dim V = Rank of T

  17. --- Content provided by​ FirstRanker.com ---

  18. 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]

      --- Content provided by‌ FirstRanker.com ---

    3. If F = C then A has real eigenvalues

      [Option ID = 40872]

    4. If F = C then A is diagonalizable

      [Option ID = 40873]

    5. --- Content provided by‌ FirstRanker.com ---

    Correct Answer :-

    • If F = C then A is diagonalizable

    [Option ID = 40873]

  19. Let T: R2 ? R2 be the linear map defined by

    --- Content provided by​ FirstRanker.com ---

    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. --- Content provided by‌ FirstRanker.com ---

    3. [-121 201] [70 -116]

      [Option ID = 40875]

    4. [-121 201] [70 116]

      [Option ID = 40876]

    5. [-121 -201] [70 116]

      [Option ID = 40877]

      --- Content provided by FirstRanker.com ---

    Correct Answer :-

    [121 201] [-70 -116]

    [Option ID = 40874]

  20. --- Content provided by FirstRanker.com ---

  21. Let f be a bounded Lebesgue measurable function on [p,3p]. Then limn?8 ?p3p 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]

      --- Content provided by‌ FirstRanker.com ---

    3. exists and is equal to 0

      [Option ID = 40880]

    4. exists and is equal to 2

      [Option ID = 40881]

    5. --- Content provided by⁠ FirstRanker.com ---

    Correct Answer :-

    • exists and is equal to 0

    [Option ID = 40880]

  22. Let A = {x ? (0,1): sin(1/x) = 0} and f:

    --- Content provided by​ FirstRanker.com ---

    f(x) = sin(1/x), if x ? (0,1) ~ A, 0, if x ? A.

  23. Which of the following statements is true?

    [Question ID = 10222]

    1. f is not Lebesgue measurable

      [Option ID = 40882]

      --- Content provided by​ FirstRanker.com ---

    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. --- Content provided by⁠ FirstRanker.com ---

    5. f is Lebesgue integrable on [0,1]

      [Option ID = 40885]

    Correct Answer :-

    • f is Lebesgue integrable on [0,1]

    [Option ID = 40885]

    --- Content provided by FirstRanker.com ---

  24. 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]

    --- Content provided by‌ FirstRanker.com ---

    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]

      --- Content provided by‌ FirstRanker.com ---

    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

    --- Content provided by‌ FirstRanker.com ---

    [Option ID = 40888]

  25. The value of the integral ?C dz/(z4 - e2z2) where C is the anticlockwise circle |z| = 3 is

    [Question ID = 10224]

    1. 2p(e4 + 2e4)i

      [Option ID = 40890]

      --- Content provided by⁠ FirstRanker.com ---

    2. 2p(e4 – e2) i

      [Option ID = 40891]

    3. 2p(e2 – e4)i

      [Option ID = 40892]

    4. --- Content provided by‍ FirstRanker.com ---

    5. 2p(e2 – 2e4)i

      [Option ID = 40893]

    Correct Answer :-

    • 2p(e4 – e2)i

    [Option ID = 40891]

    --- Content provided by​ FirstRanker.com ---

  26. For n ? N, let zn = (-1)n and wn = 0, if n is even, 1/n, if n is odd. functions such that f(1/n) = zn and g(1/n) = wn 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. --- Content provided by​ FirstRanker.com ---

    3. g can be chosen to be analytic but f cannot be analytic

      [Option ID = 40895]

    4. Neither f nor g can be analytic

      [Option ID = 40896]

    5. Both f and g can be chosen to be analytic

      [Option ID = 40897]

      --- Content provided by​ FirstRanker.com ---

    Correct Answer :-

    • Neither f nor g can be analytic

    [Option ID = 40896]

  27. --- Content provided by‍ FirstRanker.com ---

  28. 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. |an| = 1 and f is a constant

      [Option ID = 40898]

    2. |an| = 1 but f need not be a constant

      [Option ID = 40899]

      --- Content provided by​ FirstRanker.com ---

    3. 1<|an|[Option ID = 40900]
    4. 1<|an|< n and f is a constant

      [Option ID = 40901]

    Correct Answer :-

    --- Content provided by‌ FirstRanker.com ---

    • |an|=1 but f need not be a constant

    [Option ID = 40899]

