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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

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 MA S2

  1. Let {xn} and {yn} be sequences of real numbers such that xn ≤ yn for all n ≥ N, where N is some positive integer. Consider the following statements:

    (a) lim inf xn ≤ lim inf yn

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

    (b) lim sup xn ≤ lim sup yn

    Which of the above statements is(are) correct?

    1. Neither (a) nor (b)
    2. Only (a)
    3. Only (b)
    4. --- Content provided by FirstRanker.com ---

    5. Both (a) and (b)

    Correct Answer :-

    • Both (a) and (b)

  2. Which of the sequences {an} and {bn} of real numbers with nth terms

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

    an = (n2+20n +35) sin n3 / (n2 + n + 1)

    bn = 2 cosn - sin n

    has(have) convergent subsequences?

    1. Neither {an} nor {bn}
    2. Only {an}
    3. --- Content provided by FirstRanker.com ---

    4. Only {bn}
    5. Both {an} and {bn}

    Correct Answer :-

    • Both {an} and {bn}

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

  4. Consider the following series:

    (a) ∑ xn / n!, x ∈ R, n=1 to ∞

    (b) ∑ 1 / (n + sinn), n=1 to ∞

    (c) ∑ 1 / (n2√n), n=1 to ∞

    (d) ∑ sin n, n=1 to ∞

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

    Which of the above series is (are) convergent?

    1. Only (a), (c) and (d)
    2. Only (a), (c) and (d)
    3. Only (a) and (c)
    4. Only (c)
    5. --- Content provided by FirstRanker.com ---

    Correct Answer :-

    • Only (a) and (c)

  5. The union of infinitely many closed subsets of the real line is

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

    3. finite
    4. always closed
    5. need not be closed

    Correct Answer :-

    • need not be closed

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

  6. Consider the series ∑ an, where an = (2 + sin(nπ/2))nrn, r > 0. What are the values of lim inf (an+1/an) and lim sup (an+1/an)?

    1. r/2 and 2r
    2. r/3 and r
    3. 2r/3 and 3r/2
    4. --- Content provided by FirstRanker.com ---

    5. 0 and 1

    Correct Answer :-

    • r/2 and 2r

  7. Consider the following series:

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

    (a) ∑ 3-n sin 3nx on R, n=1 to ∞

    (b) ∑ n-2xn on (-2,2), n=1 to ∞

    (c) ∑ (1/n) cosnx on R, n=1 to ∞

    Which of the above series converge uniformly on the indicated domain?

    1. Only (a) and (b)
    2. --- Content provided by FirstRanker.com ---

    3. Only (b) and (c)
    4. Only (a) and (c)
    5. All of (a), (b) and (c)

    Correct Answer :-

    • Only (a) and (c)

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

  8. Let {fn} be a sequence of continuous functions on [a, b] converging uniformly to the function f. Consider the following statements:

    (a) f is bounded on [a, b]

    If fn' exists and the sequence {fn'} converges uniformly to f' on [a, b], f' is the derivative of f.

    Which of the following statements is(are) correct?

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

    1. Only (a) and (b)
    2. Only (a) and (c)
    3. Only (c)
    4. Only (b)

    Correct Answer :-

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

    • Only (a) and (b)

  9. Let G(x) be a real-valued function defined by G(x) = ∫04x2 cos √t dt. If G' is the derivative of G, then

    1. G'(π/2) = −4π
    2. G'(π/2) = −4π – 1
    3. --- Content provided by FirstRanker.com ---

    4. G'(π/2) = -π
    5. G'(π/2) = 0

    Correct Answer :-

    • G'(π/2) = −4π

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

  11. Let f(x) = {(4-x2)/2, |x| < 2; 0, |x| ≥ 2

    Consider the following statements:

    a. f is not continuous on R

    b. f is continuous on R but not differentiable at x = 2, -2

    c. f is differentiable on R but f' is not continuous on R

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

    d. f is differentiable on R and f' is continuous on R

    Which of the above statements is(are) correct?

