Question

Consider the system of linear equations $x \sin \alpha-3 y \cos \alpha- kz =0 ; 2 x + y + z =0,- x + y +$ $2 z =0$ then

• A. the number of integral values of $k$ for which system of equations has non-trivial solution are 11.
• B. for $k=-4$ the system of equation has infinite solution.
• C. for $k=6$ the system of equation has unique solution.
• D. for $k=-6$ the system of equation has trivial solution.

Answer 1

Answer: (A), (B), (C), (D)

$\begin{vmatrix}\sin \alpha & -3 \cos \alpha & - k \\ 2 & 1 & 1 \\ -1 & 1 & 2\end{vmatrix}=0$
$\sin \alpha+3 \cos \alpha(5)- k (3)=0 $
$\sin \alpha+15 \cos \alpha=3 k $
$-\sqrt{226} \leq 3 k \leq \sqrt{226} $
$-\frac{\sqrt{226}}{3} \leq k \leq \frac{\sqrt{226}}{3}$
So, 11 integral values of ' $k$ '.