Q. If u = (2, 3, 1) and v = (1, 0, -1), find the dot product u · v.
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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} and A = {1, 2}, what is A'?
A.
{3, 4, 5}
B.
{1, 2}
C.
{1, 2, 3}
D.
{}
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Solution
The complement A' consists of all elements in U that are not in A. Therefore, A' = {3, 4, 5}.
Correct Answer:
A
— {3, 4, 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}
Show solution
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 U is sitting at one end and V is sitting next to U, who is sitting at the other end if W is seated between V and X?
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Solution
The order is U, V, W, X. X is at the other end.
Correct Answer:
D
— X
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Q. If unpolarized light passes through a polarizer, what fraction of the light intensity is transmitted?
A.
0%
B.
25%
C.
50%
D.
100%
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Solution
When unpolarized light passes through a polarizer, 50% of the light intensity is transmitted.
Correct Answer:
C
— 50%
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Q. If unpolarized light passes through a polarizer, what is the intensity of the transmitted light?
A.
Zero
B.
Half of the original intensity
C.
Equal to the original intensity
D.
Twice the original intensity
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Solution
According to Malus's law, the intensity of transmitted light is half of the original intensity when unpolarized light passes through a polarizer.
Correct Answer:
B
— Half of the original intensity
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Q. If unpolarized light passes through a polarizer, what percentage of the light intensity will emerge?
A.
0%
B.
25%
C.
50%
D.
100%
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Solution
When unpolarized light passes through a polarizer, 50% of the light intensity will emerge, according to Malus's law.
Correct Answer:
C
— 50%
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Q. If unpolarized light passes through two polarizers at 90 degrees to each other, what is the intensity of the transmitted light?
A.
Same as incident light
B.
Half of the incident light
C.
Zero
D.
One quarter of the incident light
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Solution
When unpolarized light passes through two polarizers at 90 degrees, no light is transmitted, resulting in zero intensity.
Correct Answer:
C
— Zero
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Q. If unpolarized light passes through two polarizers, what is the maximum intensity of light transmitted?
A.
Zero
B.
Half of the original intensity
C.
Equal to the original intensity
D.
Dependent on the angle between the polarizers
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Solution
The maximum intensity transmitted through two polarizers is given by Malus's law, which states that I = I0 * cos²(θ), where θ is the angle between the polarizers.
Correct Answer:
D
— Dependent on the angle between the polarizers
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Q. If V is scheduled before W and X is scheduled after W, which of the following is correct? (2020)
A.
V is last
B.
W is last
C.
X is last
D.
V is first
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Solution
X is last as it is scheduled after W, which is after V.
Correct Answer:
C
— X is last
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Q. If V is the wife of W and X is the son of W, what is V's relationship to X? (2023)
A.
Mother
B.
Sister
C.
Aunt
D.
Cousin
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Solution
V is the mother of X as she is the wife of X's father, W.
Correct Answer:
A
— Mother
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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)
Show solution
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
Show solution
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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Q. If vector A = (1, 2, 3) and vector B = (4, 5, 6), what is the vector A - B?
A.
(-3, -3, -3)
B.
(3, 3, 3)
C.
(5, 7, 9)
D.
(0, 0, 0)
Show solution
Solution
A - B = (1-4, 2-5, 3-6) = (-3, -3, -3).
Correct Answer:
A
— (-3, -3, -3)
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Q. If vector A = (2, 2, 2) and vector B = (1, 1, 1), what is the scalar triple product A . (B × A)?
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Solution
A . (B × A) = 0, since B × A = 0.
Correct Answer:
A
— 0
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Q. If vector A = (3, -2, 1) and vector B = (1, 4, -3), what is the cross product A × B?
A.
(-5, -10, 14)
B.
(5, 10, -14)
C.
(10, 14, 5)
D.
(14, -5, 10)
Show solution
Solution
A × B = |i j k|\n|3 -2 1|\n|1 4 -3| = (-5, -10, 14).
Correct Answer:
A
— (-5, -10, 14)
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Q. If vector A = 2i + 3j + k and vector B = i - 2j + 4k, what is the cross product A × B?
A.
-10i + 5j + 7k
B.
10i - 5j - 7k
C.
10i + 5j + 7k
D.
-10i - 5j + 7k
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Solution
A × B = |i j k|\n|2 3 1|\n|1 -2 4| = (3*4 - 1*(-2))i - (2*4 - 1*1)j + (2*(-2) - 3*1)k = 10i - 5j - 7k.
Correct Answer:
A
— -10i + 5j + 7k
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Q. If vector A = 2i + 3j and vector B = -i + 4j, what is the resultant vector R = A + B?
A.
i + 7j
B.
i + j
C.
3i + 7j
D.
