Mathematics/General Ability MCQs
Topic Notes: Mathematics/General Ability
MCQs and preparation resources for competitive exams, covering important concepts, past papers, and detailed explanations.
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
101
As observed from the top of a 75 m high lighthouse from the sea-level, the angles of depression of two ships are 30° and 45°. If one ship is exactly behind the other on the same side of the lighthouse, find the distance between the two ships.
Answer:
75(√3 - 1) m
Step 1: Let the distances of the ships from the lighthouse be x and y (x > y). Step 2: For the closer ship (45°), tan(45°) = 75 / y ⇒ y = 75 m. For the farther ship (30°), tan(30°) = 75 / x ⇒ 1 / √3 = 75 / x ⇒ x = 75√3 m. Step 3: Distance between ships = x - y = 75√3 - 75 = 75(√3 - 1) m.
102
From the top of a 7 m high building, the angle of elevation of the top of a cable tower is 60° and the angle of depression of its foot is 45°. Determine the height of the tower.
Answer:
7(√3 + 1) m
Step 1: Let the horizontal distance be x. Angle of depression to the foot is 45°, so tan(45°) = 7 / x ⇒ x = 7 m. Step 2: Let the height of the tower above the building be h'. Angle of elevation is 60°, so tan(60°) = h' / 7 ⇒ √3 = h' / 7 ⇒ h' = 7√3. Step 3: Total height of the tower = h' + 7 = 7√3 + 7 = 7(√3 + 1) m.
103
The angle of elevation of the top of a tower from a point on the ground, which is 30 m away from the foot of the tower, is 30°. Find the height of the tower.
Answer:
10√3 m
Step 1: Let the height of the tower be h. The distance from the foot is 30 m. Step 2: tan(30°) = h / 30. Step 3: 1 / √3 = h / 30, so h = 30 / √3 = 10√3 m.
104
A tree breaks due to storm and the broken part bends so that the top of the tree touches the ground making an angle 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 standing part be h1 and the broken part be h2. Distance = 8 m. Step 2: tan(30°) = h1 / 8 ⇒ h1 = 8 / √3. cos(30°) = 8 / h2 ⇒ √3 / 2 = 8 / h2 ⇒ h2 = 16 / √3. Step 3: Total height = h1 + h2 = (8 / √3) + (16 / √3) = 24 / √3 = 8√3 m.
105
From the top of a 50 m high tower, the angle of depression of a car on the ground is 30°. Find the distance of the car from the foot of the tower.
Answer:
50√3 m
Step 1: The angle of depression is equal to the angle of elevation from the car to the top of the tower, which is 30°. Step 2: Let the distance be x. tan(30°) = height / distance = 50 / x. Step 3: 1 / √3 = 50 / x, so x = 50√3 m.
106
An observer 1.5 m tall is 28.5 m away from a chimney. The angle of elevation of the top of the chimney from her eyes is 45°. What is the height of the chimney?
Answer:
30 m
Step 1: The horizontal distance from the observer to the chimney is 28.5 m. The angle of elevation is 45°. Step 2: The height of the chimney above the observer's eye level is h'. tan(45°) = h' / 28.5, so h' = 28.5 m. Step 3: Total height of the chimney = h' + observer's height = 28.5 + 1.5 = 30 m.
107
A kite is flying at a height of 60 m above the ground. The string attached to the kite is temporarily tied to a point on the ground. The inclination of the string with the ground is 60°. Find the length of the string, assuming there is no slack.
Answer:
40√3 m
Step 1: Let the length of the string be L. The height is the opposite side (60 m). Step 2: sin(60°) = opposite / hypotenuse = 60 / L. Step 3: √3 / 2 = 60 / L. Therefore, L = 120 / √3 = 40√3 m.
108
The angle of elevation of a ladder leaning against a wall is 45°, and the foot of the ladder is 10 m away from the wall. What is the height of the wall where the ladder touches?
Answer:
10 m
Step 1: Let the height of the wall be h. The distance from the foot is 10 m. Step 2: tan(45°) = height / base = h / 10. Step 3: Since tan(45°) = 1, h / 10 = 1, resulting in h = 10 m.
109
A ladder leans against a wall making an angle of 60° with the ground. If the foot of the ladder is 5 m away from the wall, find the length of the ladder.
Answer:
10 m
Step 1: The ladder forms the hypotenuse of a right-angled triangle. Let its length be L. Step 2: The base is 5 m, and the angle with the ground is 60°. Use cos(60°) = base / hypotenuse. Step 3: cos(60°) = 1/2. So, 1/2 = 5 / L, which gives L = 10 m.
110
From a point 20 m away from the foot of a tower, the angle of elevation of the top of the tower is 30°. Find the height of the tower.
Answer:
20 / √3 m
Step 1: Let the height of the tower be h. The distance from the foot is 20 m. Step 2: tan(30°) = h / 20. Step 3: Since tan(30°) = 1 / √3, we have 1 / √3 = h / 20. Thus, h = 20 / √3 m.