All Categories MCQs
Topic Notes: All Categories
General Description
Plato
- Biography: Ancient Greek philosopher (427–347 BCE), student of Socrates and teacher of Aristotle, founder of the Academy in Athens.
- Important Ideas:
- Theory of Forms
- Philosopher-King
- Ideal State
2641
What is the value of 0!?
Answer:
1
Step 1: By mathematical convention and the definition of the gamma function, the factorial of zero is defined. Step 2: There is exactly one way to arrange zero objects (doing nothing). Step 3: Therefore, 0! = 1.
2642
What is the value of 6! (6 factorial)?
Answer:
720
Step 1: The factorial of a number n, denoted by n!, is the product of all positive integers less than or equal to n. Step 2: 6! = 6 × 5 × 4 × 3 × 2 × 1. Step 3: 6! = 30 × 24 = 720.
2643
A tree breaks due to a storm and the broken part bends so that the top of the tree touches the ground making an angle of 30° with it. The distance between the foot of the tree to the point where the top touches the ground is 8 m. Find the height of the tree.
Answer:
8√3 m
Step 1: Let the broken part be the hypotenuse (y) and the standing part be the perpendicular (x). tan(30°) = x/8 => x = 8/√3. Step 2: cos(30°) = 8/y => y = 16/√3. Step 3: Total height = x + y = 8/√3 + 16/√3 = 24/√3 = 8√3 m.
2644
If the distance of a point from the base of a pole is 20√3 m and the angle of depression from the top of the pole to the point is 30°, find the height of the pole.
Answer:
20 m
Step 1: Angle of elevation = 30°. Step 2: tan(30°) = Height / 20√3. Step 3: 1/√3 = h / 20√3 -> h = 20 m.
2645
The angle of depression of a point on the ground from the top of a 30 m monument is 30°. The distance of the point from the base is:
Answer:
30√3 m
Step 1: Angle of elevation is 30°. Step 2: tan(30°) = 30 / d. Step 3: 1/√3 = 30 / d -> d = 30√3 m.
2646
From a 100 m high tower, the angle of depression of a car is 60°. Find the distance of the car from the tower.
Answer:
100/√3 m
Step 1: Angle of elevation = 60°. Step 2: tan(60°) = 100 / d -> √3 = 100 / d. Step 3: d = 100 / √3 m.
2647
A lighthouse is 120 m tall. The angle of depression to a ship is 45°. How far is the ship from the lighthouse?
Answer:
120 m
Step 1: Angle of elevation from ship is 45°. Step 2: tan(45°) = Height / Distance = 120 / d. Step 3: 1 = 120 / d, so d = 120 m.
2648
From the top of a 50 m high cliff, the angle of depression of a boat is 30°. Find the distance of the boat from the base of the cliff.
Answer:
50√3 m
Step 1: The angle of depression equals the angle of elevation from the boat to the cliff top (30°). Step 2: tan(30°) = Height / Distance = 50 / d. Step 3: 1/√3 = 50 / d, so d = 50√3 m.
2649
An observer 2 m tall is 10√3 m away from a building. The angle of elevation from his eye to the top is 60°. Find the building's height.
Answer:
32 m
Step 1: Height above eye level x = Distance * tan(60°) = 10√3 * √3 = 30 m. Step 2: Total height = x + 2. Step 3: Total height = 30 + 2 = 32 m.
2650
An observer 1.5 m tall is 28.5 m away from a tower. The angle of elevation of the top of the tower from his eyes is 45°. What is the height of the tower?
Answer:
30 m
Step 1: Let the tower height above the observer's eye level be x. tan(45°) = x / 28.5. Step 2: 1 = x / 28.5, so x = 28.5 m. Step 3: Total height = x + observer's height = 28.5 + 1.5 = 30 m.