Q. If the vector a = (2, -1) and b = (1, 3), what is the cross product a × b?
Solution
Cross product in 2D = a1*b2 - a2*b1 = 2*3 - (-1)*1 = 6 + 1 = 7
Correct Answer: A — 5
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Q. If the vector a = (2, 2) and b = (2, -2), what is the angle between them?
-
A.
90 degrees
-
B.
45 degrees
-
C.
0 degrees
-
D.
180 degrees
Solution
Angle = cos⁻¹((a·b) / (|a||b|)) = cos⁻¹(0) = 90 degrees
Correct Answer: A — 90 degrees
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Q. If the vector a = (2, 2) is scaled by a factor of 3, what is the resulting vector?
-
A.
(6, 6)
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B.
(3, 3)
-
C.
(2, 2)
-
D.
(1, 1)
Solution
Scaled vector = 3 * a = 3 * (2, 2) = (6, 6)
Correct Answer: A — (6, 6)
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Q. If the vector a = (2, 3) and b = (4, 1), what is the cross product a × b?
Solution
Cross product a × b = 2*1 - 3*4 = 2 - 12 = -10.
Correct Answer: A — -10
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Q. If the vector a = (2, 3) and b = (4, 1), what is the resultant vector a + b?
-
A.
(6, 4)
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B.
(2, 4)
-
C.
(4, 2)
-
D.
(6, 2)
Solution
Resultant vector a + b = (2+4, 3+1) = (6, 4).
Correct Answer: A — (6, 4)
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Q. If the vector A = (2, 3) is multiplied by 2, what is the resulting vector?
-
A.
(4, 6)
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B.
(2, 3)
-
C.
(1, 1.5)
-
D.
(0, 0)
Solution
Resulting vector = 2 * A = 2 * (2, 3) = (4, 6).
Correct Answer: A — (4, 6)
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Q. If the vector A = (2, 3) is reflected across the line y = x, what is the resulting vector?
-
A.
(3, 2)
-
B.
(2, 3)
-
C.
(0, 0)
-
D.
(1, 1)
Solution
Reflection across y = x gives vector (3, 2).
Correct Answer: A — (3, 2)
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Q. If the vector A = (2, 3) is scaled by a factor of 2, what is the resulting vector?
-
A.
(4, 6)
-
B.
(2, 3)
-
C.
(1, 1.5)
-
D.
(0, 0)
Solution
Scaled vector = 2 * A = 2 * (2, 3) = (4, 6).
Correct Answer: A — (4, 6)
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Q. If the vector a = (2, 3, 4) and b = (1, 0, -1), what is a + b?
-
A.
(3, 3, 3)
-
B.
(1, 3, 3)
-
C.
(2, 3, 3)
-
D.
(2, 3, 5)
Solution
a + b = (2+1, 3+0, 4-1) = (3, 3, 3).
Correct Answer: A — (3, 3, 3)
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Q. If the vector a = (2, 3, 4) and b = (1, 0, -1), what is the scalar triple product a · (b × a)?
Solution
The scalar triple product is 0 because a · (b × a) = 0.
Correct Answer: A — 0
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Q. If the vector a = (2, 3, 4) is scaled by a factor of 2, what is the resulting vector?
-
A.
(4, 6, 8)
-
B.
(2, 3, 4)
-
C.
(1, 1.5, 2)
-
D.
(0, 0, 0)
Solution
Scaling the vector a by 2 gives (2*2, 2*3, 2*4) = (4, 6, 8).
Correct Answer: A — (4, 6, 8)
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Q. If the vector a = (3, 4) and b = (1, 2), find the cross product a × b.
Solution
In 2D, a × b = a1*b2 - a2*b1 = 3*2 - 4*1 = 6 - 4 = 2.
Correct Answer: A — -2
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Q. If the vector a = (3, 4) is scaled by a factor of 2, what is the new vector?
-
A.
(6, 8)
-
B.
(3, 4)
-
C.
(1.5, 2)
-
D.
(0, 0)
Solution
New vector = 2 * (3, 4) = (6, 8).
Correct Answer: A — (6, 8)
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Q. If the vector a = (3, 4, 0) and b = (0, 0, 5), what is the magnitude of a × b?
Solution
Magnitude of a × b = |a||b|sin(90) = |(3, 4, 0)|| (0, 0, 5)| = 5√(3^2 + 4^2) = 15.
Correct Answer: A — 15
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Q. If the vector A = (a, b) is perpendicular to B = (b, -a), what is the relationship between a and b?
-
A.
a = b
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B.
a = -b
-
C.
a + b = 0
-
D.
a - b = 0
Solution
A·B = ab - ab = 0, hence A and B are perpendicular if a = -b.
Correct Answer: B — a = -b
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Q. If the vectors A = (1, 2) and B = (2, 1) are given, what is the angle between them?
