MCQ
$\int_0^\pi {\frac{{xdx}}{{1 + \sin x}}} $ is equal to
- A$ - \pi $
- B$\frac{\pi }{2}$
- ✓$\pi $
- DNone of these
$I = \int_0^\pi {\frac{{(\pi - x)dx}}{{1 + \sin (\pi - x)}}} $
$I = \int_0^\pi {\frac{{(\pi - x)dx}}{{1 + \sin x}}} $ ... $(ii),$
$\left\{ \because \,\int_{0}^{a}{f(x)\,dx}=\int_{0}^{a}{f(a-x)\,dx} \right\}\,$
Adding $(i)$ and $(ii),$ we get
$2I = \int_0^\pi {\frac{{\pi \,dx}}{{1 + \sin x}}} $
$2I = \pi \int_0^\pi {\frac{{1 - \sin x}}{{(1 + \sin x)(1 - \sin x)}}dx} $
$2I = \pi \int_0^\pi {\frac{{1 - \sin x}}{{{{\cos }^2}x}}} dx = \pi \int_0^\pi {({{\sec }^2}x - \sec x\tan x)dx} $
$2I = \pi [\tan x - \sec x]_0^\pi = \pi [0 - ( - 1) - (0 - 1)]$, $2I = 2\pi $
$\therefore$ $I = \pi $.
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$x+2 y+3 z=\alpha$
$4 x+5 y+6 z=\beta$
$7 x+8 y+9 z=\gamma-$
is consistent. Let $| M |$ represent the determinant of the matrix
$M=\left[\begin{array}{ccc}\alpha & 2 & \gamma \\ \beta & 1 & 0 \\ -1 & 0 & 1\end{array}\right]$
Let $P$ be the plane containing all those $(\alpha, \beta, \gamma)$ for which the above system of linear equations is consistent, and $D$ be the square of the distance of the point $(0,1,0)$ from the plane $P$.
($1$) The value of $| M |$ is
($2$) The value of $D$ is