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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 :

JEE · Math · previous-year question

  1. A.$$\left( {0,{\pi \over 4}} \right)$$
  2. B.$$\left( {0,{{3\pi } \over 4}} \right)$$
  3. C.$$\left( {{\pi \over 4},{{3\pi } \over 4}} \right)$$
  4. D.$$\left( {0,{\pi \over 2}} \right)$$correct

Answer

D. $$\left( {0,{\pi \over 2}} \right)$$

Explanation

Let f(x, y) = x + y - 1 $$ \because f\left( {1,2} \right).f\left( {\sin \theta ,\cos \theta } \right) > 0$$ $$ \Rightarrow 2\left[ {\sin \theta + \cos \theta - 1} \right] > 0$$ $$ \Rightarrow \sin \theta + \cos \theta > 1$$ $$ \Rightarrow \sin \left( {\theta + {\pi \over 4}} \right) > {1 \over {\sqrt 2 }}$$ $$ \Rightarrow \theta + {\pi \over 4} \in \left( {{\pi \over 4},{{3\pi } \over 4}} \right)$$ $$ \Rightarrow \theta \in \left( {0,{\pi \over 2}} \right)$$

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