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
3551
A sector of a circle has a radius of 5 m and a perimeter of 16 m. What is the central angle of the sector in radians?
Answer:
1.2
The perimeter of a sector is given by P = 2r + l, where l is the arc length. Given P = 16 and r = 5. So, 2(5) + l = 16, which means 10 + l = 16 and l = 6 m. The central angle θ = l/r = 6/5 = 1.2 radians.
3552
A circular wire of radius 7 cm is cut and bent into a circular arc of radius 12 cm. Find the angle subtended by this arc at the center in radians.
Answer:
7π/6
The length of the wire is the circumference of the original circle: l = 2π(7) = 14π cm. This wire forms an arc of a new circle with r = 12 cm. The angle θ = l/r = 14π / 12 = 7π/6 radians.
3553
An angle measures 1.5 radians. If the radius of the circle is 8 cm, find the area of the sector.
Answer:
48 cm²
Using the sector area formula A = 0.5 * r² * θ. Substitute r = 8 and θ = 1.5. A = 0.5 * (8²) * 1.5 = 0.5 * 64 * 1.5 = 32 * 1.5 = 48 cm².
3554
What is the measure of an exterior angle of a regular pentagon in radians?
Answer:
2π/5
The sum of the exterior angles of any convex polygon is 360°, or 2π radians. For a regular pentagon (n=5), each exterior angle is 2π / 5 radians.
3555
What is the measure of each interior angle of a regular octagon in radians?
Answer:
3π/4
For a regular octagon (n=8), the sum of interior angles is (8-2)π = 6π. Each interior angle is 6π/8, which simplifies perfectly to 3π/4 radians.
3556
Find the sum of the interior angles of a regular hexagon in radians.
Answer:
4π
The formula for the sum of interior angles of an n-sided polygon is (n-2)π radians. For a hexagon, n=6. The sum is (6-2)π = 4π radians.
3557
The angles of a triangle are in AP and the greatest angle is double the least. What is the greatest angle in radians?
Answer:
4π/9
Let the angles be a-d, a, and a+d. Their sum is 3a = 180°, so a = 60°. The greatest is a+d and the least is a-d. Given a+d = 2(a-d), we get 60+d = 2(60-d), yielding 3d = 60, so d = 20°. The greatest angle is 60+20 = 80°. Convert to radians: 80 * π/180 = 4π/9 rad.
3558
The angles of a triangle are in the ratio 2:3:4. Find the smallest angle in radians.
Answer:
2π/9
The sum of the angles is π radians. The ratio sum is 2+3+4 = 9. The smallest angle corresponds to 2 parts. Its measure is (2/9) * π = 2π/9 radians.
3559
What is the sum of the interior angles of a triangle in radians?
Answer:
π
The sum of the interior angles of any planar triangle is exactly 180 degrees. In circular measure, 180° is mathematically equivalent to exactly π radians.
3560
Convert π/5 radians into grades.
Answer:
40g
First, convert radians to degrees: π/5 rad = 180°/5 = 36°. Next, convert degrees to grades by multiplying by 10/9. 36 * (10/9) = 4 * 10 = 40g. Alternatively, since π rad = 200g, (π/5) = 200g/5 = 40g.