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GATE 2017-2018 :: GATE Mathematics

  1. The possible set of eigen values of a 4*4 skew-symmetric orthogonal real matrix is
  2. A.
    {±i}
    B.
    {±i, ±1}
    C.
    {±1}
    D.
    {0 , ±i}

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  3. The coefficient of (z − π)2 in the Taylor series expansion of

    around Ï€ is
  4. A.
    1/2
    B.
    -1/2
    C.
    1/6
    D.
    -1/6

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  5. Consider ℝ2 with the usual topology. Which of the following statements are TRUE for all A,B ⊆ ℝ2?

  6. A.
    P and R only
    B.
    P and S only
    C.
    Q and R only
    D.
    Q and S only

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  7. Let f:ℝ →ℝ be a continuous function with f(1)=5 and f(3)=11. If g(x) =  then g²(0) is equal to ______
  8. A.
    3
    B.
    4
    C.
    5
    D.
    6

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  9. Let P be a 2*2 complex matrix such that trace(P) = 1 and det(P)=−6. Then, trace (P4 - P3) is ______
  10. A.
    78
    B.
    74
    C.
    79
    D.
    72

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  11. Suppose that R is a unique factorization domain and that a,b ∈R are distinct irreducible elements. Which of the following statements is TRUE?
  12. A.
    The ideal 〈1+a〉 is a prime ideal
    B.
    The ideal 〈a+b〉 is a prime ideal
    C.
    The ideal 〈1+ab〉 is a prime ideal
    D.
    The ideal 〈a〉 is not necessarily a maximal ideal

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  13. Let X be a compact Hausdorff topological space and let Y be a topological space. Let f: X --> Y be a bijective continuous mapping. Which of the following is TRUE?
  14. A.
    f is a closed mapbut not necessarily an open map
    B.
    f is an open map but not necessarily a closed map
    C.
    f is both an open map and a closed map
    D.
    f need not be an open map or a closed map

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  15. Consider the linear programming problem:
    Maximize              x + 3/2 y
    subject to             2x + 3y â‰¤ 16
                                    x + 4y â‰¤ 18
                                    x â‰¥ 0, y ≥ 0.
    If 𝑆𝑆 denotes the set of all solutions of the above problem, then
  16. A.
    s is empty
    B.
    s is a singleton
    C.
    s is a line segment
    D.
    s has positive area

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  17. Which of the following groups has a proper subgroup that is NOT cyclic?
  18. A.
    ℤ15*ℤ77
    B.
    S3
    C.
    (ℤ,+)
    D.
    (â„š,+)

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  19. The value of the integral

    is ______
  20. A.
    2
    B.
    4
    C.
    6
    D.
    8

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