Question

The system of equations x + ky + 3z = 0, 3x + ky - 2z = 0, 2x + 3y - 4z = 0 possess a non-trivial solution over the set of rationals, then 2k is an integral element of the interval

• A. [10, 20]
• B. (20, 30)
• C. [30, 40]
• D. (40, 50)

Answer 1

The correct option is C [30, 40]

For the given system to have a non-trival solution, we must have
$\left|\begin{array}{ccc}1& k& 3\\ 3& k& -2\\ 2& 3& -4\end{array}\right|=0\Rightarrow \left|\begin{array}{ccc}1& k& 3\\ 0& -2k& -11\\ 0& 3-2k& -10\end{array}\right|=0$
[Applying $R_2\to R_2-3R_1,R_3\to R_3-2R_1$]
$\Rightarrow$ 20k +11 (3-2k) = 0 $\Rightarrow k=\frac{33}{2}$