JEE MAIN - Mathematics (2020 - 2nd September Evening Slot - No. 14)
The set of all possible values of
$$\theta $$ in the interval
(0, $$\pi $$) for which the points (1, 2) and (sin $$\theta $$, cos $$\theta $$) lie
on the same side of the line x + y = 1 is :
(0, $$\pi $$) for which the points (1, 2) and (sin $$\theta $$, cos $$\theta $$) lie
on the same side of the line x + y = 1 is :
$$\left( {0,{\pi \over 4}} \right)$$
$$\left( {0,{{3\pi } \over 4}} \right)$$
$$\left( {{\pi \over 4},{{3\pi } \over 4}} \right)$$
$$\left( {0,{\pi \over 2}} \right)$$
Explanation
Let f(x, y) = x + y - 1
$$ \because f\left( {1,2} \right).f\left( {\sin \theta ,\cos \theta } \right) > 0$$
$$ \Rightarrow 2\left[ {\sin \theta + \cos \theta - 1} \right] > 0$$
$$ \Rightarrow \sin \theta + \cos \theta > 1$$
$$ \Rightarrow \sin \left( {\theta + {\pi \over 4}} \right) > {1 \over {\sqrt 2 }}$$
$$ \Rightarrow \theta + {\pi \over 4} \in \left( {{\pi \over 4},{{3\pi } \over 4}} \right)$$
$$ \Rightarrow \theta \in \left( {0,{\pi \over 2}} \right)$$
$$ \because f\left( {1,2} \right).f\left( {\sin \theta ,\cos \theta } \right) > 0$$
$$ \Rightarrow 2\left[ {\sin \theta + \cos \theta - 1} \right] > 0$$
$$ \Rightarrow \sin \theta + \cos \theta > 1$$
$$ \Rightarrow \sin \left( {\theta + {\pi \over 4}} \right) > {1 \over {\sqrt 2 }}$$
$$ \Rightarrow \theta + {\pi \over 4} \in \left( {{\pi \over 4},{{3\pi } \over 4}} \right)$$
$$ \Rightarrow \theta \in \left( {0,{\pi \over 2}} \right)$$
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