Explanation:
The angular frequencies are 6, 10, and 15 rad s⁻¹. Their greatest common divisor is 1 rad s⁻¹, so the fundamental angular frequency is 1 rad s⁻¹. Therefore, the fundamental period is 2π/1 = 2π.

Explanation:
Using superposition, x(t) = 2A cos(φ/2) cos(ωt + φ/2). The resultant amplitude is 2A|cos(φ/2)|, which becomes zero when cos(φ/2) = 0. Thus φ = π modulo 2π.

Explanation:
The three sinusoidal functions have equal amplitudes and phases separated by 120°. Their corresponding phasors form a closed equilateral triangle, so their vector sum and hence their total superposition are identically zero.

Explanation:
The original angular frequencies are 100 and 104 rad s⁻¹. A common fundamental angular frequency is their greatest common divisor, 4 rad s⁻¹. Hence the fundamental period of the sum is 2π/4 = π/2. The rapid oscillation and the envelope do not independently determine the exact period.

Explanation:
A finite common period requires the ratio of the component frequencies to be rational. The ratio √2 is irrational, so no nonzero time can simultaneously reproduce both sinusoidal components.

Explanation:
Because the two harmonics are orthogonal over a common period, x_rms² = (1² + a²)/2. Setting this equal to 2 gives (1 + a²)/2 = 2, so a² = 3 and the positive value is a = √3.

Explanation:
Before the common phase shift, the three equal-amplitude components have phases separated by 120° and cancel exactly. A common phase shift rotates all three phasors together without changing their zero resultant, so the total remains zero.

Explanation:
Using the addition formula, cos(ωt) + cos(ωt + φ) = 2cos(φ/2)cos(ωt + φ/2). For φ = 2π/3, the amplitude is 2cos(π/3) = 1 and the phase shift is φ/2 = π/3.

Explanation:
All frequencies are integer multiples of ω, so 2π/ω is a common period. Since the set of harmonic indices 2, 3, and 5 has greatest common divisor 1, no smaller common period exists for the complete superposition.