The sum of two or more sinusoids may or may not be periodic depending on the relationship of their separate frequencies. For the sum of two sinusoids, let the frequencies of the individual terms be f₁ and f₂, respectively. For the sum to be periodic, f₁ and f₂ must be commensurable; i.e., there must be a number f₀ contained in each an integral number of times. Thus, if f₀ is the largest such number, 

f₁ = n₁f₀ and f₂ = n₂f₀

where n₁ and n₂ are integers; f₀ is the fundamental frequency. Which of the signals given below are peridic? 

Find the periods of those that are periodic.

(a) x₁ (t) = 2 cos (2t) + 4 sin(6πt)

(b) x₂ (t) = cos (6πt) 7 cos(30πt)

(c) x₃ (t) = cos (4πt) + 9 sin(21πt) 

(d) x₄ (t) = 3 sin (4πt) + 5 cos(7πt) + 6 sin (11πt)

(e) x₅ (t) = cos (17πt) + 5cos(18πt)

(f) x₆ (t) = cos (2πt) + 7 sin(3πt) 

(g) x₇ (t) = 4 cos (7πt) + 5 cos(11πt) 

(h) x₈ (t) = cos (120πt) + 3 cos(377t)

(i) x₉ (t) = cos (19πt) + 2 sin (21πt) 

(j) x₁₀ (t) = 5 cos (6πt) + 6 sin(7πt)