MCQ
${2^{\sin \theta }} + {2^{\cos \theta }}$ is greater than
- A$\frac{1}{2}$
- B$\sqrt 2 $
- C${2^{\frac{1}{{\sqrt 2 }}}}$
- ✓${2^{\left( {1 - \,\frac{1}{{\sqrt 2 }}} \right)}}$
(${\rm{A}}{\rm{.M}}{\rm{.}} \ge {\rm{G}}{\rm{.M}}{\rm{.}}$)
==> ${2^{\sin \theta }} + {2^{\cos \theta }} \ge {2.2^{(\sin \theta + \cos \theta )/2}}$ .....(i)
Now $(\sin \theta + \cos \theta ) = \sqrt 2 \sin (\theta + \pi /4) \ge - \sqrt 2 $
For all real $\theta$,
${2^{\sin \theta }} + {2^{\cos \theta }} \ge {2.2^{(\sin \theta + \cos \theta )/2}} > 2\,.\,{2^{ - \sqrt 2 /2}}$
==> ${2^{\sin \theta }} + {2^{\cos \theta }} \ge {2^{1 - (1/\sqrt 2 )}}$.
Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.
$x+2 y+z=7$
$x+\alpha z=11$
$2 x-3 y+\beta z=\gamma$
Match each entry in List - $I$ to the correct entries in List-$II$
| List - $I$ | List - $II$ |
| ($P$) If $\beta=\frac{1}{2}(7 \alpha-3)$ and $\gamma=28$, then the system has | ($1$) a unique solution |
| ($Q$) If $\beta=\frac{1}{2}(7 \alpha-3)$ and $\gamma \neq 28$, then the system has | ($2$) no solution |
|
($R$) If $\beta \neq \frac{1}{2}(7 \alpha-3)$ where $\alpha=1$ and $\gamma \neq 28$, then the system has |
($3$) infinitely many solutions |
| ($S$) If $\beta \neq \frac{1}{2}(7 \alpha-3)$ where $\alpha=1$ and $\gamma=28$, then the system has | ($4$) $x=11, y=-2$ and $z=0$ as a solution |
| ($5$) $x=-15, y=4$ and $z=0$ as a solution |
Then the system has