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Total Questions : 69 | Page 5 of 7 pages
Question 41. Consider the functions f(z) = (z2 + αx)/(z + 1)2 and g(z) = sinh(z - Ï€/2α), α≠0The residue of f (z) at its pole is equal to 1. Then the value of Î± is
  1.    -1
  2.    1
  3.    2
  4.    3
 Discuss Question
Answer: Option D. -> 3


-NA-


Question 42. The possible set of eigen values of a 4*4 skew-symmetric orthogonal real matrix is
  1.    {±i}
  2.    {±i, ±1}
  3.    {±1}
  4.    {0 , ±i}
 Discuss Question
Answer: Option A. -> {±i}


-NA-


Question 43. Let P be a 2*2 complex matrix such that trace(P) = 1 and det(P)=−6. Then, trace (P4 - P3) is ______
  1.    78
  2.    74
  3.    79
  4.    72
 Discuss Question
Answer: Option A. -> 78


-NA-


Question 44. Let G be a group of order 231. The number of elements of order 11 in G is ______
  1.    10
  2.    15
  3.    20
  4.    25
 Discuss Question
Answer: Option A. -> 10


-NA-


Question 45. Suppose X is a random variable with P(X = k) = (1 - p)kp for k ∈ {0,1,2,..} and some p ∈ (0,1). For the hypothesis testing problem H0:p = 1/2 H1:p ≠ 1/2 consider the test "Reject H0 if X ≤ A or if X ≥ B", where A < B are given positive integers. The type-I error of this test is
  1.    1 + 2-B - 2-A
  2.    1 - 2-B + 2-A
  3.    1 + 2-B - 2-A-1
  4.    1 - 2-B + 2-A-1
 Discuss Question
Answer: Option C. -> 1 + 2-B - 2-A-1


-NA-


Question 46. Let f:ℝ2→ℝ2 be defined by f(x,y) = (ex+y, ex-y). The area of the image of the region {(x,y) ∈ℝ2:0 < x,y < 1} under the mapping f is
  1.    1
  2.    e - 1
  3.    e2
  4.    e2 - 1
 Discuss Question
Answer: Option D. -> e2 - 1


-NA-


Question 47. Let x0 = 0. Define xn+1 = cos xn for every n≥0. Then
  1.    {xn} is increasing and convergent
  2.    {xn} is decreasing and convergent
  3.    {xn} is convergent and x2n < limm-->∞ xm < x2n+1 for every n ∈ â„•
  4.    {xn} is not convergent
 Discuss Question
Answer: Option C. -> {xn} is convergent and x2n < limm-->∞ xm < x2n+1 for every n ∈ ℕ


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Question 48. Which of the following is a field?
  1.    â„‚[x]/〈x2+2〉
  2.    â„¤[x]/〈x2+2〉
  3.    â„š[x]/〈x2-2〉
  4.    â„[x]/〈x2-2〉
 Discuss Question
Answer: Option C. -> ℚ[x]/〈x2-2〉


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Question 49. 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
  1.    .
  2.    .
  3.    .
  4.    .
 Discuss Question
Answer: Option B. -> .


-NA-


Question 50. For each λ > 0, let Xλ be a random variable with exponential density λe-λx on(0,∞). Then, Var(log Xλ)
  1.    is strictly increasing in λ
  2.    is strictly decreasing in λ
  3.    does not depend on λ
  4.    first increases and then decreases in λ
 Discuss Question
Answer: Option C. -> does not depend on λ


-NA-


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