physics : class11 Fundamental Period : Periodic Functions : Oscillations and Waves MCQ Questions and Answers
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Fundamental Period
MCQ questions and answers.
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MCQ Questions
1. What is the fundamental period of the function f(x) = sin(3x) + cos(5x)?
A.
π
B.
2π
C.
π/15
D.
2π/15
Correct Answer:
2π
Explanation:
The individual periods are 2π/3 and 2π/5. The fundamental period of the sum is the least positive common period, which is 2π.
2. Determine the fundamental period of f(x) = |sin(6x)| + |cos(10x)|.
A.
π/10
B.
π/6
C.
π/2
D.
π
Correct Answer:
π/2
Explanation:
|sin(6x)| has period π/6 and |cos(10x)| has period π/10. The least positive common period satisfies 6T = mπ and 10T = nπ, giving T = π/gcd(6,10) = π/2.
3. What is the fundamental period of f(x) = sin²(2x) + cos(3x)?
A.
π/2
B.
2π/3
C.
π
D.
2π
Correct Answer:
2π
Explanation:
The period of sin²(2x) is π/2, while the period of cos(3x) is 2π/3. The least positive common multiple of these periods is 2π, so the sum has fundamental period 2π.
4. Find the fundamental period of f(x) = sin(4x)cos(2x).
A.
π/2
B.
π
C.
2π/3
D.
2π
Correct Answer:
π
Explanation:
Using sin(4x)cos(2x) = [sin(6x) + sin(2x)]/2, the function contains angular frequencies 6 and 2. Their greatest common divisor is 2, so the fundamental period is 2π/2 = π.
5. What is the fundamental period of f(x) = cos²(3x) + sin²(5x)?
A.
π/5
B.
π/3
C.
π/2
D.
π
Correct Answer:
π
Explanation:
The periods of cos²(3x) and sin²(5x) are π/3 and π/5, respectively. Their least positive common period is π, which is therefore the fundamental period of the sum.
6. Determine the fundamental period of f(x) = tan(2x) + tan(3x), considering the common domain of the function.
A.
π/6
B.
π/3
C.
π/2
D.
π
Correct Answer:
π
Explanation:
The periods of tan(2x) and tan(3x) are π/2 and π/3. The least positive common period is π. This also preserves the set of excluded points, so the fundamental period is π.
7. What is the fundamental period of f(x) = |sin(2x)| + |sin(3x)|?
A.
π/6
B.
π/3
C.
π/2
D.
π
Correct Answer:
π
Explanation:
The fundamental periods of |sin(2x)| and |sin(3x)| are π/2 and π/3. The least positive common period is π, so the sum has fundamental period π.
8. Find the fundamental period of f(x) = sin²(3x) + sin²(4x).
A.
π/12
B.
π/4
C.
π/3
D.
π
Correct Answer:
π
Explanation:
The two terms have periods π/3 and π/4. Their least positive common period is π. Equivalently, the function contains cosine components with angular frequencies 6 and 8, whose greatest common divisor is 2, giving period π.
9. What is the fundamental period of f(x) = cos(2x) + cos(4x) + cos(6x)?
A.
π/6
B.
π/3
C.
π/2
D.
π
Correct Answer:
π
Explanation:
The angular frequencies are 2, 4, and 6. Their greatest common divisor is 2, so the fundamental period is 2π/2 = π. No smaller positive shift preserves all three cosine terms.
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