Q. If G = {x, y, z}, how many subsets contain exactly 2 elements?
Solution
The subsets containing exactly 2 elements are {x, y}, {x, z}, and {y, z}. So, there are 3 such subsets.
Correct Answer: A — 3
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Q. If G = {x, y}, what is the number of subsets of G?
Solution
The number of subsets of a set with n elements is 2^n. Here, n=2, so the number of subsets is 2^2 = 4.
Correct Answer: C — 4
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Q. If g(x) = 3x + 2, what is g(-1)?
Solution
g(-1) = 3(-1) + 2 = -3 + 2 = -1.
Correct Answer: A — -1
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Q. If H = {1, 2, 3}, how many subsets of H have exactly 2 elements?
Solution
The number of ways to choose 2 elements from 3 is given by the combination formula C(3, 2) = 3.
Correct Answer: B — 6
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Q. If H = {x, y, z}, how many subsets of H have at least one element?
Solution
The total number of subsets is 2^3 = 8. Subtracting the empty set gives 8 - 1 = 7.
Correct Answer: A — 7
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Q. If H = {x, y}, how many subsets of H are also subsets of the power set of H?
Solution
The power set of H is {∅, {x}, {y}, {x, y}}. The subsets of H are {∅, {x}, {y}}.
Correct Answer: C — 4
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Q. If h(x) = x^3 - 3x + 2, what is the critical point?
-
A.
x = 0
-
B.
x = 1
-
C.
x = -1
-
D.
x = 2
Solution
h'(x) = 3x^2 - 3 = 0 gives x^2 = 1, so x = 1 and x = -1 are critical points.
Correct Answer: B — x = 1
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Q. If h(x) = x^3 - 3x + 2, what is the value of h(1)?
Solution
h(1) = 1^3 - 3(1) + 2 = 1 - 3 + 2 = 0.
Correct Answer: A — 0
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Q. If h(x) = x^3 - 3x, what is the value of h(1)?
Solution
h(1) = 1^3 - 3*1 = 1 - 3 = -2.
Correct Answer: B — 0
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Q. If I = (1, 1, 1) and J = (2, 2, 2), what is the scalar product I · J?
Solution
I · J = 1*2 + 1*2 + 1*2 = 2 + 2 + 2 = 6.
Correct Answer: D — 6
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Q. If I = (1, 2, 3) and J = (3, 2, 1), what is I · J?
Solution
I · J = 1*3 + 2*2 + 3*1 = 3 + 4 + 3 = 10.
Correct Answer: A — 10
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Q. If I = (1, 2, 3) and J = (4, 5, 6), calculate I · J.
Solution
I · J = 1*4 + 2*5 + 3*6 = 4 + 10 + 18 = 32.
Correct Answer: B — 30
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Q. If I = (a, b, c) and J = (2, 2, 2) such that I · J = 12, what is the relationship between a, b, c?
-
A.
a + b + c = 6
-
B.
2a + 2b + 2c = 12
-
C.
a + b + c = 12
-
D.
2a + 2b + 2c = 6
Solution
I · J = 2a + 2b + 2c = 12 gives the equation 2a + 2b + 2c = 12.
Correct Answer: B — 2a + 2b + 2c = 12
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Q. If K = {a, b, c}, what is the number of subsets of K that do not contain the element 'a'?
Solution
If 'a' is excluded, we can form subsets from {b, c}, which has 2^2 = 4 subsets.
Correct Answer: C — 3
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Q. If light of wavelength 500 nm passes through a diffraction grating with 1000 lines/mm, what is the angle for the first-order maximum?
-
A.
30 degrees
-
B.
60 degrees
-
C.
45 degrees
-
D.
15 degrees
Solution
Using the grating equation d sin(θ) = mλ, where d = 1/1000 mm = 1 x 10^-6 m, for m=1, we find θ ≈ 30 degrees.
Correct Answer: A — 30 degrees
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Q. If log(x) + log(2) = 3, what is the value of x?
-
A.
1000
-
B.
2000
-
C.
500
-
D.
300
Solution
log(2x) = 3 => 2x = 10^3 => x = 1000.
Correct Answer: A — 1000
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Q. If log_10(1000) = x, what is the value of x?
Solution
Since 1000 = 10^3, log_10(1000) = 3, thus x = 3.
Correct Answer: B — 3
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Q. If log_10(2) = a, what is log_10(20) in terms of a?
-
A.
2a
-
B.
a + 1
-
C.
a + 2
-
D.
2 + a
Solution
log_10(20) = log_10(2 * 10) = log_10(2) + log_10(10) = a + 1.
Correct Answer: B — a + 1
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Q. If log_10(x) = 2, what is the value of x?
-
A.
100
-
B.
200
-
C.
300
-
D.
400
Solution
log_10(x) = 2 implies x = 10^2 = 100.
Correct Answer: A — 100
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Q. If log_2(x + 1) - log_2(x) = 1, what is the value of x?
Solution
log_2((x + 1)/x) = 1 implies (x + 1)/x = 2 => x + 1 = 2x => x = 1.
Correct Answer: A — 1
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Q. If log_2(x + 1) = 3, what is the value of x?
Solution
log_2(x + 1) = 3 implies x + 1 = 2^3 = 8 => x = 7.
Correct Answer: A — 6
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Q. If log_2(x) + log_2(4) = 5, find x.
Solution
log_2(x) + 2 = 5 => log_2(x) = 3 => x = 2^3 = 8.
Correct Answer: B — 32
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Q. If log_2(x) + log_2(x - 3) = 3, what is the value of x?
Solution
log_2(x(x - 3)) = 3 => x(x - 3) = 2^3 = 8 => x^2 - 3x - 8 = 0. Solving gives x = 6.
Correct Answer: B — 6
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Q. If log_2(x) + log_2(x-1) = 3, what is the value of x?
Solution
log_2(x(x-1)) = 3 => x(x-1) = 2^3 = 8 => x^2 - x - 8 = 0. Solving gives x = 5.
Correct Answer: B — 5
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Q. If log_2(x) = 5, what is the value of x?
Solution
log_2(x) = 5 implies x = 2^5 = 32.
Correct Answer: C — 64
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Q. If log_3(9) + log_3(27) = x, what is the value of x?
Solution
log_3(9) = 2 and log_3(27) = 3, thus x = 2 + 3 = 5.
Correct Answer: C — 4
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Q. If log_3(9) = x, what is the value of x?
Solution
log_3(9) = log_3(3^2) = 2.
Correct Answer: B — 2
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Q. If log_3(x + 1) = 2, what is the value of x?
Solution
log_3(x + 1) = 2 implies x + 1 = 3^2 = 9 => x = 8.
Correct Answer: B — 8
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Q. If log_3(x) + log_3(4) = 2, find x.
Solution
log_3(4x) = 2 => 4x = 3^2 = 9 => x = 9/4 = 2.25.
Correct Answer: C — 9
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Q. If log_3(x) = 2, what is the value of x?
Solution
log_3(x) = 2 implies x = 3^2 = 9.
Correct Answer: B — 9
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