Q. A tangent to a circle is drawn from a point outside the circle. If the distance from the point to the center of the circle is 10 cm and the radius of the circle is 6 cm, what is the length of the tangent?
A.
8 cm
B.
10 cm
C.
12 cm
D.
14 cm
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Solution
Using the Pythagorean theorem, the length of the tangent is √(10^2 - 6^2) = √(100 - 36) = √64 = 8 cm.
Correct Answer:
A
— 8 cm
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Q. A tangent to a circle is drawn from a point outside the circle. If the radius of the circle is 3 cm and the distance from the center to the point is 5 cm, what is the length of the tangent?
A.
4 cm
B.
6 cm
C.
5 cm
D.
3 cm
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Solution
Length of tangent = √(d² - r²) = √(5² - 3²) = √(25 - 9) = √16 = 4 cm.
Correct Answer:
A
— 4 cm
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Q. A train travels 60 km at a certain speed and returns at 90 km/h. If the total time for the journey is 4 hours, what is the speed of the train on the way to the destination?
A.
30 km/h
B.
40 km/h
C.
50 km/h
D.
60 km/h
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Solution
Let the speed of the train on the way to the destination be x km/h. The time taken to travel to the destination is 60/x hours and the time taken to return is 60/90 hours. The total time is given by: 60/x + 60/90 = 4. Solving for x gives x = 40 km/h.
Correct Answer:
B
— 40 km/h
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Q. A train travels 60 km in 1 hour. How far will it travel in 3 hours?
A.
120 km
B.
180 km
C.
150 km
D.
200 km
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Solution
If the train travels 60 km in 1 hour, in 3 hours it will travel 60 km × 3 = 180 km.
Correct Answer:
B
— 180 km
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Q. A train travels 60 km in 1 hour. How far will it travel in t hours?
A.
60t
B.
60/t
C.
t/60
D.
t+60
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Solution
Distance = Speed × Time. Therefore, Distance = 60 km/h × t h = 60t km.
Correct Answer:
A
— 60t
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Q. A trapezoid has bases of lengths 10 cm and 6 cm, and a height of 4 cm. What is its area?
A.
32 cm²
B.
40 cm²
C.
24 cm²
D.
28 cm²
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Solution
Area = 1/2 * (base1 + base2) * height = 1/2 * (10 + 6) * 4 = 32 cm².
Correct Answer:
A
— 32 cm²
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Q. A tree casts a shadow of 10 meters when the angle of elevation of the sun is 30°. How tall is the tree?
A.
5 m
B.
10 m
C.
15 m
D.
20 m
Show solution
Solution
Using tan(30°) = height/shadow, height = 10 * tan(30°) = 10 * (1/√3) = 10 m.
Correct Answer:
B
— 10 m
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Q. A tree casts a shadow of 10 meters when the angle of elevation of the sun is 30°. What is the height of the tree?
A.
5√3
B.
10
C.
10√3
D.
15
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Solution
Using tan(30°) = height/shadow, height = 10 * tan(30°) = 10 * (1/√3) = 10√3/3.
Correct Answer:
A
— 5√3
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Q. A tree casts a shadow of 15 meters when the angle of elevation of the sun is 30 degrees. How tall is the tree?
A.
5√3 meters
B.
15 meters
C.
10 meters
D.
7.5 meters
Show solution
Solution
Using the tangent function, tan(30) = height / 15. Therefore, height = 15 * tan(30) = 15 * (1/√3) = 5√3 meters.
Correct Answer:
A
— 5√3 meters
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Q. A triangle has a base of 10 cm and a height of 6 cm. What is its area?
A.
30 cm²
B.
60 cm²
C.
20 cm²
D.
50 cm²
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Solution
Area = 1/2 × base × height = 1/2 × 10 cm × 6 cm = 30 cm².
Correct Answer:
A
— 30 cm²
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Q. A triangle has an area of 30 cm² and a base of 10 cm. What is the height?
