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
$\therefore \begin{vmatrix}1&k&3\\ 3&k&-2\\ 2&3&-4\end{vmatrix} = 0$
$\Rightarrow \quad1 \left(- 4k + 6\right) - k\left(-12 + 4\right) + 3 \left(9 - 2k\right) = 0$
$\Rightarrow \quad 4k+33-6k=0 \Rightarrow k = \frac{33}{2}$
Hence, statement -1 is false.
Statement-2 is the property.
It is a true statement.