If the system of equations $3x - 2y + z = 0$, $\lambda x - 14y + 15z = 0$, and $x + 2y - 3z = 0$ has a solution other than $x = y = z = 0$, then $\lambda = $

Give the correct order of initials $T$ or $F$ for following statements. Use $T$ if statement is true and $F$ if it is false.
Statement $-1$ : If the graphs of two linear equations in two variables are neither parallel nor the same,then there is a unique solution to the system.
Statement $-2$ : If the system of equations $ax + by = 0, cx + dy = 0$ has a non-zero solution,then it has infinitely many solutions.
Statement $-3$ : The system $x + y + z = 1, x = y, y = 1 + z$ is inconsistent.
Statement $-4$ : If two of the equations in a system of three linear equations are inconsistent,then the whole system is inconsistent.

Let $n$ be the number obtained on rolling a fair die. If the probability that the system of equations
$x-ny+z=6$
$x+(n-2)y+(n+1)z=8$
$(n-1)y+z=1$
has a unique solution is $\frac{k}{6}$, then the sum of $k$ and all possible values of $n$ is:

Let $A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}$ be a $2 \times 2$ real matrix with $\det(A) = 1$. If the equation $\det(A - \lambda I_2) = 0$ has imaginary roots (where $I_2$ is the identity matrix of order $2$), then:

Consider the system of equations: $x + ay = 0$,$y + az = 0$,and $z + ax = 0$. The set of all real values of $a$ for which the system has a unique solution is:

If the system of linear equations
$7x + 11y + \alpha z = 13$
$5x + 4y + 7z = \beta$
$175x + 194y + 57z = 361$
has infinitely many solutions,then $\alpha + \beta + 2$ is equal to