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Question

If cos(θα), cosθ, cos(θ+α) are in H.P. and cosα1, then the angle α cannot lie in the

A
I quadrant
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B
II quadrant
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C
III quadrant
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D
IV quadrant
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Solution

The correct option is C IV quadrant
cos(θα),cosθ,cos(θ+α) are in the H.P
2cosθ=1cos(θα)+1cos(θ+α)
cosθ=cos2θ+cos2θ2cosθcosα
2cos2θcosα=2cos2θ1+2cos2α1
2cos2θcosα=2cos2θ+2cos2α2
cos2θ(cosα1)=2cos2α2
cos2αcosα(cos2θ)+cos2θ=0
cosα=cos2θ±cos4θ4cos2θ+42
cosα=cos2θ±(cos2θ2)22
cosα=cos2θcos2θ+22
cosα=1
cosα=cos0
α=0
cosα=cos2θ+cos2θ22
cosα=2(cos2θ1)2
cosα=sin2θ
cosα is always negative
cosα does not lie in 1st and 4th quadrant.

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