physics : class11 Function Symmetry : Periodic Functions : Oscillations and Waves MCQ Questions and Answers
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Function Symmetry
MCQ questions and answers.
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MCQ Questions
1. For the periodic function f(x) = sin(3x) + cos(2x), which statement about its symmetry is correct?
A.
It is even because both terms have periods that are integer multiples of π
B.
It is odd because sin(3x) is odd and cos(2x) is even
C.
It is neither even nor odd because the sum contains both an odd and an even function
D.
It is both even and odd because its fundamental period is 2π
Correct Answer:
It is neither even nor odd because the sum contains both an odd and an even function
Explanation:
f(-x) = -sin(3x) + cos(2x), which is neither equal to f(x) nor to -f(x). Therefore, the function has neither even nor odd symmetry.
2. The function f(t) = A cos(ωt + φ) represents a periodic oscillation. For which values of φ is f(t) even about t = 0?
A.
φ = nπ, where n is any integer
B.
φ = (2n + 1)π/2, where n is any integer
C.
φ = nπ/4, where n is any integer
D.
Any value of φ, because cosine is always even
Correct Answer:
φ = nπ, where n is any integer
Explanation:
For even symmetry, f(-t) must equal f(t). Since cos(ωt + φ) and cos(ωt - φ) are equal for all t only when sinφ = 0, φ must be an integer multiple of π.
3. Consider f(x) = sin x + a cos x, where a is a real constant. For which value of a is f(x) odd?
A.
a = 1
B.
a = -1
C.
a = 0
D.
Every real value of a
Correct Answer:
a = 0
Explanation:
Under x → -x, f(-x) = -sin x + a cos x. For f(-x) = -f(x) = -sin x - a cos x, the cosine terms require a = 0.
4. Which symmetry does the function f(x) = cos x have when considered over one complete period from -π to π?
A.
Reflection symmetry about the y-axis only
B.
Reflection symmetry about the x-axis only
C.
Point symmetry about the origin
D.
No symmetry because the function is periodic
Correct Answer:
Reflection symmetry about the y-axis only
Explanation:
Since cos(-x) = cos x, the graph is symmetric about the y-axis. It is not odd, so it does not possess point symmetry about the origin.
5. The function f(x) = sin x is defined on the interval [-π, π]. Which integral must vanish solely because of function symmetry?
A.
∫ from -π to π of sin x dx
B.
∫ from -π to π of cos x dx
C.
∫ from 0 to π of sin x dx
D.
∫ from -π to 0 of sin x dx
Correct Answer:
∫ from -π to π of sin x dx
Explanation:
sin x is an odd function, so its integral over the symmetric interval [-π, π] is zero. The other listed integrals are not forced to vanish by odd symmetry.
6. For the periodic function f(x) = sin x cos x, which parity does f possess?
A.
Even, because it is a product of two trigonometric functions
B.
Odd, because sin x is odd and cos x is even
C.
Neither even nor odd, because its period is π
D.
Both even and odd, because f(0) = 0
Correct Answer:
Odd, because sin x is odd and cos x is even
Explanation:
f(-x) = sin(-x)cos(-x) = (-sin x)(cos x) = -f(x). Hence, the product is odd. Its period does not determine its parity.
7. A function satisfies f(x + T/2) = -f(x) for every x, where T is its fundamental period. Which conclusion necessarily follows?
A.
The function is even
B.
The function is odd
C.
The function has half-wave antisymmetry but need not be even or odd
D.
The function cannot be periodic
Correct Answer:
The function has half-wave antisymmetry but need not be even or odd
Explanation:
The relation f(x + T/2) = -f(x) describes antisymmetry under a half-period translation. It does not imply f(-x) = f(x) or f(-x) = -f(x), so parity is not determined.
8. For f(x) = cos(2x) + sin(4x), which statement is correct?
A.
f is even because both terms have even arguments
B.
f is odd because both terms have even angular frequencies
C.
f is neither even nor odd because the cosine term is even and the sine term is odd
D.
f is both even and odd because its fundamental period is π
Correct Answer:
f is neither even nor odd because the cosine term is even and the sine term is odd
Explanation:
f(-x) = cos(2x) - sin(4x), which is not equal to f(x) or -f(x). Thus, the function lacks both even and odd symmetry.
9. The waveform f(t) = A sin(ωt + π/2) is rewritten using a standard trigonometric identity. What symmetry does the resulting function possess about t = 0?
A.
Odd symmetry, because sine is an odd function
B.
Even symmetry, because f(t) = A cos(ωt)
C.
Neither symmetry, because the original expression contains a phase shift
D.
Both even and odd symmetry, because the function is periodic
Correct Answer:
Even symmetry, because f(t) = A cos(ωt)
Explanation:
Using sin(θ + π/2) = cos θ, the function becomes f(t) = A cos(ωt). Since cos(-ωt) = cos(ωt), the waveform is even about t = 0.
10. Let f(x) be a periodic function satisfying f(-x) = f(x) and f(x + T/2) = -f(x), where T is its period. Which value must f(T/4) have in relation to f(T/4)?
A.
It must equal zero
B.
It must equal f(0)
C.
It must equal -f(0)
D.
It can have any value independently of f(0)
Correct Answer:
It must equal zero
Explanation:
Using even symmetry, f(-T/4) = f(T/4). Using half-wave antisymmetry, f(-T/4) = f(T/4 - T/2) = f(-T/4) and periodicity gives f(-T/4) = f(3T/4) = -f(T/4). Therefore f(T/4) = -f(T/4), so f(T/4) = 0.
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