Q. If log_2(x) + log_2(4) = 6, what is the value of x?
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Solution
log_2(x) + 2 = 6 implies log_2(x) = 4, so x = 2^4 = 16.
Correct Answer: C — 32
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Q. If log_3(x) = 4, what is the value of x?
A.
27
B.
81
C.
243
D.
729
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Solution
log_3(x) = 4 implies x = 3^4 = 81.
Correct Answer: C — 243
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Q. If log_4(16) = x, what is the value of x?
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Solution
log_4(16) = log_4(4^2) = 2.
Correct Answer: B — 2
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Q. If log_4(256) = x, what is the value of x? (2022)
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Solution
log_4(256) = log_4(4^4) = 4.
Correct Answer: D — 8
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Q. If log_a(16) = 4, what is the value of a? (2021)
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Solution
log_a(16) = 4 implies a^4 = 16. Since 16 = 2^4, we have a^4 = 2^4, thus a = 2.
Correct Answer: A — 2
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Q. If log_a(3) = 0.5, what is the value of a? (2022)
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Solution
log_a(3) = 0.5 implies a^0.5 = 3, thus a = 3^2 = 9.
Correct Answer: A — 9
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Q. If log_b(25) = 2, what is the value of b?
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Solution
log_b(25) = 2 implies b^2 = 25. Therefore, b = 5.
Correct Answer: A — 5
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Q. If log_x(64) = 6, what is the value of x? (2023)
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Solution
log_x(64) = 6 implies x^6 = 64. Since 64 = 2^6, we have x = 2.
Correct Answer: C — 8
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Q. If one root of the quadratic equation x^2 + px + q = 0 is 3, and the other root is -1, what is the value of p? (2021)
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Solution
The sum of the roots is 3 + (-1) = 2, hence p = -2.
Correct Answer: A — 2
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Q. If one root of the quadratic equation x^2 - 4x + k = 0 is 2, what is the value of k? (2021)
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Solution
Substituting x = 2 into the equation gives 2^2 - 4*2 + k = 0, which simplifies to k = 4.
Correct Answer: C — 4
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Q. If one root of the quadratic equation x^2 - 7x + k = 0 is 3, what is the value of k? (2023)
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Solution
Using the root 3 in the equation: 3^2 - 7*3 + k = 0, we get k = 6.
Correct Answer: A — 6
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Q. If sin 2A = 2 sin A cos A, what is the double angle formula for cosine?
A.
cos 2A = cos²A - sin²A
B.
cos 2A = 2 sin A cos A
C.
cos 2A = sin²A - cos²A
D.
cos 2A = 1 - 2 sin²A
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Solution
The double angle formula for cosine is cos 2A = cos²A - sin²A.
Correct Answer: A — cos 2A = cos²A - sin²A
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Q. If sin 2A = 2 sin A cos A, what is the value of sin 2A when sin A = 1/2?
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Solution
Using the double angle formula, sin 2A = 2(1/2)(√(1 - (1/2)²)) = 2(1/2)(√(3/4)) = 1.
Correct Answer: B — 1
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Q. If sin A = 0, what is the value of A? (2019)
A.
0°
B.
90°
C.
180°
D.
360°
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Solution
Sin A = 0 at A = 0°, 180°, and 360°. The principal value is 0°.
Correct Answer: A — 0°
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Q. If sin A = 0.6, what is the value of cos A to two decimal places?
A.
0.8
B.
0.6
C.
0.4
D.
0.2
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Solution
Using the Pythagorean identity, cos A = √(1 - sin²A) = √(1 - 0.36) = √(0.64) = 0.8.
Correct Answer: A — 0.8
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Q. If sin A = 1/2, what is the value of A in radians?
A.
π/6
B.
π/4
C.
π/3
D.
π/2
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Solution
The angle A for which sin A = 1/2 is π/6 radians.
Correct Answer: A — π/6
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Q. If sin A = 1/2, what is the value of A in the first quadrant?
A.
30°
B.
45°
C.
60°
D.
90°
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Solution
In the first quadrant, sin A = 1/2 corresponds to A = 30°.
Correct Answer: A — 30°
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Q. If sin A = 1/√2, what is the value of tan A?
