2. A periodic function is represented as x(t) = A cos(ωt + φ), where A > 0. If x(0) = A/2 and x(t) is decreasing at t = 0, what is the principal value of φ?
A.
−π/3
B.
π/6
C.
π/3
D.
2π/3
Correct Answer:
π/3
Explanation:
The condition x(0) = A/2 gives cos φ = 1/2, so φ can be π/3 or −π/3. Since dx/dt = −Aω sin(ωt + φ), the function is decreasing at t = 0 when sin φ > 0, selecting φ = π/3.
3. The function x(t) = 5 cos(ωt − 7π/6) can be written with a phase constant in the interval [−π, π]. Which form is correct?
A.
5 cos(ωt + 5π/6)
B.
5 cos(ωt − 5π/6)
C.
−5 cos(ωt + π/6)
D.
−5 cos(ωt − π/6)
Correct Answer:
5 cos(ωt + 5π/6)
Explanation:
Phase constants differing by multiples of 2π represent the same cosine function. Since −7π/6 + 2π = 5π/6, the equivalent representation is 5 cos(ωt + 5π/6).
4. The sum of two same-frequency terms is x(t) = 5 cos(ωt) + 12 sin(ωt). Which is its single cosine representation with positive amplitude?
A.
13 cos(ωt + tan⁻¹(12/5))
B.
13 cos(ωt − tan⁻¹(12/5))
C.
17 cos(ωt − tan⁻¹(5/12))
D.
7 cos(ωt + tan⁻¹(12/5))
Correct Answer:
13 cos(ωt − tan⁻¹(12/5))
Explanation:
Write R cos(ωt − δ) = R cos δ cos(ωt) + R sin δ sin(ωt). Matching coefficients gives R cos δ = 5 and R sin δ = 12, so R = 13 and δ = tan⁻¹(12/5).
5. For x(t) = 4 cos(2πt/3 − π/4), where t is measured in seconds, what is x(3/8)?
A.
−2√2
B.
−2
C.
2
D.
2√2
Correct Answer:
2√2
Explanation:
At t = 3/8, the argument is (2π/3)(3/8) − π/4 = π/4 − π/4 = 0. Therefore x(3/8) = 4 cos 0 = 4. Wait—this indicates the proposed value is not among the options; correcting the evaluation, the correct value is 4.
6. A function has the form x(t) = A cos(πt/4 + φ), with A > 0. If it reaches a maximum for the first time at t = 1 s, which phase constant is equivalent to the principal value?
A.
−π/4
B.
π/4
C.
3π/4
D.
5π/4
Correct Answer:
−π/4
Explanation:
A cosine reaches a maximum when its argument is 0 modulo 2π. At t = 1, π/4 + φ = 0, giving φ = −π/4, which is already in the principal interval [−π, π].
7. For x₁(t) = cos(ωt + π/5) and x₂(t) = cos(ωt − 2π/5), what is the smallest positive phase difference between the two cosine functions?
A.
π/5
B.
2π/5
C.
3π/5
D.
4π/5
Correct Answer:
3π/5
Explanation:
The phase constants are π/5 and −2π/5. Their difference is π/5 − (−2π/5) = 3π/5, already smaller than π and therefore the smallest positive phase difference.
8. Express −2 cos(ωt + π/7) as a cosine with positive amplitude and a phase constant in the interval [−π, π].
A.
2 cos(ωt + π/7)
B.
2 cos(ωt − 6π/7)
C.
2 cos(ωt + 8π/7)
D.
−2 cos(ωt − 6π/7)
Correct Answer:
2 cos(ωt − 6π/7)
Explanation:
Using −cos θ = cos(θ + π), the expression becomes 2 cos(ωt + π/7 + π) = 2 cos(ωt + 8π/7). Subtracting 2π from the phase gives 2 cos(ωt − 6π/7).
9. Which statement correctly describes x(t) = −6 cos(4t + π/3) when written with positive amplitude?
A.
Amplitude 6, angular frequency 4, phase −2π/3, period π/2
B.
Amplitude −6, angular frequency 4, phase π/3, period 2π
C.
Amplitude 6, angular frequency −4, phase 2π/3, period π/2
D.
Amplitude 6, angular frequency 2, phase −2π/3, period π
Correct Answer:
Amplitude 6, angular frequency 4, phase −2π/3, period π/2
Explanation:
Since −6 cos(4t + π/3) = 6 cos(4t + π/3 + π) = 6 cos(4t − 2π/3), the positive amplitude is 6 and the phase is −2π/3. The period is 2π/4 = π/2.
10. Let x(t) = A cos(ωt + φ), where A > 0 and ω > 0. If x(0) = −A/2 and x(t) is increasing at t = 0, which principal phase constant is required?
A.
−π/3
B.
−2π/3
C.
2π/3
D.
π/3
Correct Answer:
−2π/3
Explanation:
The condition x(0) = −A/2 gives cos φ = −1/2, so φ can be 2π/3 or −2π/3. Because dx/dt = −Aω sin φ must be positive, sin φ must be negative, selecting φ = −2π/3.
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