Q. In triangle ABC, if the lengths of the sides are 8, 15, and 17, what is the type of triangle?
-
A.
Acute
-
B.
Obtuse
-
C.
Right
-
D.
Equilateral
Solution
Since 8² + 15² = 64 + 225 = 289 = 17², triangle ABC is a right triangle.
Correct Answer: C — Right
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Q. In triangle ABC, if the lengths of the sides are a = 5, b = 12, and c = 13, what is the perimeter of the triangle?
Solution
The perimeter of a triangle is the sum of its sides. Therefore, perimeter = a + b + c = 5 + 12 + 13 = 30.
Correct Answer: B — 25
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Q. In triangle ABC, if the lengths of the sides are a = 8, b = 15, and c = 17, what is the perimeter?
Solution
The perimeter of a triangle is the sum of its sides: a + b + c = 8 + 15 + 17 = 40.
Correct Answer: A — 30
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Q. In triangle ABC, if the lengths of the sides are a = 8, b = 15, and c = 17, what is the value of cos A?
-
A.
0.5
-
B.
0.6
-
C.
0.8
-
D.
0.9
Solution
Using the cosine rule, cos A = (b² + c² - a²) / (2bc) = (15² + 17² - 8²) / (2 * 15 * 17) = 0.8.
Correct Answer: C — 0.8
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Q. In triangle ABC, if the lengths of the sides are in the ratio 3:4:5, what type of triangle is it?
-
A.
Acute
-
B.
Obtuse
-
C.
Right
-
D.
Equilateral
Solution
Since the sides are in the ratio of a Pythagorean triplet (3, 4, 5), triangle ABC is a right triangle.
Correct Answer: C — Right
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Q. In triangle ABC, if the sides are in the ratio 3:4:5, what is the nature of the triangle?
-
A.
Equilateral
-
B.
Isosceles
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C.
Right
-
D.
Scalene
Solution
The sides satisfy the Pythagorean theorem, hence it is a right triangle.
Correct Answer: C — Right
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Q. In triangle ABC, if the sides are in the ratio 3:4:5, what type of triangle is it?
-
A.
Acute
-
B.
Obtuse
-
C.
Right
-
D.
Equilateral
Solution
A triangle with sides in the ratio 3:4:5 is a right triangle, as it satisfies the Pythagorean theorem.
Correct Answer: C — Right
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Q. In triangle MNO, if angle M = 45 degrees and angle N = 45 degrees, what is angle O?
-
A.
90 degrees
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B.
45 degrees
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C.
60 degrees
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D.
30 degrees
Solution
Angle O = 180 - (angle M + angle N) = 180 - (45 + 45) = 90 degrees.
Correct Answer: A — 90 degrees
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Q. In triangle PQR, if PQ = 10 cm, QR = 24 cm, and PR = 26 cm, what is the area of the triangle?
-
A.
120 cm²
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B.
120√3 cm²
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C.
240 cm²
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D.
48 cm²
Solution
Using Heron's formula, s = (10 + 24 + 26)/2 = 30. Area = √(30(30-10)(30-24)(30-26)) = √(30*20*6*4) = 120 cm².
Correct Answer: A — 120 cm²
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Q. In triangle XYZ, if XY = 8 cm, YZ = 15 cm, and XZ = 17 cm, is it a right triangle?
-
A.
Yes
-
B.
No
-
C.
Cannot be determined
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D.
Only if XY is the hypotenuse
Solution
Since 8^2 + 15^2 = 17^2, triangle XYZ is a right triangle.
Correct Answer: A — Yes
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Q. Solve the equation 2sin(x) + √3 = 0 for x in the interval [0, 2π].
-
A.
5π/3
-
B.
π/3
-
C.
2π/3
-
D.
4π/3
Solution
Rearranging gives sin(x) = -√3/2, so x = 4π/3 and x = 5π/3.
Correct Answer: A — 5π/3
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Q. Solve the equation 2sin(x) - 1 = 0 for x in the interval [0, 2π].
-
A.
π/6
-
B.
5π/6
-
C.
π/2
-
D.
7π/6
Solution
The solution is x = π/2.
Correct Answer: C — π/2
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Q. Solve the equation 3cos^2(x) - 1 = 0.
-
A.
x = π/3, 2π/3
-
B.
x = π/4, 3π/4
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C.
x = 0, π
-
D.
x = π/6, 5π/6
Solution
Rearranging gives cos^2(x) = 1/3, so x = π/3 and 2π/3.
Correct Answer: A — x = π/3, 2π/3
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Q. Solve the equation 3sin(x) - 4 = 0 for x in the interval [0, 2π].
-
A.
π/6
-
B.
π/3
-
C.
2π/3
-
D.
5π/6
Solution
The solution is x = π/3.
Correct Answer: B — π/3
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Q. Solve the equation cos(x) + sin(x) = 1 for x in the interval [0, 2π].
-
A.
π/4
-
B.
π/2
-
C.
3π/4
-
D.