    1. Only (a) and (d)
    2. Only (b) and (c)
    3. Only (c)
    4. --- Content provided by FirstRanker.com ---

    5. Only (d)

    Correct Answer :-

    • Only (d)

  12. Let f(x) be a real-valued function defined on R lie on the interval

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

    1. (-1, 1)
    2. [3, 4]
    3. [-2, -1]
    4. [-5, -3]

    Correct Answer :-

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

    • [-2, -1]

  13. The Wronskian of cosx, sin x and ex at x = 0 is

    1. 1
    2. 2
    3. --- Content provided by FirstRanker.com ---

    4. -1
    5. -2

    Correct Answer :-

    • 2

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

  15. The solution of the initial value problem y' = 1 + y2, y(0) = 1, is:-

    1. y = cosec(x + π/4)
    2. y = tan(x + π/4)
    3. y = sec(x + π/4)
    4. y = cot(x + π/4)
    5. --- Content provided by FirstRanker.com ---

    Correct Answer :-

    • y = tan(x + π/4)

  16. How many solution(s) does the initial value problem y' - 2y = 0, y(0) = 0 have?

    1. No solution
    2. --- Content provided by FirstRanker.com ---

    3. Unique solution
    4. Two solutions
    5. Infinitely many solutions

    Correct Answer :-

    • Infinitely many solutions

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

  17. The general solution of the equation y'' + y = x cosx is (c1 and c2 are arbitrary constants)

    1. c1 cosx + c2 sin x - x cosx + sin x ln(sin x)
    2. c1 cosx + c2 sin x + x cosx + sin x ln(sin x)
    3. c1 cosx + c2 sin x - x sinx + cos x ln(sin x)
    4. --- Content provided by FirstRanker.com ---

    5. c1 cosx + c2 sin x + x sin x + cosx ln(sin x)

    Correct Answer :-

    • c1 cosx + c2 sin x + x cosx + sin x ln(sin x)

  18. The particular integral of the differential equation is y'' + y = x3 is

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

    1. x2 + 6x
    2. x2-6x
    3. x3 + 6x
    4. x3-6x

    Correct Answer :-

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

    • x3-6x

  19. The complete integral of the partial differential equation p2z2 + q2 = 1, where p = ∂z/∂x, q = ∂z/∂y is (a, b are arbitrary constants)

    1. z + √(z2 + a2) + a2 ln (z + √(z2 + a2) / a) = 0
    2. a2z + by + x2 = 0
    3. --- Content provided by FirstRanker.com ---

    4. z + √(z2 + a2) + a2 ln (z + √(z2 + a2) / a) = 2x + 2ay + 2b
    5. z2 + y2 = x2 + 2x + 2ay + 2b

    Correct Answer :-

    • z + √(z2 + a2) + a2 ln (z + √(z2 + a2) / a) = 2x + 2ay + 2b

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

  21. The complete integral of the partial differential equation z = px+qy - sin(pq) where p = ∂z/∂x, q = ∂z/∂y is

    1. z = ax + by - sm(ab)
    2. z = ax + y + sin b
    3. z = x + by - sin a

    Correct Answer :-

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

    • z = ax + by - sin(ab)

  22. The partial differential equation y uxx + 2xy uxy + x uyy = ux + uy, is

    1. Hyperbolic in {(x,y) | 0 < xy < 1}
    2. Hyperbolic in {(x,y) | xy > 1}
    3. --- Content provided by FirstRanker.com ---

    4. Elliptic in {(x, y) | xy > 1}
    5. Elliptic in {(x, y) | xy < 0}

    Correct Answer :-

    • Hyperbolic in {(x,y) | xy > 1}

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

  24. The general solution of the equation ∂2z/∂y2 = x - y is

    1. (1/4) x(x - y)2 + Ø1 (x2 + y) + Ø2 (x - y)
    2. (1/4) x(x - y)2 + Ø1 (x + y) + Ø2 (x - y)
    3. Ø1 (x + y) + Ø2 (x2 - y)
    4. Ø1 (x2 + y) + Ø2 (x2 - y) - (1/4) x(x + y)
    5. --- Content provided by FirstRanker.com ---

    Correct Answer :-

    • (1/4) x(x - y)2 + Ø1 (x + y) + Ø2 (x - y)

