NIMCET 2023 Question Paper with solutions

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Question : 16 of 120

Marks: +1, -0

The range of values of

θ

in the interval

(0, π)

such that the points

(3,2)

and

(\cos θ, sin \;θ)

lie on the same side of the line

x+y-1=0

is:

$(0, π \/ 4)$
$(0,3 π \/ 4)$
$(0, π \/ 2)$
$(0, π \/ 3)$

Solution:

{\L}: x+y-1=0

As

(3,2)

and

(\cos θ, sin \;θ)

lies on same side of line

{\L}:(3,2) 3+2-1=4>0

So,

{\L}:(\cos θ, sin \;θ) ≡ sin \;θ+\cos θ-1>0

⇒ {√{2}} [\;{1}/{{√{2}}} \cos θ+\;{1}/{{√{2}}} sin \;θ] >1

\; sin \; (θ+\;{π}/{4}) \;>\;{1}/{{√{2}}}

⇒ \;{π}/{4}\;< θ+\;{π}/{4}< \;{3 π}/{4}

⇒\;0< θ< \;{π}/{2}

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