Q. What is the range of the function y = tan^(-1)(x)?
-
A.
(-π/2, π/2)
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B.
(0, π)
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C.
(0, 1)
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D.
(-1, 1)
Solution
The range of the function y = tan^(-1)(x) is (-π/2, π/2).
Correct Answer: A — (-π/2, π/2)
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Q. What is the rank of the matrix A = [[1, 2, 3], [4, 5, 6], [7, 8, 9]]?
Solution
The rows of A are linearly dependent, hence the rank is 2.
Correct Answer: B — 2
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Q. What is the rank of the matrix [[1, 2, 3], [4, 5, 6], [7, 8, 9]]?
Solution
The rank of the matrix is 2, as the rows are linearly dependent.
Correct Answer: B — 2
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Q. What is the real part of the complex number 5 - 7i?
Solution
The real part of 5 - 7i is 5.
Correct Answer: A — 5
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Q. What is the real part of the complex number z = 2e^(iπ/3)?
Solution
The real part is 2 * cos(π/3) = 2 * 1/2 = 1.
Correct Answer: B — 2
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Q. What is the real part of the complex number z = 4e^(iπ/3)?
Solution
The real part is Re(z) = 4 * cos(π/3) = 4 * 1/2 = 2.
Correct Answer: C — 2√3
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Q. What is the real part of the complex number z = 5 - 2i?
Solution
The real part of z = 5 - 2i is 5.
Correct Answer: A — 5
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Q. What is the real part of the complex number z = 5 - 4i?
Solution
The real part of z = 5 - 4i is 5.
Correct Answer: A — 5
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Q. What is the real part of the complex number z = 5 - 6i?
Solution
The real part of z = 5 - 6i is 5.
Correct Answer: A — 5
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Q. What is the real part of the complex number z = 5e^(iπ/3)?
-
A.
5/2
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B.
5/√3
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C.
5√3/2
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D.
5
Solution
The real part is Re(z) = 5cos(π/3) = 5 * 1/2 = 5/2.
Correct Answer: A — 5/2
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Q. What is the resultant of the vectors (2, 3) and (-1, 4)?
-
A.
(1, 7)
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B.
(3, 1)
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C.
(1, 1)
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D.
(2, 4)
Solution
Resultant = (2 + (-1), 3 + 4) = (1, 7).
Correct Answer: A — (1, 7)
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Q. What is the resultant of vectors (1, 2) and (-1, -2)?
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A.
(0, 0)
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B.
(1, 2)
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C.
(2, 4)
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D.
(1, 1)
Solution
Resultant = (1 - 1, 2 - 2) = (0, 0)
Correct Answer: A — (0, 0)
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Q. What is the scalar product of A = (3, 4, 0) and B = (0, 0, 5)?
Solution
A · B = 3*0 + 4*0 + 0*5 = 0.
Correct Answer: A — 0
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Q. What is the scalar product of the unit vectors i and j?
Solution
i · j = 0, since they are orthogonal.
Correct Answer: B — 0
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Q. What is the scalar product of the vectors (3, 4) and (4, 3)?
Solution
The scalar product is 3*4 + 4*3 = 12 + 12 = 24.
Correct Answer: B — 25
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Q. What is the scalar product of the vectors (4, -3, 2) and (1, 1, 1)?
Solution
Scalar product = 4*1 + (-3)*1 + 2*1 = 4 - 3 + 2 = 3.
Correct Answer: D — 6
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Q. What is the scalar product of the vectors (5, -3) and (-2, 4)?
Solution
Scalar product = 5*(-2) + (-3)*4 = -10 - 12 = -22.
Correct Answer: A — -6
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Q. What is the scalar product of the vectors (5, 5, 5) and (1, 2, 3)?
Solution
Scalar product = 5*1 + 5*2 + 5*3 = 5 + 10 + 15 = 30.
Correct Answer: A — 30
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Q. What is the scalar product of the vectors A = (0, 1, 0) and B = (1, 0, 1)?
Solution
A · B = 0*1 + 1*0 + 0*1 = 0.
Correct Answer: A — 0
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Q. What is the scalar product of the vectors A = (1, 1, 1) and B = (1, 1, 1)?
Solution
A · B = 1*1 + 1*1 + 1*1 = 1 + 1 + 1 = 3.
Correct Answer: C — 3
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Q. What is the scalar product of the vectors A = (1, 2, 3) and B = (4, 5, 6)?
Solution
A · B = 1*4 + 2*5 + 3*6 = 4 + 10 + 18 = 32.
Correct Answer: B — 30
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Q. What is the scalar product of the vectors A = (2, -1, 3) and B = (0, 4, -2)?
Solution
A · B = 2*0 + (-1)*4 + 3*(-2) = 0 - 4 - 6 = -10.
Correct Answer: A — -10
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Q. What is the scalar product of the vectors A = (4, 0, -3) and B = (0, 5, 2)?
Solution
A · B = 4*0 + 0*5 + (-3)*2 = 0 - 6 = -6.
Correct Answer: B — 0
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Q. What is the scalar product of the vectors K = (0, 1, 0) and L = (1, 0, 1)?
Solution
K · L = 0*1 + 1*0 + 0*1 = 0 + 0 + 0 = 0.
Correct Answer: A — 0
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Q. What is the scalar projection of vector (3, 4) onto (1, 0)?
Solution
Scalar projection = (3*1 + 4*0) / √(1^2) = 3
Correct Answer: A — 3
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Q. What is the scalar projection of vector A = (3, 4) onto vector B = (1, 0)?
Solution
Scalar projection = (A · B) / |B| = (3*1 + 4*0) / 1 = 3.
Correct Answer: A — 3
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Q. What is the scalar triple product of the vectors (1, 2, 3), (4, 5, 6), and (7, 8, 9)?
Solution
Scalar triple product = (1, 2, 3) · ((4, 5, 6) × (7, 8, 9)) = 0.
Correct Answer: A — 0
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Q. What is the scalar triple product of vectors A = (1, 0, 0), B = (0, 1, 0), C = (0, 0, 1)?
Solution
Scalar triple product = A · (B × C) = 1.
Correct Answer: A — 1
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Q. What is the scalar triple product of vectors a = (1, 2, 3), b = (4, 5, 6), c = (7, 8, 9)?
Solution
Scalar triple product = a · (b × c). Since b and c are linearly dependent, b × c = 0, hence the scalar triple product is 0.
Correct Answer: A — 0
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Q. What is the second derivative of f(x) = e^x sin(x)?
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A.
e^x (sin(x) + cos(x))
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B.
e^x (2sin(x) + cos(x))
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C.
e^x (sin(x) - cos(x))
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D.
e^x (sin(x) + 2cos(x))
Solution
Using the product rule and chain rule, the second derivative is f''(x) = e^x (sin(x) + cos(x)).
Correct Answer: A — e^x (sin(x) + cos(x))
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