Question

Statement 1 : If the system of equations $x+ky+3z=0,3x+ky-2z=0,2x+3ty-4z=0$ has a non- trivial solution, then the value of 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.

• A. Statement 1 is false, Statement 2 is true
• B. Statement 1 is true, Statement 2 is true, Statement 2 is a correct explanation for Statement 1
• C. Statement 1 is true, Statement 2 is true, , Statement 2 is not a correct explanation for Statement 1
• D. Statement 1 is true, Statement 2 is false

Answer 1

Answer: (A)

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