Question bank/JEE Main/Mathematics

The positive real value of k so that planes
$x - y + z + 1 = 0,kx + 3y + 2z - 3 = 0,3x + \ \ \ \ \ \ \ \ \ \ \ \ \ \ ky + z - 2 = 0\$form
a triangular prism, is equal to

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[IMAGE] For triangular prism
$D = 0,D_{1} \neq 0,D_{2} \neq 0,D_{3} \neq 0$
$D = \left| \begin{matrix}
1 & - 1 & 1 \\
k & 3 & 2 \\
3 & k & 1
\end{matrix} \right| = (k + 3)(k - 4)$
$D_{1} = \left| \begin{matrix}
- 1 & - 1 & 1 \\
3 & 3 & 2 \\
2 & k & 1
\end{matrix} \right| = 5(k - 2),D_{2} = \left| \begin{matrix}
1 & - 1 & 1 \\
k & 3 & 2 \\
3 & 2 & 1
\end{matrix} \right|,D_{2} = 3k - 16$
$D_{3} = \left| \begin{matrix}
- 1 & - 1 & 1 \\
3 & 3 & 2 \\
2 & k & 1
\end{matrix} \right| = 5(k - 2)$
$\ If\ \ k = 4, \Rightarrow D = 0,D_{1} \neq 0,D_{2} \neq 0,D_{3} \neq 0$

Source: Grass-G / M-28 Mathematics Paper FST 16 Jan.docx