Verified Solution • 2025–26 Session moderate

QJEEM21Madde3: JEE Main 2021 Mathematics - Class 11 JEEM21Madde3 Solution: All possible values of [0, 2 ] for which sin 2 + tan 2 > 0 lie…

All possible values of

\theta

\in

[0, 2

\pi

] for which sin 2

\theta

+ tan 2

\theta

> 0 lie in :

Multiple Choice Options

A $\left( {0,{\pi \over 4}} \right) \cup \left( {{\pi \over 2},{{3\pi } \over 4}} \right) \cup \left( {{{3\pi } \over 2},{{11\pi } \over 6}} \right)$
B $\left( {0,{\pi \over 2}} \right) \cup \left( {\pi ,{{3\pi } \over 2}} \right)$
C $\left( {0,{\pi \over 2}} \right) \cup \left( {{\pi \over 2},{{3\pi } \over 4}} \right) \cup \left( {\pi ,{{7\pi } \over 6}} \right)$
D $\left( {0,{\pi \over 4}} \right) \cup \left( {{\pi \over 2},{{3\pi } \over 4}} \right) \cup \left( {\pi ,{{5\pi } \over 4}} \right) \cup \left( {{{3\pi } \over 2},{{7\pi } \over 4}} \right)$

Step-by-Step Verified Solution

\sin 2\theta + \tan 2\theta &gt; 0

\Rightarrow \sin 2\theta + {{\sin 2\theta } \over {\cos 2\theta }} &gt; 0

\Rightarrow \sin 2\theta {{(\cos 2\theta + 1)} \over {\cos 2\theta }} &gt; 0 \Rightarrow \tan 2\theta (2{\cos ^2}\theta ) &gt; 0

Note :

\cos 2\theta \ne 0

\Rightarrow 1 - 2{\sin ^2}\theta \ne \theta \Rightarrow \sin \theta \ne \pm {1 \over {\sqrt 2 }}

Now,

\tan 2\theta (1 + \cos 2\theta ) &gt; 0

\Rightarrow \tan 2\theta &gt; 0

(as

\cos 2\theta + 1 &gt; 0

)

\Rightarrow 2\theta \in \left( {0,{\pi \over 2}} \right) \cup \left( {\pi ,{{3\pi } \over 2}} \right) \cup \left( {2\pi ,{{5\pi } \over 2}} \right) \cup \left( {3\pi ,{{7\pi } \over 2}} \right)

\Rightarrow \theta \in \left( {0,{\pi \over 4}} \right) \cup \left( {{\pi \over 2},{{3\pi } \over 4}} \right) \cup \left( {\pi ,{{5\pi } \over 4}} \right) \cup \left( {{{3\pi } \over 2},{{7\pi } \over 4}} \right)

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