0
Solution
The only solution is x = π/2.
Correct Answer: B — π/2
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Q. Solve the equation cos(x) = -1/2 for x in the interval [0, 2π].
-
A.
2π/3, 4π/3
-
B.
π/3, 5π/3
-
C.
π/2, 3π/2
-
D.
0, π
Solution
The solutions are x = 2π/3 and x = 4π/3 in the interval [0, 2π].
Correct Answer: A — 2π/3, 4π/3
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Q. Solve the equation sin(2x) = 0 for x in the interval [0, 2π].
-
A.
0, π, 2π
-
B.
π/2, 3π/2
-
C.
π/4, 3π/4
-
D.
π/6, 5π/6
Solution
The solutions are x = 0, π, 2π.
Correct Answer: A — 0, π, 2π
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Q. Solve the equation sin(2x) = 1 for x in the interval [0, 2π].
-
A.
π/4
-
B.
3π/4
-
C.
π/2
-
D.
5π/4
Solution
The equation sin(2x) = 1 gives 2x = π/2 + 2nπ, hence x = π/4 + nπ/2. In [0, 2π], the solutions are π/4 and 5π/4.
Correct Answer: C — π/2
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Q. Solve the equation sin(2x) = √3/2 for x in the interval [0, 2π].
-
A.
π/12
-
B.
5π/12
-
C.
7π/12
-
D.
11π/12
Solution
The solutions are x = π/12, 5π/12, 7π/12, and 11π/12.
Correct Answer: A — π/12
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Q. Solve the equation sin(3x) = 0 for x in the interval [0, 2π].
-
A.
0, π, 2π
-
B.
0, π/3, 2π/3
-
C.
0, π/2, π
-
D.
0, π/4, π/2
Solution
The solutions are x = 0, π, 2π, and x = nπ/3 for n = 0, 1, 2, 3, 4, 5.
Correct Answer: A — 0, π, 2π
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Q. Solve the equation sin(x) = 0.5 for x in the interval [0, 2π].
-
A.
π/6
-
B.
5π/6
-
C.
7π/6
-
D.
11π/6
Solution
The solutions are x = π/6 and x = 5π/6 in the interval [0, 2π].
Correct Answer: A — π/6
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Q. Solve the equation tan(x) = √3 for x in the interval [0, 2π].
-
A.
π/3
-
B.
2π/3
-
C.
4π/3
-
D.
5π/3
Solution
The solutions are x = π/3 and x = 4π/3 in the interval [0, 2π].
Correct Answer: A — π/3
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Q. Solve the equation tan^2(x) = 3 for x in the interval [0, 2π].
-
A.
π/3
-
B.
2π/3
-
C.
4π/3
-
D.
5π/3
Solution
The solutions are x = π/3 and x = 4π/3.
Correct Answer: A — π/3
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Q. Solve the equation tan^2(x) = 3.
-
A.
x = π/3
-
B.
x = 2π/3
-
C.
x = 4π/3
-
D.
x = 5π/3
Solution
The solutions are x = π/3 and x = 4π/3.
Correct Answer: A — x = π/3
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Q. The area of triangle ABC is 24 cm², and the base BC = 8 cm. What is the height from A to BC?
-
A.
6 cm
-
B.
8 cm
-
C.
4 cm
-
D.
3 cm
Solution
Area = 1/2 * base * height => 24 = 1/2 * 8 * height => height = 6 cm.
Correct Answer: A — 6 cm
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Q. The area of triangle ABC is 30 square units, and the base BC is 10 units. What is the height from A to BC?
-
A.
3 units
-
B.
6 units
-
C.
5 units
-
D.
4 units
Solution
Area = 1/2 * base * height => 30 = 1/2 * 10 * height => height = 6 units.
Correct Answer: B — 6 units
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Q. The lengths of the sides of triangle ABC are 7 cm, 24 cm, and 25 cm. What type of triangle is it?
-
A.
Acute
-
B.
Obtuse
-
C.
Right
-
D.
Equilateral
Solution
Since 7^2 + 24^2 = 25^2, triangle ABC is a right triangle.
Correct Answer: C — Right
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Q. What are the solutions of the equation cos(x) + sin(x) = 1?
-
A.
x = 0
-
B.
x = π/4
-
C.
x = π/2
-
D.
x = π
Solution
The only solution is x = 0.
Correct Answer: A — x = 0
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Q. What are the solutions of the equation cos(x) = -1/2 in the interval [0, 2π]?
-
A.
2π/3, 4π/3
-
B.
π/3, 5π/3
-
C.
π/2, 3π/2
-
D.
0, π
Solution
The solutions are x = 2π/3 and 4π/3.
Correct Answer: A — 2π/3, 4π/3
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Q. What are the solutions of the equation cos(x) = -1/2?
-
A.
2π/3
-
B.
4π/3
-
C.
π/3
-
D.
5π/3
Solution
The solutions are x = 2π/3 and x = 4π/3.
Correct Answer: A — 2π/3
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