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
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).
22
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.
23
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.
24
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°.
25
If sin(x) = 1/3, what is the value of cos(2x)?
Answer:
7/9
Using the double-angle identity cos(2x) = 1 - 2*sin²(x), we substitute sin(x) = 1/3. This gives 1 - 2*(1/3)² = 1 - 2*(1/9) = 1 - 2/9 = 7/9.
26
Which identity represents the half-angle formula for sin(A/2)?
Answer:
±√((1 - cosA)/2)
Derived directly from the double-angle cosine formula cos(A) = 1 - 2sin²(A/2), we can isolate sin²(A/2) = (1 - cosA)/2. Taking the square root gives the standard half-angle identity: ±√((1 - cosA)/2).
27
If tan(θ) = 0, what is a possible value for θ?
Answer:
180°
Tangent is defined as sin(θ)/cos(θ). For tangent to be 0, the sine of the angle must be 0 (and cosine non-zero). Sine is 0 at integer multiples of 180° (0°, 180°, 360°, etc.). Thus, 180° is a correct answer.
28
What is the principal value of arccos(-1/2)?
Answer:
120°
The principal range for arccosine is [0°, 180°]. A negative cosine value means the angle must be in the second quadrant. The reference angle for cos = 1/2 is 60°. Therefore, the principal angle is 180° - 60° = 120°.
29
If sin(x) = a and cos(x) = b, what is tan(x) in terms of a and b?
Answer:
a/b
By fundamental trigonometric definition, the tangent of an angle is strictly the ratio of its sine to its cosine. Therefore, substituting the given variables, tan(x) equals a / b.
30
A 10m long ladder reaches a window that is 5m above the ground. What is the angle the ladder makes with the ground?
Answer:
30°
Let the angle be θ. We know the hypotenuse (ladder) is 10m and the opposite side (window height) is 5m. Using sine: sin(θ) = opposite/hypotenuse = 5/10 = 1/2. The angle whose sine is 1/2 is exactly 30°.