  25. The general solution of ∂2u/∂t2 = c22u/∂x2 with u(0,t) = u(2,t) = 0, u(x, 0) = sin3(πx/2) and ut(x, 0) = 0 is

    1. (3/4) sin(πx/2) sin(πct/2) - (1/4) sin(3πx/2) sin(3πct/2)
    2. --- Content provided by FirstRanker.com ---

    3. (3/4) sin(πx/2) cos(πct/2) - (1/4) sin(3πx/2) cos(3πct/2)
    4. (3/4) cos(πx/2) sin(πct/2) - (1/4) cos(3πx/2) sin(3πct/2)
    5. (3/4) sin(πx/2) cos(πct/2) - (1/4) cos(3πx/2) sin(3πct/2)
  26. Let f: R2 → R be given by f(x) = {(x2 + y2) ln(x2 + y2), if (x, y) ≠ (0,0); 0, if (x,y) = (0,0)}

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

    Then,

    1. fxy and fyx are continuous at (0, 0), and fxy(0,0) = fyx (0,0)
    2. fxy and fyx are discontinuous at (0, 0), but fxy (0,0) = fyx (0,0)
    3. fxy and fyx are continuous at (0, 0), but fxy (0,0) ≠ fyx (0,0)
    4. fxy and fyx are discontinuous at (0, 0) and fxy(0,0) ≠ fyx (0,0)
    5. --- Content provided by FirstRanker.com ---

    Correct Answer :-

    • fxy and fyx are discontinuous at (0, 0), but fxy (0,0) = fyx (0,0)

  27. The directional derivative of f(x,y,z) = xy2 + yz2 + zx2 defined on R3 along the tangent to the curve x = t, y = t2, z = t3 at the point (1, 1, 1) is

    1. 18/√14
    2. --- Content provided by FirstRanker.com ---

    3. 13/√14
    4. 13/√14
    5. 18/√14

    Correct Answer :-

    • 18/√14

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

  28. The unique polynomial of degree 2 passing through (1, 1), (3, 27) and (4, 64) obtained by Lagrange interpolation is

    1. 8x2-17x + 12
    2. 8x2-19x- 12
    3. 8x2+14x- 12
    4. --- Content provided by FirstRanker.com ---

    5. 8x2-19x + 12

    Correct Answer :-

    • 8x2-19x + 12

  29. The approximate value of ∫01 dx/(1+x)2 by Simpson's 1/3-rd rule, using the least number of equal subintervals, is

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

    1. 0.8512
    2. 0.8125
    3. 0.7625
    4. 0.6702

    Correct Answer :-

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

    • 0.8512

  30. Consider the differential equation, dy/dx = y - x, y(0) = 2. The absolute value of the difference in the solutions obtained by Euler method and Runge-Kutta second order method at y(0.1) using step size 0.1 is

    1. 2.205
    2. 2.252
    3. --- Content provided by FirstRanker.com ---

    4. 0.005
    5. 0.055

    Correct Answer :-

    • 0.005

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

  32. The approximate value of (17)1/3 obtained after two iterations of Newton-Raphson method starting with initial approximation x0 = 2 is

    1. 2.7566
    2. 2.5826
    3. 2.6713
    4. 2.4566
    5. --- Content provided by FirstRanker.com ---

    Correct Answer :-

    • 2.5826

  33. For an infinite discrete metric space (X, d), which of the following statements is correct?

    1. x is compact
    2. --- Content provided by FirstRanker.com ---

    3. For every A ⊆ X, A° ∪ Ā = X, where A and A° denote respectively the closure and interior of A in X
    4. x is connected
    5. x is not totally bounded

    Correct Answer :-

    • y is not totally bounded

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

  34. Consider the metric space (l2,d) of square summable sequences with the Euclidean metric. Y = {e1, e2, ...} ⊆ l2 where ei is the sequence of 1 at the i – th place and 0 elsewhere. Then,

    1. y is not compact and has no limit point
    2. y is compact and each ei is a limit point of y
    3. y is not compact and has a limit point
    4. --- Content provided by FirstRanker.com ---

    5. y is compact and has no limit point

    Correct Answer :-

    • y is not compact and has no limit point

  35. Let C[0, 1] be the set of real valued continuous functions on [0, 1] with sup-metric. Let A = {f ∈ C[0, 1] | f(0) = 0} and B = {f ∈ C[0, 1] | f(0) > 0} be the subspaces of C[0, 1]. Then,

