Let f i x = s i n 2 p i x for i = 1,2 , 3 & p i ∈ N . It is given that the fundamental periods of f 1 x + f 2 x + f 3 x , f 1 x + f 2 x and f 1 x + f 3 x are π , π 2 and π 3 respectively, then the minimum value of p 1 + p 2 + p 3 is

Question

Let f i x = s i n 2 p i x for i = 1,2 , 3 & p i ∈ N . It is given that the fundamental periods of f 1 x + f 2 x + f 3 x , f 1 x + f 2 x and f 1 x + f 3 x are π , π 2 and π 3 respectively, then the minimum value of p 1 + p 2 + p 3 is,

MARKS App · Accepted Answer

f 1 x = sin ⁡ 2 p 1 x f 2 x = sin ⁡ 2 p 2 x f 3 x = sin ⁡ 2 p 3 x Now, π = L C M 2 π 2 p 1 , 2 π 2 p 2 , 2 π 2 p 3 ⇒ H C F p 1 , p 2 , p 3 = 1 π 2 = L C M 2 π 2 p 1 , 2 π 2 p 2 ⇒ H C F p 1 , p 2 = 2 ⇒ p 1 = 2 a p 2 = 2 b π 3 = L C M 2 π 2 p 1 , 2 π 2 p 3 ⇒ H C F p 1 , p 3 = 3 ⇒ p 1 = 3 x p 3 = 3 y Hence, p 1 = 6 , p 2 = 2 , p 3 = 3 (for minimum value)

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