3i + 4j
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Solution
R = A + B = (2 - 1)i + (3 + 4)j = 1i + 7j.
Correct Answer:
A
— i + 7j
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Q. If vector A = 2i + 3j and vector B = 3i + 4j, what is the angle between them? (2023)
A.
0 degrees
B.
90 degrees
C.
45 degrees
D.
60 degrees
Show solution
Solution
cos(θ) = (A · B) / (|A| |B|) = (2*3 + 3*4) / (√(2^2 + 3^2) * √(3^2 + 4^2)) = 0.6, θ = 53.13 degrees.
Correct Answer:
D
— 60 degrees
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Q. If vector A = 2i + 3j and vector B = 4i + 5j, what is the angle between A and B? (2023)
A.
0 degrees
B.
90 degrees
C.
45 degrees
D.
60 degrees
Show solution
Solution
cos(θ) = (A · B) / (|A||B|) = (8 + 15) / (√(13) * √(41)). θ = 60 degrees.
Correct Answer:
D
— 60 degrees
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Q. If vector A = 2i + 3j and vector B = 4i + 5j, what is the angle between them?
A.
0 degrees
B.
90 degrees
C.
45 degrees
D.
60 degrees
Show solution
Solution
cos(θ) = (A · B) / (|A||B|) = (2*4 + 3*5) / (√(2^2 + 3^2) * √(4^2 + 5^2)) = 0.5, θ = 60 degrees.
Correct Answer:
D
— 60 degrees
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Q. If vector A = 2i + 3j and vector B = 4i + 6j, are the vectors A and B parallel?
A.
Yes
B.
No
C.
Cannot be determined
D.
Only if scaled
Show solution
Solution
Vectors A and B are parallel because B = 2A.
Correct Answer:
A
— Yes
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Q. If vector A = 2i + 3j and vector B = 4i + k, what is the angle between A and B?
A.
90 degrees
B.
60 degrees
C.
45 degrees
D.
30 degrees
Show solution
Solution
cos(θ) = (A · B) / (|A| |B|). A · B = 8 + 0 + 3(0) = 8; |A| = √(2^2 + 3^2) = √13; |B| = √(4^2 + 1^2) = √17. θ = cos^(-1)(8/(√13 * √17)).
Correct Answer:
B
— 60 degrees
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Q. If vector A = 3i + 4j and vector B = 2i - j, what is the dot product A · B?
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Solution
A · B = (3)(2) + (4)(-1) = 6 - 4 = 2.
Correct Answer:
A
— 10
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Q. If vector A = 3i + 4j and vector B = 4i + 3j, what is the cross product A × B?
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Solution
A × B = |i j k| |3 4 0| |4 3 0| = (0 - 0)i - (0 - 0)j + (9 - 12)k = -3k.
Correct Answer:
B
— 1k
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Q. If vector A = 4i + 3j and vector B = -3i + 4j, what is the angle between them?
A.
90 degrees
B.
45 degrees
C.
60 degrees
D.
30 degrees
Show solution
Solution
cos(θ) = (A · B) / (|A||B|) = (4*-3 + 3*4) / (5*5) = 0, θ = 90 degrees.
Correct Answer:
A
— 90 degrees
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Q. If vector A = 4i + 3j and vector B = -i + 2j, what is the resultant vector A + B? (2019)
A.
3i + 5j
B.
5i + j
C.
3i + j
D.
5i + 5j
Show solution
Solution
A + B = (4 - 1)i + (3 + 2)j = 3i + 5j.
Correct Answer:
A
— 3i + 5j
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Q. If vector A = 4i + 3j and vector B = -i + 2j, what is the resultant vector R = A + B? (2019)
A.
3i + 5j
B.
5i + j
C.
3i + j
D.
5i + 5j
Show solution
Solution
R = A + B = (4 - 1)i + (3 + 2)j = 3i + 5j.
Correct Answer:
A
— 3i + 5j
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Q. If vector A = 4i + 3j and vector B = 3i + 4j, what is the angle between A and B?
A.
45 degrees
B.
60 degrees
C.
90 degrees
D.
135 degrees
Show solution
Solution
cos(θ) = (A · B) / (|A||B|) = (12 + 12) / (5 * 5) = 24/25, θ = cos^(-1)(24/25) ≈ 60 degrees.
Correct Answer:
B
— 60 degrees
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Q. If vector A = 4i + 3j and vector B = 4i - 3j, what is the angle between A and B? (2019)
A.
0 degrees
B.
90 degrees
C.
180 degrees
D.
45 degrees
Show solution
Solution
A · B = 16 - 9 = 7. |A| = 5, |B| = 5. cos(θ) = 7/(5*5) = 0.28, θ = 180 degrees.
Correct Answer:
C
— 180 degrees
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