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

    1. Both A and B are complete
    2. A is complete but B is incomplete
    3. A is incomplete but B is complete
    4. Neither A nor B is complete

    Correct Answer :-

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

    • A is complete but B is incomplete

  36. Let (R, d) and (R, u) be the metric spaces with the discrete metric space d and usual metric u respectively. Let f: (R, d) → (R, u) and g: (R, u) → (R, d) be the functions given by f(x) = g(x) = {x + 1, x ≤ 0; x > 0}

    Then,

    1. Both f and g are continuous
    2. --- Content provided by FirstRanker.com ---

    3. Neither f nor g is continuous
    4. f is continuous but g is not
    5. g is continuous but f is not

    Correct Answer :-

    • f is continuous but g is not

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

  37. Let Y1 = {(x,y) ∈ R2 | y = sin(1/x), 0 < x ≤ π} and Y2 = {(0, y) ∈ R2 | y ∈ [-2,2]} be subspaces of the metric space (R2) being the Euclidean metric. For any A ⊆ R2, Ā denotes the closure of A

    1. Y1 ∪ Y2 is connected
    2. Y1 ∪ Y2 is connected
    3. Y1 ∩ Y2 is disconnected
    4. --- Content provided by FirstRanker.com ---

    5. Y1 ∩ Y2 is a non-empty bounded subset of R2

    Correct Answer :-

    • Y1 ∪ Y2 is connected

  38. Let R be the set of all real-valued Riemann integrable functions on and let f be the function given by f(x) = {0 if x = 0; 1/(n+1) if 1/(n+1) < x ≤ 1/n for n ∈ N}

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

    Which of the following statements is correct?

    1. f is monotonically decreasing on [0, 1] but f ∉ R[0, 1]
    2. f is monotonically decreasing on [0, 1] and f ∈ R[0, 1]
    3. f is discontinuous at infinitely many points in [0, 1] but f ∉ R[0, 1]
    4. f is discontinuous at infinitely many points in [0, 1] and f ∈ R[0, 1]
    5. --- Content provided by FirstRanker.com ---

    Correct Answer :-

    • f is discontinuous at infinitely many points in [0, 1] and f ∈ R[0, 1]

  39. The improper integral ∫-∞ dx/(x2+1)

    1. Converges to π
    2. --- Content provided by FirstRanker.com ---

    3. Converges to π/2
    4. Converges to 0
    5. Diverges

    Correct Answer :-

    • Converges to π

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

  40. Consider the functions f(x) = (x2-1)/(x-1) and g(x) = |x-1|/(x-1), x ≠ 1. Then

    1. Both f and g have removable discontinuity at x = 1
    2. f has a removable discontinuity at x = 1, while g has a jump discontinuity at x = 1
    3. f has a jump discontinuity at x = 1 while g has a removable discontinuity at x = 1
    4. --- Content provided by FirstRanker.com ---

    Correct Answer :-

    • f has a removable discontinuity at x = 1, while g has a jump discontinuity at x = 1

  41. What is the length of the interval on which the function f(x) = x3 - 6x2 + 15x + 8 is decreasing?

    1. 8
    2. --- Content provided by FirstRanker.com ---

    3. 6
    4. 4
    5. 2

    Correct Answer :-

    • 6

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

  42. Let f: [a, b] → R be a monotonic function. Consider the following statements:

    a. The function f obeys the maximum principle

    b. The function f is Riemann integrable on [a, b]

    Which of the above statement(s) is (are) true?

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

    1. Only (a)
    2. Only (b)
    3. Both (a) and (b)
    4. Neither (a) nor (b)

    Correct Answer :-

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

    • Both (a) and (b)

  43. Consider the following:

    a. ((a,b), (c, d)) = ac - bd, (a, b), (c, d) ∈ R2

    b. (f(x), g(x)) = ∫01 f'(x)g(x) dx, where f(x), g(x) are polynomials over R

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

    Which of the above is(are) an inner product?

