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Question

The equation k cosx3 sinx=k+1 is solvable only if k belongs to the intarval

A
[4,]
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B
[4,4]
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C
(4,4)
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D
[,4]
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Solution

The correct option is D [,4]
kcosx3sinx=k+1
kcosx(k+1)=3sinx
[kcosx(k+1)]2=(3sinx)2
k2cos2x2k(k+1)cosx+(k+1)2=9sin2x
k2cos2x(2k2+2k)cosx+(k+1)2=9(1cos2x)
(k2)cos2x(2k2+2k)cosx+(k2+2k+1)+9cos2x9=0
(k2+9)cos2x(2k2+2k)cosx+(k2+2k8)=0

This is the quadratic equation in terms of cosx
b24ac0
[(2k2+2k)]24(k2+9)(k2+2k8)0
(4k4+8k3+4k2)(4k4+8k3+4k2+72k288)0
72k+2880
72k2880
72k288
k4
k lies [,4]
Hence, the answer is [,4].

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