A.
6 cm
B.
5 cm
C.
3 cm
D.
4 cm
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Solution
Area = 1/2 * base * height. Therefore, height = (2 * Area) / base = (2 * 30) / 10 = 6 cm.
Correct Answer:
A
— 6 cm
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Q. A triangle has an area of 48 cm² and a base of 8 cm. What is the height?
A.
12 cm
B.
6 cm
C.
8 cm
D.
10 cm
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Solution
Area = 1/2 * base * height. Therefore, 48 = 1/2 * 8 * height. Solving gives height = 48 / 4 = 12 cm.
Correct Answer:
B
— 6 cm
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Q. A triangle has an area of 50 cm² and a base of 10 cm. What is the height?
A.
5 cm
B.
10 cm
C.
8 cm
D.
12 cm
Show solution
Solution
Area = 1/2 * base * height. Therefore, height = (2 * Area) / base = (2 * 50) / 10 = 10 cm.
Correct Answer:
A
— 5 cm
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Q. A triangle has angles measuring 30°, 60°, and 90°. If the shortest side is 5 cm, what is the area?
A.
12.5 cm²
B.
15 cm²
C.
10 cm²
D.
20 cm²
Show solution
Solution
Area = 1/2 * base * height. Height = 5 * sin(60°) = 5 * (√3/2) = 5√3/2. Area = 1/2 * 5 * (5√3/2) = 12.5 cm².
Correct Answer:
A
— 12.5 cm²
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Q. A triangle has angles measuring 50 degrees and 60 degrees. What is the measure of the third angle?
A.
70 degrees
B.
80 degrees
C.
90 degrees
D.
100 degrees
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Solution
The sum of angles in a triangle is 180 degrees. Therefore, the third angle is 180 - 50 - 60 = 70 degrees.
Correct Answer:
B
— 80 degrees
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Q. A triangle has sides of lengths 5 cm, 12 cm, and 13 cm. Is it a right triangle?
A.
Yes
B.
No
C.
Cannot be determined
D.
Only if angles are known
Show solution
Solution
Yes, because 5² + 12² = 25 + 144 = 169 = 13².
Correct Answer:
A
— Yes
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Q. A triangle has sides of lengths 5 cm, 12 cm, and 13 cm. Is this triangle a right triangle?
A.
Yes
B.
No
C.
Cannot be determined
D.
Only if angles are known
Show solution
Solution
Yes, because 5² + 12² = 25 + 144 = 169 = 13².
Correct Answer:
A
— Yes
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Q. A triangle has sides of lengths 7 cm, 24 cm, and 25 cm. What is its area?
A.
84 cm²
B.
96 cm²
C.
70 cm²
D.
120 cm²
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Solution
Using Heron's formula, s = (7 + 24 + 25)/2 = 28. Area = √(s(s-a)(s-b)(s-c)) = √(28(28-7)(28-24)(28-25)) = √(28*21*4*3) = 84 cm².
Correct Answer:
B
— 96 cm²
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Q. A triangle has sides of lengths 9 cm, 12 cm, and 15 cm. Is it a right triangle?
A.
Yes
B.
No
C.
Cannot be determined
D.
Only if angles are known
Show solution
Solution
Yes, because 9² + 12² = 81 + 144 = 225 = 15².
Correct Answer:
A
— Yes
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Q. A triangle has two sides measuring 6 cm and 8 cm. If the included angle is 60 degrees, what is the area of the triangle?
A.
24 cm²
B.
18 cm²
C.
20 cm²
D.
30 cm²
Show solution
Solution
Area = 1/2 * a * b * sin(C) = 1/2 * 6 * 8 * sin(60°) = 24√3/2 = 20.78 cm² (approximately 18 cm²).
Correct Answer:
B
— 18 cm²
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Q. A triangle has two sides measuring 8 cm and 15 cm. If the angle between them is 60 degrees, what is the area of the triangle?
A.
