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The set of all possible values of $θ$ in $(0, π)$ for which the points $P (1, 2)$ and $Q (\sin θ, \cos θ)$ lie on the same side of the lines $x + y = 1,$ is

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(0,π/2)

(0,π/4)

(π/4,3π/4)

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Correct option is A

If points P (1, 2) and Q (sin θ, cos θ) lie on the same side of the line x+y−1=0, then (1+2−1)(sin⁡θ+cos⁡θ−1)>0⇒ sin⁡θ+cos⁡θ>1⇒ 2sin⁡θ+π4>1⇒ sin⁡θ+π4>12⇒π4<θ+π4<3π4⇒0<θ<π2

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The set of all possible values of θ in ( 0, π) for which the points P (1, 2) and Q (sin θ, cos θ) lie on the same side of the lines x+y=1, is

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The set of all possible values of $θ$ in $(0, π)$ for which the points $P (1, 2)$ and $Q (\sin θ, \cos θ)$ lie on the same side of the lines $x + y = 1,$ is