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Question:

Let A and B denote the statements A: cosα + cosβ + cosγ = 0, B: sinα + sinβ + sinγ = 0. If cos(β − γ) + cos(γ − α) + cos(α − β) = 3/2, then

A is false and B is true

both A and B are true

both A and B are false

A is true and B is false

Solution:

cos(β − γ) + cos(γ − α) + cos(α − β) = 3/2
⇒ 2cos(β − γ) + 2cos(γ − α) + 2cos(α − β) = 3
⇒ 2Σcosβcosγ + 2Σsinαsinβ + 3 = 0
⇒ (cosα + cosβ + cosγ)² + (sinα + sinβ + sinγ)² = 0
⇒ cosα + cosβ + cosγ = 0 and sinα + sinβ + sinγ = 0
Hence, option 'B' is correct.