The system of linear equations $x + y + z = 6$, $2x + 5y + az = 36$, and $x + 2y + 3z = b$ has:

Let the system of linear equations $x+y+\alpha z=2$,$3x+y+z=4$,and $x+2z=1$ have a unique solution $(x^{*}, y^{*}, z^{*})$. If $(\alpha, x^{*}), (y^{*}, \alpha)$ and $(x^{*}, -y^{*})$ are collinear points,then the sum of absolute values of all possible values of $\alpha$ is

Let $a, \lambda, \mu \in \mathbb{R}$. Consider the system of linear equations:
$a x + 2 y = \lambda$
$3 x - 2 y = \mu$
Which of the following statement$(s)$ is(are) correct?
$(A)$ If $a = -3$,then the system has infinitely many solutions for all values of $\lambda$ and $\mu$.
$(B)$ If $a \neq -3$,then the system has a unique solution for all values of $\lambda$ and $\mu$.
$(C)$ If $\lambda + \mu = 0$,then the system has infinitely many solutions for $a = -3$.
$(D)$ If $\lambda + \mu \neq 0$,then the system has no solution for $a = -3$.

The system of equations $\lambda x - y + (\cos\theta) z = 0$,$3x + y + 2z = 0$,and $(\cos\theta) x + y + 2z = 0$ for $0 < \theta < 2\pi$ has non-trivial solution$(s)$:

In a legislative assembly election,a political group hired a public relations firm to promote its candidate in three ways: telephone,house calls,and letters. The cost per contact (in paise) is given in matrix $A$ as $A = \begin{bmatrix} 40 \\ 100 \\ 50 \end{bmatrix} \begin{matrix} \text{Telephone} \\ \text{Housecall} \\ \text{Letter} \end{matrix}$. The number of contacts of each type made in two cities $X$ and $Y$ is given by $B = \begin{bmatrix} 1000 & 500 & 5000 \\ 3000 & 1000 & 10000 \end{bmatrix} \begin{matrix} \text{Telephone} & \text{Housecall} & \text{Letter} \\ \to X \\ \to Y \end{matrix}$. Find the total amount spent by the group in the two cities $X$ and $Y$.

The set of real values of $\alpha$ for which the system of linear equations
$\begin{aligned}
& x+(\sin \alpha) y+(\cos \alpha) z=0 \\
& x+(\cos \alpha) y+(\sin \alpha) z=0 \\
& -x+(\sin \alpha) y-(\cos \alpha) z=0
\end{aligned}$
has a non-trivial solution is