q

Topic:- MATHS MA S2

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

   (a) lim inf x_n = lim inf y_n

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

   (b) lim sup x_n = lim sup y_n

   Which of the above statements is(are) correct?

   1. Neither (a) nor (b)
   2. Only (a)
   3. Only (b)

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

   4. Both (a) and (b)

   Correct Answer :-

   • Both (a) and (b)

2. Which of the sequences {a_n} and {b_n} of real numbers with n^th terms

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

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

   b_n = 2 cosn - sin n

   has(have) convergent subsequences?

   1. Neither {a_n} nor {b_n}
   2. Only {a_n}

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

   3. Only {b_n}
   4. Both {a_n} and {b_n}

   Correct Answer :-

   • Both {a_n} and {b_n}

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

3. Consider the following series:

   (a) ? xⁿ / n!, x ? R, n=1 to 8

   (b) ? 1 / (n + sinn), n=1 to 8

   (c) ? 1 / (n²vn), n=1 to 8

   (d) ? sin n, n=1 to 8

   --- 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)

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

   Correct Answer :-

   • Only (a) and (c)

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

   1. uncountable

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

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

   Correct Answer :-

   • need not be closed

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

5. Consider the series ? a_n, where a_n = (2 + sin(np/2))ⁿrⁿ, r > 0. What are the values of lim inf (a_n+1/a_n) and lim sup (a_n+1/a_n)?

   1. r/2 and 2r
   2. r/3 and r
   3. 2r/3 and 3r/2

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

   4. 0 and 1

   Correct Answer :-

   • r/2 and 2r

6. Consider the following series:

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

   (a) ? 3^-n sin 3ⁿx on R, n=1 to 8

   (b) ? n^-2xⁿ on (-2,2), n=1 to 8

   (c) ? (1/n) cosnx on R, n=1 to 8

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

   1. Only (a) and (b)

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

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

   Correct Answer :-

   • Only (a) and (c)

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

7. Let {f_n} 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 f_n' exists and the sequence {f_n'} 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)

8. Let G(x) be a real-valued function defined by G(x) = ?₀^4x2 cos vt dt. If G' is the derivative of G, then

   1. G'(p/2) = -4p
   2. G'(p/2) = -4p – 1

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

   3. G'(p/2) = -p
   4. G'(p/2) = 0

   Correct Answer :-

   • G'(p/2) = -4p

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

9. 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)

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

   4. Only (d)

   Correct Answer :-

   • Only (d)

10. 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]

11. The Wronskian of cosx, sin x and e^x at x = 0 is

    1. 1
    2. 2

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

    3. -1
    4. -2

    Correct Answer :-

    • 2

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

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

    1. y = cosec(x + p/4)
    2. y = tan(x + p/4)
    3. y = sec(x + p/4)
    4. y = cot(x + p/4)

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

    Correct Answer :-

    • y = tan(x + p/4)

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

    1. No solution

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

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

    Correct Answer :-

    • Infinitely many solutions

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

14. 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)

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

    4. 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)

15. 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

16. 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 + v(z² + a²) + a² ln (z + v(z² + a²) / a) = 0
    2. a²z + by + x² = 0

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

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

    Correct Answer :-

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

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

17. 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)

18. The partial differential equation y u_xx + 2xy u_xy + x u_yy = u_x + u_y, is

    1. Hyperbolic in {(x,y) | 0 < xy < 1}
    2. Hyperbolic in {(x,y) | xy > 1}

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

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

    Correct Answer :-

    • Hyperbolic in {(x,y) | xy > 1}

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

19. 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)

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

    Correct Answer :-

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

20. The general solution of ?²u/?t² = c² ?²u/?x² with u(0,t) = u(2,t) = 0, u(x, 0) = sin³(px/2) and u_t(x, 0) = 0 is

    1. (3/4) sin(px/2) sin(pct/2) - (1/4) sin(3px/2) sin(3pct/2)

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

    2. (3/4) sin(px/2) cos(pct/2) - (1/4) sin(3px/2) cos(3pct/2)
    3. (3/4) cos(px/2) sin(pct/2) - (1/4) cos(3px/2) sin(3pct/2)
    4. (3/4) sin(px/2) cos(pct/2) - (1/4) cos(3px/2) sin(3pct/2)

21. 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. f_xy and f_yx are continuous at (0, 0), and f_xy(0,0) = f_yx (0,0)
    2. f_xy and f_yx are discontinuous at (0, 0), but f_xy (0,0) = f_yx (0,0)
    3. f_xy and f_yx are continuous at (0, 0), but f_xy (0,0) ? f_yx (0,0)
    4. f_xy and f_yx are discontinuous at (0, 0) and f_xy(0,0) ? f_yx (0,0)

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

    Correct Answer :-

    • f_xy and f_yx are discontinuous at (0, 0), but f_xy (0,0) = f_yx (0,0)

22. 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/v14

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

    2. 13/v14
    3. 13/v14
    4. 18/v14

    Correct Answer :-

    • 18/v14

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

23. 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

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

    4. 8x²-19x + 12

    Correct Answer :-

    • 8x²-19x + 12

24. 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

25. 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

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

    3. 0.005
    4. 0.055

    Correct Answer :-

    • 0.005

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

26. 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

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

    Correct Answer :-

    • 2.5826

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

    1. x is compact

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

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

    Correct Answer :-

    • y is not totally bounded

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

28. Consider the metric space (l²,d) of square summable sequences with the Euclidean metric. Y = {e₁, e₂, ...} ? l² where e_i 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 e_i is a limit point of y
    3. y is not compact and has a limit point

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

    4. y is compact and has no limit point

    Correct Answer :-

    • y is not compact and has no limit point

29. 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

30. 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

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

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

    Correct Answer :-

    • f is continuous but g is not

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

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

    1. Y₁ ? Y₂ is connected
    2. Y₁ ? Y₂ is connected
    3. Y₁ n Y₂ is disconnected

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

    4. Y₁ n Y₂ is a non-empty bounded subset of R²

    Correct Answer :-

    • Y₁ ? Y₂ is connected

32. 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]

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

    Correct Answer :-

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

33. The improper integral ?_-8⁸ dx/(x²+1)

    1. Converges to p

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

    2. Converges to p/2
    3. Converges to 0
    4. Diverges

    Correct Answer :-

    • Converges to p

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

34. 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

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

    Correct Answer :-

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

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

    1. 8

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

    2. 6
    3. 4
    4. 2

    Correct Answer :-

    • 6

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

36. 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)

37. 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)

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

38. 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

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

    4. -10T-4I

    Correct Answer :-

    • 10T-4I

39. 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)

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

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

    Correct Answer :-

    • Neither (a) nor (b)

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

40. Let P_n(R) be the set of all polynomials over R of degree at most n. Let T: P_n(R) ? P_n+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

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

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

    Correct Answer :-

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

41. 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

42. 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

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

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

    Correct Answer :-

    • Order of H is even

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

43. 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

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

    Correct Answer :-

    • 1

44. 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 ---

45. 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

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

    4. G = HKL

    Correct Answer :-

    • HK is a non-abelian subgroup of G

46. 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

47. 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

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

    3. 122
    4. 62

    Correct Answer :-

    • 62

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

48. 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

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

49. 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 x^p-1 + x^p-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