9. A plane is parallel to two lines whose direction ratios are $( 1,0 , - 1 )$ and $( - 1,1,0 )$ and it contains the point $( 1,1,1 )$. If it cuts coordinate axis at $\mathrm { A } , \mathrm { B } , \mathrm { C }$, then find the volume of the tetrahedron OABC . Sol. Let $( l , m , n )$ be the direction ratios of the normal to the required plane so that $l - n = 0$ and $- l + m = 0$ $\Rightarrow \mathrm { l } = \mathrm { m } = \mathrm { n }$ and hence the equation of the plane containing $( 1,1,1 )$ is $\frac { \mathrm { x } } { 3 } + \frac { \mathrm { y } } { 3 } + \frac { \mathrm { z } } { 3 } = 1$. Its intercepts with the coordinate axes are $\mathrm { A } ( 3,0,0 ) ; \mathrm { B } ( 0,3,0 ) ; \mathrm { C } ( 0,0,3 )$. Hence the volume of OABC $= \frac { 1 } { 6 } \left| \begin{array} { l l l } 3 & 0 & 0 \\ 0 & 3 & 0 \\ 0 & 0 & 3 \end{array} \right| = \frac { 27 } { 6 } = \frac { 9 } { 2 }$ cubic units.
If $\left( \frac { 2 + \sin x } { 1 + y } \right) \frac { d y } { d x } = - \cos x , y ( 0 ) = 1$, then $y \left( \frac { \pi } { 2 } \right) =$
9. A plane is parallel to two lines whose direction ratios are $( 1,0 , - 1 )$ and $( - 1,1,0 )$ and it contains the point $( 1,1,1 )$. If it cuts coordinate axis at $\mathrm { A } , \mathrm { B } , \mathrm { C }$, then find the volume of the tetrahedron OABC .
Sol. Let $( l , m , n )$ be the direction ratios of the normal to the required plane so that $l - n = 0$ and $- l + m = 0$\\
$\Rightarrow \mathrm { l } = \mathrm { m } = \mathrm { n }$ and hence the equation of the plane containing $( 1,1,1 )$ is $\frac { \mathrm { x } } { 3 } + \frac { \mathrm { y } } { 3 } + \frac { \mathrm { z } } { 3 } = 1$.\\
Its intercepts with the coordinate axes are $\mathrm { A } ( 3,0,0 ) ; \mathrm { B } ( 0,3,0 ) ; \mathrm { C } ( 0,0,3 )$. Hence the volume of OABC $= \frac { 1 } { 6 } \left| \begin{array} { l l l } 3 & 0 & 0 \\ 0 & 3 & 0 \\ 0 & 0 & 3 \end{array} \right| = \frac { 27 } { 6 } = \frac { 9 } { 2 }$ cubic units.\\