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GATE 2017-2018 :: GATE Mathematics

  1. In a topological space, which of the following statements is NOT always true :
  2. A.
    Union of any finite family of compact sets is compact.
    B.
    Union of any family of closed sets is closed.
    C.
    Union of any family of connected sets having a non empty intersection is connected.
    D.
    Union of any family of dense subsets is dense.

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  3. Consider the following statements: 
    P: The family of subsets {An = (-1/n, 1/n), n = 1, 2, ...} satisfies the finite intersection property. 
    Q: On an infinite set X , a metric d : X * X --> R is defined as d(x,y) = [0 , x = y and 1, x â‰  y] The metric space (X,d) is compact. 
    R: In a Frechet (T1) topological space, every finite set is closed. 
    S: If f : R --> X is continuous, where R is given the usual topology and (X, t) is a Hausdorff (T2) space, then f is a one-one function. 
    Which of the above statements are correct?
  4. A.
    P and R
    B.
    P and S
    C.
    R and S
    D.
    Q and S

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  5. A simple random sample of size 10 from 2 N(μ,σ2) gives 98% confidence interval (20.49, 23.51). Then the null hypothesis H0 : μ = 20.5 against HA : μ ≠ 20.5
  6. A.
    can be rejected at 2% level of significance
    B.
    cannot be rejected at 5% level of significance
    C.
    can be rejected at 10% level of significance
    D.
    cannot be rejected at any level of significance

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  7. For the linear programming problem 
    Maximize z = x1 + 2x2 + 3x3 - 4x4 
    Subject to      2x1 + 3x2 - x3 - x4 = 15  
                           6x1 + x2 + x3 - 3x4 = 21 
                           8x1 + 2x2 + 3x3 - 4x4 = 30 
                           x1, x2, x3, x4 ≥ 0, 
    x1 = 4, x2 = 3, x3 = 0, x4 = 2 is
  8. A.
    an optimal solution
    B.
    a degenerate basic feasible solution
    C.
    a non-degenerate basic feasible solution
    D.
    a non-basic feasible solution

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  9. Which one of the following statements is TRUE?
  10. A.
    A convex set cannot have infinite many extreme points.
    B.
    A linear programming problem can have infinite many extreme points.
    C.
    A linear programming problem can have exactly two different optimal solutions.
    D.
    A linear programming problem can have a non-basic optimal solution.

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  11. Let V = ℂ2 be the vector space over the field of complex numbers and B{(1, i), (i,1)}be a given ordered basis of V. Then for which of the following, B* = {f1, f2}is a dual basis of B over ℂ?
  12. A.
    f1(z1, z2) = 1/2 (z1 - iz2), f2(z1, z2) = 1/2 (z1 + iz2)
    B.
    f1(z1, z2) = 1/2 (z1 + iz2), f2(z1, z2) = 1/2 (iz1 + z2)
    C.
    f1(z1, z2) = 1/2 (z1 - iz2), f2(z1, z2) = 1/2 (-iz1 + z2)
    D.
    f1(z1, z2) = 1/2 (z1 + iz2), f2(z1, z2) = 1/2 (-iz1 - z2)

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  13. Let R = ℤ*ℤ*ℤ and I = ℤ*ℤ*{0}. Then which of the following statement is correct?
  14. A.
    I is a maximal ideal but not a prime ideal of R .
    B.
    I is a prime ideal but not a maximal ideal of R .
    C.
    I is both maximal ideal as well as a prime ideal of R .
    D.
    I is neither a maximal ideal nor a prime ideal of R .

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  15. The function u(r, θ) satisfying the Laplace equation

    subject to the conditions u(e, θ) = 1, u(e2 ,θ) = 0 is
  16. A.
    ln(e/r)
    B.
    ln(e/r2)
    C.
    ln(e2/r)
    D.
    .

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  17. The functional

    is path independent if k equals
  18. A.
    1
    B.
    2
    C.
    3
    D.
    4

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  19. If a transformation y = uv transforms the given differential equation 
    f(x)y'' - 4f'(x)y' + g(x)y = 0 into the equation of the form v'' + h(x)v = 0, then u must be
  20. A.
    1/f2
    B.
    xf
    C.
    1/2f
    D.
    f2

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