60 cm²
B.
30 cm²
C.
40 cm²
D.
70 cm²
Show solution
Solution
Area = 1/2 * a * b * sin(C) = 1/2 * 8 * 15 * sin(60°) = 60 cm².
Correct Answer:
A
— 60 cm²
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Q. A triangle has two sides measuring 8 cm and 15 cm. If the included angle is 60 degrees, what is the area of the triangle?
A.
60 cm²
B.
30 cm²
C.
40 cm²
D.
70 cm²
Show solution
Solution
Area = 1/2 * a * b * sin(C) = 1/2 * 8 * 15 * sin(60) = 60 cm².
Correct Answer:
A
— 60 cm²
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Q. A triangle has vertices at (0, 0), (4, 0), and (0, 3). What is its area?
A.
6 cm²
B.
12 cm²
C.
8 cm²
D.
10 cm²
Show solution
Solution
Area = 1/2 * base * height = 1/2 * 4 * 3 = 6 cm².
Correct Answer:
A
— 6 cm²
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Q. A triangle has vertices at (1, 2), (4, 6), and (1, 6). What is the area of the triangle?
A.
6 cm²
B.
8 cm²
C.
10 cm²
D.
12 cm²
Show solution
Solution
Area = 1/2 * |x1(y2-y3) + x2(y3-y1) + x3(y1-y2)| = 1/2 * |1(6-6) + 4(6-2) + 1(2-6)| = 1/2 * |0 + 16 - 4| = 1/2 * 12 = 6 cm².
Correct Answer:
A
— 6 cm²
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Q. A triangle is inscribed in a circle of radius 5 cm. What is the maximum area of the triangle?
A.
12.5 cm²
B.
25 cm²
C.
20 cm²
D.
15 cm²
Show solution
Solution
Maximum area = (1/2) * r^2 * sin(θ) = (1/2) * 5^2 * 1 = 25 cm².
Correct Answer:
B
— 25 cm²
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Q. At what value of x does the function y = tan(x) have a vertical asymptote?
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Solution
The tangent function has vertical asymptotes at x = π/2 + nπ, where n is an integer. The first asymptote in [0, π] is at π/2.
Correct Answer:
C
— π/2
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Q. Determine the solution for the inequality: 3(x - 1) < 2(x + 2).
A.
x < 5
B.
x > 5
C.
x < 1
D.
x > 1
Show solution
Solution
Step 1: Expand: 3x - 3 < 2x + 4. Step 2: Subtract 2x: x - 3 < 4. Step 3: Add 3: x < 7.
Correct Answer:
A
— x < 5
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Q. Determine the solution set for the inequality: 2x^2 - 8 < 0.
A.
(-2, 2)
B.
(2, -2)
C.
(-∞, -2) ∪ (2, ∞)
D.
(-2, ∞)
Show solution
Solution
Step 1: Factor the inequality: 2(x^2 - 4) < 0. Step 2: Roots are x = -2 and x = 2. Step 3: The solution is between the roots: (-2, 2).
Correct Answer:
A
— (-2, 2)
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Q. Determine the solution set for the inequality: x^2 + 4x + 3 < 0.
A.
(-3, -1)
B.
(-1, 3)
C.
(-∞, -3)
D.
(-∞, -1)
Show solution
Solution
Step 1: Factor the quadratic: (x + 3)(x + 1) < 0. Step 2: The critical points are x = -3 and x = -1. Step 3: Test intervals: The solution set is (-3, -1).
Correct Answer:
A
— (-3, -1)
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Q. Determine the solution set for the inequality: x^2 - 6x + 8 ≤ 0.
A.
[2, 4]
B.
(2, 4)
C.
[4, 2]
D.
(-∞, 2) ∪ (4, ∞)
Show solution
Solution
Step 1: Factor: (x - 2)(x - 4) ≤ 0. Step 2: The solution is between the roots: [2, 4].
Correct Answer:
A
— [2, 4]
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