MCQ
If $A> 0, B > 0$ and $A + B = \frac{\pi }{6}$, then the minimum value of $tan\,A + tan\,B$ is
- A$\sqrt 3 - \sqrt 2 $
- ✓$ 4 - 2\sqrt 3 $
- C$\frac{2}{{\sqrt 3 }}$
- D$2 - \sqrt 3 $
$ \Rightarrow \,\frac{1}{{\sqrt 3 }}\, = \,\frac{y}{{1 - \tan \,A\,\tan \,B}}$ where $y\, = \tan \,A\, + \tan \,B$
$ \Rightarrow \tan \,A\,\tan \,B\, = \,1 - \sqrt 3 y$ Also $AM \geqslant GM$
$ \Rightarrow \,\frac{{\tan \,A\, + \,\tan \,B}}{2}\, \geqslant \,\sqrt {\tan \,A\,\tan \,B} $
$ \Rightarrow \,y\, \geqslant \,2\sqrt {1 - \sqrt 3 y} $
$ \Rightarrow \,{y^2}\, \geqslant \,4 - 4\sqrt 3 y$
$ \Rightarrow \,{y^2}\, + \,4\sqrt 3 y - 4 \geqslant 0$
$ \Rightarrow \,y\, \leqslant \, - \,2\sqrt 3 - 4$ or $ \Rightarrow \,y\, \geqslant \, - \,2\sqrt 3 + 4$
( $y\, \leqslant \, - \,2\sqrt 3 - 4$ is not possible as $\tan A\,\tan B\, > 0$
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[Note: Here $z$ takes the values in the complex plane and $\operatorname{Im} z$ and $\operatorname{Re} z$ denote, respectively, the imaginary part and the real part of $z]$
| column-$I$ | column-$II$ |
| $(A)$ The set of points $z$ satisfying $|z-i| z||=|z+i| z||$ is contained in or equal to | $(p)$ an ellipse with eccentricity $\frac{4}{5}$ |
| $(B)$ The set of points $z$ satisfying $|z+4|+|z-4|=10$ is contained in or equal to | $(q)$ the set of points $z$ satisfying $\operatorname{Im} z=0$ |
| $(C)$ If $|\omega|=2$, then the set of points $z=\omega-1 / \omega$ is contained in or equal to | $(r)$ the set of points $z$ satisfying $|\operatorname{Im} z| \leq 1$ |
| $(D)$ If $|\omega|=1$, then the set of points $z=\omega+1 / \omega$ is contained in or equal to | $(s)$ the set of points $z$ satisfying $|\operatorname{Re} z| \leq 1$ |
| $(t)$ the set of points $z$ satisfying $|z| \leq 3$ |