Question

The value(s) of $k\in R$ for which the system of equations
$x+ky+3z=0,$
$kx+2y+2z=0$ and
$2x+3y+4z=0$
admits a non-trivial solution, is

• A. $2$
  
• B. $\frac{5}{2}$
  
• C. $3$
  
• D. $\frac{5}{4}$

Answer 1

Answer: (A), (D)

$\frac{5}{4}$

$\left|\begin{array}{ccc}1& k& 3\\ k& 2& 2\\ 2& 3& 4\end{array}\right|=0$
$\Rightarrow 8+4k+9k-12-4k^2-6=0$
$\Rightarrow 4k^2-13k+10=0$
$\Rightarrow 4k^2-8k-5k+10=0$
$\Rightarrow (4k-5)(k-2)=0$
$\Rightarrow k=\frac{5}{4},2$