Homogeneous System of Linear Equations - Practice Questions & MCQ

Let $\lambda$ and $\alpha$ be real. Find the set of all values $\lambda$ for which the system of linear equations

$
\begin{aligned}
& \lambda x+\sin \alpha \cdot \mathrm{y}+\cos \alpha \cdot \mathrm{z}=0 \\
& \mathrm{x}+\cos \alpha \cdot \mathrm{y}+\sin \alpha \cdot \mathrm{z}=0 \\
& -\mathrm{x}+\sin \alpha \cdot \mathrm{y}-\cos \alpha \cdot \mathrm{z}=0
\end{aligned}
$

Has a non-trivial solution. For $\lambda=1$, The possible values of $\alpha$ are

An ordered pair $(\alpha, \beta)$ for which the system of linear equations

$
\begin{aligned}
& (1+\alpha) x+\beta y+z=2 \\
& \alpha x+(1+\beta) y+z=3 \\
& \alpha x+\beta y+2 z=2
\end{aligned}
$

has a unique solution, is :