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Vectors, matrices and complex numbers
8 NET Mathematics questions with answers and explanations.
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What is the eccentricity of the ellipse x^2 / 25 + y^2 / 16 = 1?
- 1 3/5
- 2 4/5
- 3 5/4
- 4 9/25
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Correct answer: 3/5
Here a^2 = 25 and b^2 = 16. e = sqrt(1 - b^2 / a^2) = sqrt(9/25) = 3/5.
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What is the angle between the vectors (1, 0) and (1, 1)?
- 1 30 degrees
- 2 45 degrees
- 3 60 degrees
- 4 90 degrees
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Correct answer: 45 degrees
cos(theta) = (1 x 1 + 0 x 1) / (1 x sqrt(2)) = 1 / sqrt(2), so theta = 45 degrees.
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What is the magnitude of the cross product of a = (2, 0, 0) and b = (0, 3, 0)?
- 1 5
- 2 6
- 3 0
- 4 1
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Correct answer: 6
The vectors are perpendicular, so |a x b| = |a||b| = 2 x 3 = 6.
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What are the eigenvalues of the matrix [[2, 0], [0, 3]]?
- 1 0 and 5
- 2 1 and 6
- 3 2 and 3
- 4 2 and -3
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Correct answer: 2 and 3
The eigenvalues of a diagonal matrix are its diagonal entries.
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If A is a 2 x 2 matrix with det A = 3, what is det(2A)?
- 1 6
- 2 12
- 3 9
- 4 24
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Correct answer: 12
For a 2 x 2 matrix, det(kA) = k^2 det(A) = 4 x 3 = 12.
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What is (1 + i) / (1 - i)?
- 1 1
- 2 -1
- 3 -i
- 4 i
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Correct answer: i
Multiply top and bottom by (1 + i): (1 + i)^2 / 2 = 2i / 2 = i.
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What is the argument of the complex number -1?
- 1 0
- 2 pi / 2
- 3 pi
- 4 -pi / 2
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Correct answer: pi
The number -1 lies on the negative real axis, at an angle of pi from the positive real axis.
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What is the sum of the three cube roots of unity?
- 1 1
- 2 0
- 3 -1
- 4 3
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Correct answer: 0
The roots of z^3 = 1 sum to the negative of the coefficient of z^2, which is 0.