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Statement $1$ : If the system of equations $x + ky + 3z = 0, 3x+ ky - 2z = 0, 2x + 3y - 4z = 0$ 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 equation
$x + ky + 3z = 0$
$3x + ky – 2z = 0$
$2x + 3y – 4z = 0$
Since, system has non-trivial solution
$\therefore \left| {\begin{array}{*{20}{c}}
1&k&3\\
3&k&{ – 2}\\
2&3&{ – 4}
\end{array}} \right| = 0$
$ \Rightarrow 1\left( { – 4k + 6} \right) – k\left( { – 12 + 4} \right) + 3\left( {9 – 2k} \right) = 0$
$ \Rightarrow 4k + 33 – 6k = 0 \Rightarrow k = \frac{{33}}{2}$
Hence, statement – $1$ is false.
Statement – $2$ is the property.
It is a true statement .
