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
21
What is the relationship between the radius r, the arc length s, and the area A of a circular sector?
Answer:
A = 0.5 * r * s
The standard area of a sector is A = 0.5 * r² * θ. Since the arc length is defined as s = r * θ, we can substitute (r * θ) with s in the area equation, yielding A = 0.5 * r * (r * θ) = 0.5 * r * s.
22
Express the angle 72° in terms of radians as a fraction of π.
Answer:
2π/5
To convert degrees to radians, multiply by π/180. Thus, 72 * (π/180) = 72π / 180. Dividing the numerator and denominator by 36 simplifies this exactly to 2π/5.
23
Through what angle in degrees does the minute hand of a clock move between 10:15 AM and 10:40 AM?
Answer:
150°
The time elapsed is exactly 25 minutes. Since the minute hand moves a full 360° in 60 minutes, its speed is 360/60 = 6° per minute. In 25 minutes, it moves 25 * 6° = 150°.
24
A pendulum length is 60 cm. It swings such that its tip describes an arc of 10π cm. Find the angle of the swing in degrees.
Answer:
30°
Using the radian formula θ = l/r, we get θ = 10π / 60 = π/6 radians. Converting this precisely to degrees by substituting 180° for π gives 180° / 6 = 30°.
25
The difference between two acute angles of a right-angled triangle is π/9 radians. Find the smallest angle in degrees.
Answer:
35°
Convert the difference to degrees: π/9 rad = 180°/9 = 20°. Let the two acute angles be A and B. We know A + B = 90° and A - B = 20°. Adding them gives 2A = 110°, so A = 55°. Subtracting gives 2B = 70°, so B = 35°. The smallest angle is 35°.
26
What is the equivalent of 120 grades in radians?
Answer:
3π/5
We know that 200 grades exactly equal π radians. Therefore, to convert 120 grades to radians, we multiply by (π / 200). Thus, 120 * (π / 200) = 120π / 200. Dividing both by 40 yields exactly 3π/5 radians.
27
A rotating sprinkler waters a circular sector of radius 12 m and central angle 2.5 radians. Find the perimeter of the watered region.
Answer:
54 m
The perimeter of a sector is composed of the arc length plus the two straight radii: P = l + 2r. First, find the arc length: l = rθ = 12 * 2.5 = 30 m. Now add the two radii: P = 30 + 12 + 12 = 54 m.
28
A regular polygon has an interior angle of 3π/4 radians. How many sides does the polygon have?
Answer:
8
The interior angle is 3π/4. The corresponding exterior angle is π - 3π/4 = π/4 radians. The sum of the exterior angles of any polygon is 2π. The number of sides n = (Total exterior angle sum) / (one exterior angle) = 2π / (π/4) = 8 sides.
29
A sector's area is exactly equal to the square of its radius. What is the central angle in radians?
Answer:
2 rad
We are given Area A = r². The formula for the sector area is A = 0.5 * r² * θ. Setting these strictly equal: r² = 0.5 * r² * θ. Dividing by r² leaves 1 = 0.5 * θ. Multiplying by 2 gives θ = 2 radians.
30
Find the ratio of the areas of two sectors of a circle if their central angles are in the ratio 2:5.
Answer:
2:5
The formula for the area of a sector is A = 0.5 * r² * θ. For a given circle with a constant radius r, the area is strictly directly proportional to the central angle θ. Thus, the ratio of areas matches the angle ratio, which is 2:5.