Questions and answers for exam Preparation

MCQs

Total Questions : 69 | Page 6 of 7 pages

Question 51. The number of non-isomorphic abelian groups of order 24 is ______

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Answer: Option A. -> 3

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Question 52. Suppose X is a real-valued random variable. Which of the following values CANNOT be attained by E[X] and E[X2], respectively?

0 and 1
2 and 3
1/2 and 1/3
2 and 5

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Answer: Option B. -> 2 and 3

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Question 53. Consider the following linear programming problem: Maximize x + 3y + 6z - wsubject to 5x + y + 6z + 7w â‰¤ 20, 6x + 2y + 2z + 9w â‰¤ 40, x â‰¥ 0, y â‰¥ 0, z â‰¥ 0, w â‰¥ 0.Then the optimal value is ______

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Answer: Option D. -> 60

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Question 54. Which of the following subsets of â„2 is NOT compact?

{(x,y) âˆˆ â„2 : -1 â‰¤ x â‰¤ 1, y = sinx}
{(x,y) âˆˆ â„2 : -1 â‰¤ y â‰¤ 1, y = x8 - x3 - 1}
{(x,y) âˆˆ â„2 : y = 0, sin(ex) = 0}
{(x,y) âˆˆ â„2 : x > 0, y = sin(1/x)}â‹‚{(x,y) âˆˆ â„2 : x > 0, y = 1/x}

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Answer: Option C. -> {(x,y) âˆˆ â„2 : y = 0, sin(ex) = 0}

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Question 55. Let â„‹ be a Hilbert space and let {en : n â‰¥ 1} be an orthonormal basis of â„‹. Suppose T:â„‹ â†’ â„‹ is a bounded linear operator. Which of the following CANNOT be true?

T(en) = e1 for all n â‰¥ 1
T(en) = en+1 for all n â‰¥ 1
T(en) = √(n+1)/n en for all n â‰¥ 1
T(en) = en-1 for all n â‰¥ 2 and T(e1) = 0

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Answer: Option A. -> T(en) = e1 for all n â‰¥ 1

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Question 56. Let f : â„‚\{3i}â†’â„‚ be defined by f(z) = (z-i)/(iz+3). Which of the following statements about f is FALSE?

f is conformal on â„‚\{3i}
f maps circles in â„‚\{3i} onto circles in â„‚
All the fixed points of f are in the region {z âˆˆ â„‚âˆ¶Im(z)>0}
There is no straight line in â„‚\{3i} which is mapped onto a straight line in â„‚ by f

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Answer: Option C. -> All the fixed points of f are in the region {z âˆˆ â„‚âˆ¶Im(z)>0}

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Question 57. The expression 1/(Dx2 - Dy2) sin(x - y) is equal to

-x/2 cos(x - y)
-x/2 sin(x - y) + cos(x - y)
-x/2 cos(x - y) + sin(x - y)
3x/2 sin(x - y)

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Answer: Option A. -> -x/2 cos(x - y)

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Question 58. 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

1/f2
xf
1/2f
f2

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Answer: Option D. -> f2

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Question 59. If a random variable X assumes only positive integral values, with the probability P(X = x) = 2/3 (1/3)x-1, x = 1,2,3,... then E(X) is

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Answer: Option D. -> 3/2

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Question 60. The order of the smallest possible non trivial group containing elements x and y such that x7 = y2 = e and yx = x4y is

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Answer: Option B. -> 2

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