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GATE Mathematics - Practice Test Paper 1
GATE Mathematics - Practice Test Paper 2
  1. For each λ > 0, let Xλ be a random variable with exponential density λe-λx on(0,∞). Then, Var(log Xλ)
    2. A.
    is strictly increasing in λ
    B.
    is strictly decreasing in λ
    C.
    does not depend on λ
    D.
    first increases and then decreases in λ

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  2. Let {an} be the sequence of consecutive positive solutions of the equation tan x = x and let {bn} be the sequence of consecutive positive solutions of the equation tan √x = x. Then
    3. A.
    .
    B.
    .
    C.
    .
    D.
    .

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  3. Let f be an analytic function on Then, which of the following is NOT a possible value of (ef)''(0)?
    4. A.
    2
    B.
    6
    C.
    √2
    D.
    √2 + i√2

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  4. The number of non-isomorphic abelian groups of order 24 is ______
    5. A.
    3
    B.
    6
    C.
    12
    D.
    24

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  5. Let V be the real vector space of all polynomials in one variable with real coefficients and having degree at most 20. Define the subspaces

    Then the dimension of W1∩W2 is ______
    6. A.
    5
    B.
    10
    C.
    15
    D.
    20

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  6. Let f, g : [0,1] →ℝ be defined by

    then
    7. A.
    Both f and g are Riemann integrable
    B.
    f is Riemann integrable and g is Lebesgue integrable
    C.
    g is Riemann integrable and f is Lebesgue integrable
    D.
    Neither f nor g is Riemann integrable

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  7. Consider the following linear programming problem: 
    Maximize             x + 3y + 6z - w
    subject to            5x + y + 6z + 7w â‰¤ 20,
                                  6x + 2y + 2z + 9w â‰¤ 40,
                                  x â‰¥ 0, y ≥ 0, z ≥ 0, w ≥ 0.
    Then the optimal value is ______
    8. A.
    20
    B.
    40
    C.
    50
    D.
    60

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  8. Suppose X is a real-valued random variable. Which of the following values CANNOT be attained by E[X] and E[X2], respectively?
    9. A.
    0 and 1
    B.
    2 and 3
    C.
    1/2 and 1/3
    D.
    2 and 5

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  9. Which of the following subsets of ℝ2 is NOT compact?
    10. A.
    {(x,y) ∈ ℝ2 : -1 ≤ x ≤ 1, y = sinx}
    B.
    {(x,y) ∈ ℝ2 : -1 ≤ y ≤ 1, y = x8 - x3 - 1}
    C.
    {(x,y) ∈ ℝ2 : y = 0, sin(ex) = 0}
    D.
    {(x,y) ∈ ℝ2 : x > 0, y = sin(1/x)}⋂{(x,y) ∈ ℝ2 : x > 0, y = 1/x}

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  10. Let ℋ be a Hilbert space and let {en : n ≥ 1} be an orthonormal basis of ℋ. Suppose T:ℋ → ℋ is a bounded linear operator. Which of the following CANNOT be true?
    11. A.
    T(en) = e1 for all n ≥ 1
    B.
    T(en) = en+1 for all n ≥ 1
    C.
    T(en) = √(n+1)/n en for all n ≥ 1
    D.
    T(en) = en-1 for all n ≥ 2 and T(e1) = 0

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