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Solution
tan A = sin A / cos A = (1/√2) / (1/√2) = 1.
Correct Answer: A — 1
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Q. If sin A = 5/13, what is the value of cos A?
A.
12/13
B.
5/12
C.
13/5
D.
1/5
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Solution
Using the Pythagorean identity, cos A = √(1 - sin²A) = √(1 - (5/13)²) = √(1 - 25/169) = √(144/169) = 12/13.
Correct Answer: A — 12/13
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Q. If sin C = 0.8, what is the value of cos C?
A.
0.6
B.
0.8
C.
0.4
D.
0.2
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Solution
Using the Pythagorean identity, cos C = √(1 - sin²C) = √(1 - 0.8²) = √(1 - 0.64) = √(0.36) = 0.6.
Correct Answer: A — 0.6
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Q. If sin θ = 0.8, what is the value of cos θ?
A.
0.6
B.
0.8
C.
0.4
D.
0.2
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Solution
Using the Pythagorean identity, cos θ = √(1 - sin²θ) = √(1 - 0.8²) = √(1 - 0.64) = √(0.36) = 0.6.
Correct Answer: A — 0.6
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Q. If sin²A + cos²A = 1, what is the value of sin²A when cos A = 0?
A.
0
B.
1
C.
1/2
D.
Undefined
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Solution
If cos A = 0, then sin²A = 1 - cos²A = 1 - 0 = 1.
Correct Answer: B — 1
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Q. If the angle between two vectors A and B is 60 degrees and |A| = 5, |B| = 10, what is A · B?
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Solution
A · B = |A||B|cos(60°) = 5 * 10 * 0.5 = 25.
Correct Answer: B — 30
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Q. If the angle between two vectors A and B is 60 degrees and |A| = 5, |B| = 10, what is the scalar product A · B? (2020)
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Solution
A · B = |A||B|cos(60°) = 5 * 10 * 0.5 = 25
Correct Answer: B — 30
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Q. If the angle of elevation of a hill from a point on the ground is 30 degrees and the distance from the point to the base of the hill is 100 m, what is the height of the hill? (2021)
A.
50 m
B.
30 m
C.
20 m
D.
10 m
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Solution
Height = distance * tan(30) = 100 * (1/√3) ≈ 57.74 m, which rounds to 50 m.
Correct Answer: A — 50 m
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Q. If the angle of elevation of the sun is 30 degrees, how tall is a 10 m pole casting a shadow? (2023)
A.
5 m
B.
10 m
C.
15 m
D.
20 m
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Solution
Height = shadow * tan(angle) = 10 * tan(30) = 10 * (1/√3) ≈ 10 m.
Correct Answer: B — 10 m
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Q. If the angles of triangle JKL are in the ratio 2:3:4, what is the measure of the largest angle? (2023)
A.
40 degrees
B.
60 degrees
C.
80 degrees
D.
90 degrees
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Solution
Let the angles be 2x, 3x, and 4x. Then, 2x + 3x + 4x = 180 degrees. Therefore, 9x = 180, x = 20. The largest angle is 4x = 80 degrees.
Correct Answer: C — 80 degrees
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Q. If the angles of triangle MNO are in the ratio 2:3:4, what is the measure of the largest angle? (2023)
A.
40 degrees
B.
60 degrees
C.
80 degrees
D.
90 degrees
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Solution
Let the angles be 2x, 3x, and 4x. Then, 2x + 3x + 4x = 180 degrees. Thus, 9x = 180 degrees, x = 20 degrees. The largest angle = 4x = 80 degrees.
Correct Answer: C — 80 degrees
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Q. If the area of a rectangle is 48 square units and the length is 8 units, what is the width? (2022)
A.
4 units
B.
6 units
C.
8 units
D.
10 units
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Solution
Area = Length × Width. Therefore, Width = Area / Length = 48 / 8 = 6 units.
Correct Answer: A — 4 units
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Q. If the area of a triangle is 30 cm² and the base is 10 cm, what is the height? (2020)
A.
3 cm
B.
6 cm
C.
9 cm
D.
12 cm
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Solution
Area = 1/2 * base * height. Thus, 30 = 1/2 * 10 * height. Solving gives height = 6 cm.
Correct Answer: B — 6 cm
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