Statement 1: If the system of equations $x+k y+$ $3 z=0,3 x+k y-2 z=0,2 x+3 y-4 z=0$ has a nontrivial…
Statement 1: If the system of equations $x+k y+$ $3 z=0,3 x+k y-2 z=0,2 x+3 y-4 z=0$ has a nontrivial solution, then the value of $k$ is $\frac{31}{2}$.
Statement 2: A system of three homogeneous equations in three variables has a non trivial solution if the determinant of the coefficient matrix is zero.
Statement 1 is false, Statement 2 is true.
Statement 1 is true, Statement 2 is true, Statement 2 is a correct explanation for Statement 1.
Statement 1 is true, Statement 2 is true,, Statement 2 is not a correct explanation for Statement 1.
Statement 1 is true, Statement 2 is false.
Solution
Given system of equations is
$
\begin{aligned}
& x+k y+3 z=0 \\
& 3 x+k y-2 z=0 \\
& 2 x+3 y-4 z=0
\end{aligned}
$
Since, system has non-trivial solution
$
\begin{aligned}
& \therefore\left|\begin{array}{ccc}
1 & k & 3 \\
3 & k & -2 \\
2 & 3 & -4
\end{array}\right|=0 \\
\Rightarrow & 1(-4 k+6)-k(-12+4)+3(9-2 k)=0 \\
\Rightarrow & 4 k+33-6 k=0 \Rightarrow k=\frac{33}{2}
\end{aligned}
$
Hence, statement - 1 is false.
Statement- 2 is the property.
It is a true statement