-
A.
90 degrees
-
B.
45 degrees
-
C.
60 degrees
-
D.
30 degrees
Solution
Cosine of angle θ = (A · B) / (|A| |B|) = (1*2 + 2*1) / (√5 * √5) = 4/5, θ = cos⁻¹(4/5).
Correct Answer: B — 45 degrees
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Q. If the vectors A = (2, 3) and B = (4, 5) are given, what is the scalar product A · B?
Solution
A · B = 2*4 + 3*5 = 8 + 15 = 23.
Correct Answer: C — 20
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Q. If the vectors A = (3, -2, 1) and B = (k, 4, -2) are orthogonal, find the value of k.
Solution
A · B = 3k - 8 - 2 = 0; 3k - 10 = 0; k = 10/3.
Correct Answer: A — -1
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Q. If the vectors A = (x, 2, 3) and B = (4, y, 6) are orthogonal, what is the value of y?
Solution
A · B = x*4 + 2*y + 3*6 = 0. Thus, 4x + 2y + 18 = 0. If x = 0, then y = -9. If x = 1, y = -10. The only integer solution is y = 3.
Correct Answer: B — 3
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Q. If the vectors A = (x, 2, 3) and B = (4, y, 6) are orthogonal, what is the value of x + y?
Solution
A · B = x*4 + 2*y + 3*6 = 0. Thus, 4x + 2y + 18 = 0. Solving gives x + y = -9/2, which is not an option. Correcting gives x + y = 0.
Correct Answer: A — -2
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Q. If the vertex of the parabola y = ax^2 + bx + c is at (1, -2), what is the value of a if b = 4 and c = -6?
Solution
The vertex form of a parabola is given by x = -b/(2a). Here, 1 = -4/(2a) => 2a = -4 => a = -2.
Correct Answer: A — 1
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Q. If two coins are tossed, what is the probability of getting at least one head?
-
A.
1/4
-
B.
1/2
-
C.
3/4
-
D.
1
Solution
Total outcomes = 4 (HH, HT, TH, TT). At least one head = 3 outcomes. Probability = 3/4.
Correct Answer: C — 3/4
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Q. If two dice are rolled, what is the probability of getting a sum of 7?
-
A.
1/6
-
B.
1/12
-
C.
1/36
-
D.
1/18
Solution
Possible combinations for sum of 7: (1,6), (2,5), (3,4), (4,3), (5,2), (6,1) = 6 combinations. Total outcomes = 36. Probability = 6/36 = 1/6.
Correct Answer: A — 1/6
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Q. If u = (1, 2) and v = (3, 4), what is the dot product u · v?
Solution
Dot product u · v = 1*3 + 2*4 = 3 + 8 = 11.
Correct Answer: A — 10
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Q. If u = (1, 2) and v = (3, 4), what is u + v?
-
A.
(4, 6)
-
B.
(2, 3)
-
C.
(1, 2)
-
D.
(3, 4)
Solution
u + v = (1 + 3, 2 + 4) = (4, 6)
Correct Answer: A — (4, 6)
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Q. If u = (1, 2, 3) and v = (4, 5, 6), what is the dot product u · v?
Solution
Dot product u · v = 1*4 + 2*5 + 3*6 = 4 + 10 + 18 = 32.
Correct Answer: B — 27
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Q. If u = (2, 3, 1) and v = (1, 0, -1), find the dot product u · v.
Solution
u · v = 2*1 + 3*0 + 1*(-1) = 2 + 0 - 1 = 1.
Correct Answer: A — 5
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Q. If U = {1, 2, 3, 4, 5}, A = {1, 2}, and B = {2, 3}, what is A ∪ B?
-
A.
{1, 2}
-
B.
{1, 2, 3}
-
C.
{1, 2, 3, 4, 5}
-
D.
{2, 3, 4, 5}
Solution
The union A ∪ B includes all elements from both sets, which are {1, 2, 3}.
Correct Answer: B — {1, 2, 3}
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Q. If vector A = (1, 2, 3) and vector B = (4, 5, 6), what is A + B?
-
A.
(5, 7, 9)
-
B.
(4, 5, 6)
-
C.
(1, 2, 3)
-
D.
(0, 0, 0)
Solution
A + B = (1+4, 2+5, 3+6) = (5, 7, 9).
Correct Answer: A — (5, 7, 9)
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Q. If vector A = (1, 2, 3) and vector B = (4, 5, 6), what is the angle between them?
-
A.
0 degrees
-
B.
30 degrees
-
C.
60 degrees
-
D.
90 degrees
Solution
Cosine of angle θ = (A . B) / (|A| |B|) = (1*4 + 2*5 + 3*6) / (√14 * √77) = 0, hence θ = 90 degrees.
Correct Answer: D — 90 degrees
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