MCQ
$\int\limits_0^{\frac{\pi }{2}} {\,\,\frac{{d\,x}}{{{{\cos }^6}x + \,{{\sin }^6}\,x}}}$ is equal to :
- Azero
- ✓$\pi$
- C$\pi /2$
- D$2 \pi $
$= \int\limits_0^{\frac{\pi }{2}} {\,\,\frac{{d\,x}}{{1\,\, - \,\,{\textstyle{3 \over 4}}\,{{\sin }^2}\,2x}}}$
$= 2 \int\limits_0^\pi {\,\,\frac{{d\,t}}{{4\,\, - \,\,3\,{{\sin }^2}\,t}}}$
where $2x = t$
$= 4 \, \int\limits_0^{\frac{\pi }{2}} {\,\,\frac{{d\,t}}{{4\,\, - \,\,3\,{{\sin }^2}\,t}}}$ etc.
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$x+2 y+z=7$
$x+\alpha z=11$
$2 x-3 y+\beta z=\gamma$
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