    1. Neither (a) nor (b)
    2. Both (a) and (b)
    3. Only (a)
    4. Only (b)
    5. --- Content provided by FirstRanker.com ---

  44. Let T = ((0,1), (1,0), (0,0), (0,2)). Then T3 + 4T2 + 5T - 2I is equal to

    1. 10T +4I
    2. 10T-4I
    3. -10T +4I
    4. --- Content provided by FirstRanker.com ---

    5. -10T-4I

    Correct Answer :-

    • 10T-4I

  45. Let V be an infinite dimensional vector space over a field F.

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

    Consider the following statements:

    a. Any one-one linear transformation from V to itself is onto

    b. Any onto linear transformation from V to itself must be one-one

    Which of the above statements is (are) correct?

    1. Both (a) and (b)
    2. --- Content provided by FirstRanker.com ---

    3. Only (a)
    4. Only (b)
    5. Neither (a) nor (b)

    Correct Answer :-

    • Neither (a) nor (b)

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

  46. Let Pn(R) be the set of all polynomials over R of degree at most n. Let T: Pn(R) → Pn+1(R) be given by T(f(x)) = xf(x). Then

    1. T is one-one and onto linear transformation
    2. T is an onto function but neither a linear transformation nor one-one
    3. T is not onto but a one-one linear transformation
    4. --- Content provided by FirstRanker.com ---

    5. T is one-one but neither a linear transformation nor onto

    Correct Answer :-

    • T is not onto but a one-one linear transformation

  47. Let Z is the set of integers. The inverse of a is

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

    1. a-6
    2. a-4
    3. 4-a
    4. 6-a

    Correct Answer :-

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

    • 4-a

  48. Let G be a group of even order. Suppose that exactly half of G consists of elements of order 2 and the rest forms a subgroup H of G. Which of the following statements is incorrect?

    1. H is a normal subgroup of G
    2. Order of H is even
    3. --- Content provided by FirstRanker.com ---

    4. H is abelian
    5. G: H = 2

    Correct Answer :-

    • Order of H is even

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

  50. Let G and K be finite groups such that |G| = 21 and |K| = 49. Suppose G does not have a normal subgroup of order 3. Let be the set of all group homomorphism from G to K. Then the number of elements in is

    1. 1
    2. 3
    3. 5
    4. 7
    5. --- Content provided by FirstRanker.com ---

    Correct Answer :-

    • 1

  51. Let G be a finite group of a ∈ G has exactly two conjugates. Suppose that C(a) = {x-1ax | x ∈ G} and N(a) = {x ∈ G | ax = xa}.

    Which of the following statements is incorrect?

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

    1. The number of elements in C(a) is a prime number
    2. G is a simple group
    3. N(a) is a subgroup of G

    Correct Answer :-

    • G is a simple group

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

  52. Let G be a finite group of order 385. Let H, K and L be p-Sylow subgroups of G for p = 5,7 and 11, respectively. Which of the following statements is incorrect?

    1. K is a normal subgroup of G
    2. L is normal subgroup of G
    3. HK is a non-abelian subgroup of G
    4. --- Content provided by FirstRanker.com ---

    5. G = HKL

    Correct Answer :-

    • HK is a non-abelian subgroup of G

  53. The remainder when 20202020 is divided by 12 is

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

    1. 0
    2. 2
    3. 4
    4. 8

    Correct Answer :-

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

    • 4

  54. The smallest integer a > 2 such that 2|a, 3|(a+1), 4|(a + 2), 5|(a + 3) and 6|(a + 4) is

    1. 14
    2. 56
    3. --- Content provided by FirstRanker.com ---

    4. 122
    5. 62

    Correct Answer :-

    • 62

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

  56. Let R = {(a, b) | a, b ∈ Z} be a ring and f: R → Z be given by f((a, b)) = a - b. Which of the following statements is incorrect?

    1. f is a ring homomorphism
    2. ker f is a prime ideal but not maximal
    3. ker f is maximal ideal
  57. --- Content provided by FirstRanker.com ---

  58. Consider the following statements

    a. A polynomial is irreducible over a field F if it has no zeros in F

    b. Let f(x) ∈ Z[x]. If f(x) is reducible over Q, then it is reducible over Z

    c. For any prime p, the polynomial xp-1 + xp-2 + ... + x +

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

    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