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Question
Let α,β ε[−π,π] be such that cos(α−β)=1 and cos(α+β)=1e, the number of pairs of α,β satisfying the above system of equations is:
Let α,β ε[−π,π] be such that cos(α−β)=1 and cos(α+β)=1e, the number of pairs of α,β satisfying the above system of equations is:
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Solution
The correct option is D 4
Given −π≤α,β≤π
⇒−2π≤α−β≤2π
Also given cos(α−β)=1
⇒α−β=−2π,0,2π
⇒β=α−2π,α,α+2π
As given cos(α+β)=1e
⇒cos2α=1e
Since, −π≤α≤π
⇒−2π≤2α≤2π
So, there are four values of α and hence at least four pairs (α,β)
Given −π≤α,β≤π
⇒−2π≤α−β≤2π
Also given cos(α−β)=1
⇒α−β=−2π,0,2π
⇒β=α−2π,α,α+2π
As given cos(α+β)=1e
⇒cos2α=1e
Since, −π≤α≤π
⇒−2π≤2α≤2π
So, there are four values of α and hence at least four pairs (α,β)
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