  29. Let S be compact subset of a metric space. Then,

    [Question ID = 10227]

    --- Content provided by‍ FirstRanker.com ---

    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]

      --- Content provided by‌ FirstRanker.com ---

    4. S is totally bounded but S need not be totally bounded

      [Option ID = 40905]

    Correct Answer :-

    • S is complete and totally bounded

    --- Content provided by‍ FirstRanker.com ---

    [Option ID = 40902]

  30. 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]

      --- Content provided by⁠ FirstRanker.com ---

    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. --- Content provided by FirstRanker.com ---

    5. 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]

  31. Let {Xa: a ? ?} be a family of topological spaces. Which of the following statements is false?

    [Question ID = 10229]

    1. Product topology on ?a?Xa is finer than the box topology on ?a?Xa

      [Option ID = 40910]

      --- Content provided by⁠ FirstRanker.com ---

    2. If ?a?Xa has product topology and Ea ? Xa then ?a?Ea = ?a?Ea

      [Option ID = 40911]

    3. If each Xa is completely regular then ?a?Xa under product topology is completely regular

      [Option ID = 40912]

    4. --- Content provided by⁠ FirstRanker.com ---

    5. If Ba is a basis for Xa then the family of sets ?a?Ba, Ba? Ba is a basis for ?a?Xa under box topology

      [Option ID = 40913]

    Correct Answer :-

    • Product topology on ?a?Xa is finer than the box topology on ?a?Xa

    [Option ID = 40910]

    --- Content provided by‌ FirstRanker.com ---

  32. 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. --- Content provided by‍ FirstRanker.com ---

    3. The space Rl, (R with lower limit topology) is separable but not second countable.

      [Option ID = 40915]

    4. Let X and Y be topological spaces, f: X ? Y be such that xn ? x in X implies f(xn) ? f(x) in Y, then f is continuous.

      [Option ID = 40916]

    5. Both axioms of first countability and second countability are hereditary.

      [Option ID = 40917]

      --- Content provided by‍ FirstRanker.com ---

    Correct Answer :-

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

    [Option ID = 40916]

  33. --- Content provided by‌ FirstRanker.com ---

  34. 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. --- Content provided by FirstRanker.com ---

    4. If (An) n?N is a sequence of nowhere dense sets in a complete metric space X then X ? ?n?NAn

      [Option ID = 40920]

    5. Any continuous function from [a, b] to R is limit sequence of polynomials.

      [Option ID = 40921]

  35. --- Content provided by‌ FirstRanker.com ---

  36. For a metric space (X, d) and x?X, {y ? X: d(x,y)

    [Option ID = 40919]

  37. Which of the following statements is false?

    [Question ID = 10232]

    --- Content provided by⁠ FirstRanker.com ---

    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]

      --- Content provided by‍ FirstRanker.com ---

    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.

    --- Content provided by FirstRanker.com ---

    [Option ID = 40925]

  38. 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||X1/2 + ||y||Y1/2)2 ¡|| (x,y)||3 = (||x||X3 + ||y||Y3)1/3.

    Consider the following statements:

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

    --- Content provided by FirstRanker.com ---

    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]

      --- Content provided by​ FirstRanker.com ---

    2. Only a) and c) are correct.

      [Option ID = 40927]

    3. Only b) and c) are correct.

      [Option ID = 40928]

    4. --- Content provided by‌ FirstRanker.com ---

    5. None of a), b) and c) is correct.

      [Option ID = 40929]

    Correct Answer :-

    • Only a) and c) are correct.

    [Option ID = 40927]

    --- Content provided by‍ FirstRanker.com ---

  39. For a normed space X consider the following statements:

    a. For a sequence (xn) n?N in X if ?m=18||xn|| < 8, then the series ?m=18xn converges in X.

    b. If X is complete and for a sequence (xn) n?N in X if ?m=18||xn|| < 8, then the series ?m=18xn converges in X.

    c. If ?m=18||xn|| < 8 implies the series ?m=18xn converges for any sequence (xn) n?N in X, then X is complete.

    --- Content provided by FirstRanker.com ---

    d. For a sequence (xn) n?N in X if the series ?m=18xn converges in X, then ?m=18||xn|| < 8.

    Which of the following options is correct?

    [Question ID = 10234]

    1. Only a)

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      --- Content provided by⁠ FirstRanker.com ---