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Question

If z=αsinθ+βcosθ, where α and β are positive constants. The maximum value of z is

A
α
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B
β
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C
α2+β2
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D
α+β
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Solution

The correct option is C α2+β2
given,

z=αsinθ+βcosθ

z=α2+β2[αα2+β2sinθ+βα2+β2cosθ]

let,

cosϕ=αα2+β2

sinϕ=1cos2ϕ=1α2α2+β2=βα2+β2

z=α2+β2[cosϕsinθ+sinϕcosθ]

z=α2+β2sin(ϕ+θ)

1sin(ϕ+θ)1

α2+β2α2+β2sin(ϕ+θ)α2+β2

α2+β2zα2+β2

Therefore the maximum and minimum values of z are α2+β2 and α2+β2 respectively.


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