physics : class11 Average Value over a Period : Periodic Functions : Oscillations and Waves MCQ Questions and Answers
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Average Value over a Period
MCQ questions and answers.
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MCQ Questions
1. For the periodic function f(x) = |sin x|, what is its average value over one period?
A.
1/2
B.
2/π
C.
π/2
D.
0
Correct Answer:
2/π
Explanation:
The period of |sin x| is π. Its average value is (1/π)∫₀^π sin x dx = (1/π)[−cos x]₀^π = 2/π.
2. The function f(t) = 7 − 3 cos(ωt) + 5 sin(2ωt) is periodic. What is its average value over any complete period?
A.
−3
B.
2
C.
5
D.
7
Correct Answer:
7
Explanation:
Over a complete period, the average values of cos(ωt) and sin(2ωt) are both zero. Therefore, only the constant term contributes, giving an average value of 7.
3. What is the average value of f(x) = sin²(3x) over the interval 0 ≤ x ≤ π?
A.
1/4
B.
1/2
C.
2/π
D.
1
Correct Answer:
1/2
Explanation:
The function sin²(3x) has period π/3, so the interval 0 to π contains three complete periods. The average of sin² over any complete period is 1/2.
4. For f(t) = (1 + cos ωt)², what is the average value over one complete period 2π/ω?
A.
1
B.
3/2
C.
2
D.
5/2
Correct Answer:
3/2
Explanation:
Expanding gives f(t) = 1 + 2 cos(ωt) + cos²(ωt). The average values of cos(ωt) and cos(2ωt) are zero, while the average of cos²(ωt) is 1/2. Hence the average is 1 + 1/2 = 3/2.
5. A periodic function of period 6 s is defined by f(t) = 3 for 0 ≤ t < 2 s and f(t) = −1 for 2 ≤ t < 6 s. What is its average value over one period?
A.
−1/3
B.
0
C.
1/3
D.
2/3
Correct Answer:
1/3
Explanation:
The average is the time-weighted value: [(3)(2) + (−1)(4)]/6 = (6 − 4)/6 = 1/3.
6. What is the average value of f(x) = sin x cos 2x over any complete period?
A.
−1/4
B.
0
C.
1/4
D.
1/2
Correct Answer:
0
Explanation:
Using sin x cos 2x = [sin(3x) − sin x]/2, the function is a sum of sinusoidal terms whose averages over a complete common period are both zero. Therefore, the average is zero.
7. For the periodic function f(t) = max(0, A sin ωt), where A > 0, what is its average value over one period?
A.
A/(2π)
B.
A/π
C.
2A/π
D.
A/2
Correct Answer:
A/π
Explanation:
The function equals A sin ωt during the positive half-cycle and zero during the negative half-cycle. Its average is [1/(2π)]∫₀^π A sin θ dθ = 2A/(2π) = A/π.
8. The average value of f(x) = 1/(2 + cos x) over 0 ≤ x ≤ 2π is
A.
1/2
B.
1/√3
C.
1/3
D.
2/√3
Correct Answer:
1/√3
Explanation:
The standard complete-period integral is ∫₀^{2π} dx/(a + b cos x) = 2π/√(a² − b²) for a > |b|. With a = 2 and b = 1, the integral is 2π/√3, so dividing by 2π gives 1/√3.
9. If f(t) = 4 sin(ωt) + 3 cos(2ωt) − 2 sin(3ωt) + 6, what is its average value over the least common complete period?
A.
−2
B.
0
C.
4
D.
6
Correct Answer:
6
Explanation:
Each sine and cosine term has zero average over the common period 2π/ω. The constant term 6 remains unchanged, so the average value is 6.
10. A periodic waveform of period T is given by f(t) = A sin(2πt/T) for 0 ≤ t < T/2 and f(t) = 0 for T/2 ≤ t < T, where A > 0. What is its average value?
A.
A/(2π)
B.
A/π
C.
2A/π
D.
A/2
Correct Answer:
A/π
Explanation:
The average is (1/T)∫₀^{T/2} A sin(2πt/T) dt. Substituting θ = 2πt/T gives (A/2π)∫₀^π sin θ dθ = A/π.
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