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
1761
Find the value of sec(0°).
Answer:
1
The secant function is defined as 1 / cos(θ). Since cos(0°) equals exactly 1, taking the reciprocal gives sec(0°) = 1 / 1 = 1.
1762
If tan(x) = 4/3, evaluate sin(x) assuming x is an acute angle.
Answer:
4/5
Since tangent is opposite/adjacent, we have opp=4 and adj=3. The hypotenuse is found using Pythagoras: √(3² + 4²) = √25 = 5. Sine is defined as opposite/hypotenuse, so sin(x) = 4/5.
1763
Evaluate: sin(15°) * cos(75°) + cos(15°) * sin(75°)
Answer:
1
This expression matches the sine addition identity: sin(A)*cos(B) + cos(A)*sin(B) = sin(A + B). Substituting A=15° and B=75°, the expression becomes sin(15° + 75°) = sin(90°). We know sin(90°) is exactly 1.
1764
From the top of a 100m cliff, the angle of depression to a ship is 30°. How far is the ship from the base of the cliff?
Answer:
100√3 m
The angle of depression is equal to the angle of elevation from the ship (30°). Using tangent: tan(30°) = height / distance. 1/√3 = 100 / distance. Solving for distance yields 100√3 meters.
1765
What is the maximum value of 5*sin(x) - 12*cos(x)?
Answer:
13
The maximum value of any function in the form a*sin(x) + b*cos(x) is mathematically proven to be √(a² + b²). Substituting a=5 and b=-12, we get √(5² + (-12)²) = √(25 + 144) = √169 = 13.
1766
If cos(θ) = -1/2 and θ is in the third quadrant, find the value of θ.
Answer:
240°
The reference angle for cos(α) = 1/2 is 60°. In the third quadrant, the angle is formed by adding the reference angle to 180°. Therefore, θ = 180° + 60° = exactly 240°.
1767
Simplify: cot(θ) * sin(θ)
Answer:
cos(θ)
Using the quotient identity, we know cot(θ) = cos(θ) / sin(θ). Multiplying this by sin(θ) yields (cos(θ)/sin(θ)) * sin(θ). The sin(θ) terms cancel out perfectly, leaving just cos(θ).
1768
Find the area of a circular sector with radius 4 cm and central angle π/2 radians.
Answer:
4π cm²
The formula for the area of a sector with angle in radians is Area = (1/2)*r²*θ. Substituting r = 4 and θ = π/2: Area = (1/2) * (4²) * (π/2) = (1/2) * 16 * (π/2) = 8 * (π/2) = 4π cm².
1769
What is the length of an arc cut off by a central angle of 2 radians in a circle of radius 5 cm?
Answer:
10 cm
The formula for arc length when the angle is given in radians is simply s = r * θ. Substituting the given values: radius r = 5 cm and angle θ = 2 radians, the arc length s = 5 * 2 = 10 cm.
1770
Convert 3π/4 radians to degrees.
Answer:
135°
To convert radians to degrees, we multiply the value by (180/π). Substituting the given value: (3π/4) * (180/π) = 3 * (180/4) = 3 * 45 = exactly 135°.