physics : class11 Sinusoidal Functions : Periodic Functions : Oscillations and Waves MCQ Questions and Answers
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Sinusoidal Functions
MCQ questions and answers.
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MCQ Questions
1. A sinusoidal function is given by x(t) = 7 sin(6πt + π/3) − 2. Which pair correctly gives its period and phase constant, respectively?
A.
1/3 s and π/3 rad
B.
1/6 s and π/3 rad
C.
1/3 s and 2π/3 rad
D.
1/6 s and 2π/3 rad
Correct Answer:
1/3 s and π/3 rad
Explanation:
For x(t) = A sin(ωt + φ), the period is T = 2π/ω = 2π/(6π) = 1/3 s, and the phase constant is φ = π/3 rad.
2. The function y(x) = 4 cos(3x − π/2) + 5 is rewritten in the form y(x) = 4 sin(3x + φ) + 5. What is the value of φ?
A.
−π/2
B.
0
C.
π/2
D.
π
Correct Answer:
0
Explanation:
Using cos θ = sin(θ + π/2), we get cos(3x − π/2) = sin(3x). Therefore, y(x) = 4 sin(3x) + 5, so φ = 0.
3. The sinusoidal function f(t) = 3 sin(ωt) + 4 cos(ωt) is expressed as R sin(ωt + φ), where R > 0 and 0 < φ < π/2. Which values of R and φ are correct?
A.
R = 5 and φ = tan⁻¹(3/4)
B.
R = 5 and φ = tan⁻¹(4/3)
C.
R = 7 and φ = tan⁻¹(4/3)
D.
R = 1 and φ = tan⁻¹(3/4)
Correct Answer:
R = 5 and φ = tan⁻¹(4/3)
Explanation:
Expanding R sin(ωt + φ) gives R cosφ sin(ωt) + R sinφ cos(ωt). Thus R cosφ = 3 and R sinφ = 4, so R = √(3² + 4²) = 5 and tanφ = 4/3.
4. For f(t) = 5 sin(8πt + π/6), the time interval between any two consecutive zeros of f(t) is
A.
1/16 s
B.
1/8 s
C.
1/4 s
D.
1/2 s
Correct Answer:
1/8 s
Explanation:
Zeros of sine occur when its argument changes by π. Therefore, 8π Δt = π, giving Δt = 1/8 s. This is half the period, since the period is 2π/(8π) = 1/4 s.
5. A sinusoidal quantity is written as f(t) = 5 sin(ωt + φ), with 0 < φ < π/2. If f(0) = 3 and f′(0) = 4ω, what is φ?
6. Which sinusoidal function has amplitude 6, period π/2, maximum value at x = π/8, and midline y = −2?
A.
y = 6 sin(4x) − 2
B.
y = 6 cos(4x) − 2
C.
y = 6 sin(2x + π/2) − 2
D.
y = 6 cos(2x − π/4) − 2
Correct Answer:
y = 6 sin(4x) − 2
Explanation:
A period of π/2 requires angular coefficient 4 because 2π/4 = π/2. The function 6 sin(4x) − 2 reaches its maximum when 4x = π/2, or x = π/8, and has amplitude 6 and midline −2.
7. The functions f(t) = 8 sin(5t − π/3) and g(t) = 8 sin(5t + π/6) have what phase relationship?
A.
f leads g by π/2
B.
f lags g by π/2
C.
f leads g by π/6
D.
f and g are in phase
Correct Answer:
f lags g by π/2
Explanation:
The phase constants are −π/3 for f and π/6 for g. The phase difference is π/6 − (−π/3) = π/2, so f has the smaller phase and therefore lags g by π/2.
8. For the sinusoidal function y = 2 cos(6x + π/4) − 1, at which value of x in the interval 0 ≤ x ≤ π/3 does y attain its maximum for the first time?
A.
π/24
B.
π/8
C.
5π/24
D.
π/3
Correct Answer:
π/8
Explanation:
The maximum occurs when cos(6x + π/4) = 1. Thus 6x + π/4 = 2πn. The first nonnegative solution is obtained with n = 1: x = (2π − π/4)/6 = π/8.
9. A sinusoidal function has maximum value 9, minimum value −3, and completes 5 cycles in an interval of length 2π. Which equation is consistent with these properties and has a maximum at x = π/10?
A.
y = 6 cos(5x) + 3
B.
y = 6 sin(5x) + 3
C.
y = 3 cos(5x) + 6
D.
y = 6 cos(10x) + 3
Correct Answer:
y = 6 sin(5x) + 3
Explanation:
The amplitude is (9 − (−3))/2 = 6 and the midline is (9 + (−3))/2 = 3. Five cycles over 2π gives angular coefficient 5. For y = 6 sin(5x) + 3, the first maximum occurs when 5x = π/2, giving x = π/10.
10. For h(t) = A sin(ωt + φ), where A > 0 and ω > 0, the graph passes through its midline with positive slope at t = t₀. Which phase condition must hold?
A.
ωt₀ + φ = π/2 + 2πn
B.
ωt₀ + φ = π + 2πn
C.
ωt₀ + φ = 2πn
D.
ωt₀ + φ = 3π/2 + 2πn
Correct Answer:
ωt₀ + φ = 2πn
Explanation:
The midline occurs when sin(ωt + φ) = 0. The slope is h′(t) = Aω cos(ωt + φ), which is positive when the cosine is positive. Therefore the phase must be 2πn, not π + 2πn.
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