Consider the system of equations $$\begin{aligned} & x - 2 y + 3 z = - 1 \\ & - x + y - 2 z = k \\ & x - 3 y + 4 z = 1 . \end{aligned}$$ STATEMENT-1 : The system of equations has no solution for $k \neq 3$. and STATEMENT-2 : The determinant $\left| \begin{array} { c c c } 1 & 3 & - 1 \\ - 1 & - 2 & k \\ 1 & 4 & 1 \end{array} \right| \neq 0$, for $k \neq 3$.
(A) STATEMENT-1 is True, STATEMENT-2 is True; STATEMENT-2 is a correct explanation for STATEMENT-1
(B) STATEMENT-1 is True, STATEMENT-2 is True; STATEMENT-2 is NOT a correct explanation for STATEMENT-1
(C) STATEMENT-1 is True, STATEMENT-2 is False
(D) STATEMENT-1 is False, STATEMENT-2 is True

The set of all values of $\lambda$ for which the system of linear equations: $2x_1 - 2x_2 + x_3 = \lambda x_1$ $2x_1 - 3x_2 + 2x_3 = \lambda x_2$ $-x_1 + 2x_2 = \lambda x_3$ has a non-trivial solution:
(1) is an empty set
(2) is a singleton
(3) contains two elements
(4) contains more than two elements

The set of all values of $\lambda$ for which the system of linear equations: $2 x _ { 1 } - 2 x _ { 2 } + x _ { 3 } = \lambda x _ { 1 }$ $2 x _ { 1 } - 3 x _ { 2 } + 2 x _ { 3 } = \lambda x _ { 2 }$ $- x _ { 1 } + 2 x _ { 2 } = \lambda x _ { 3 }$ has a non-trivial solution,
(1) Contains more than two elements.
(2) Is an empty set.
(3) Is a singleton.
(4) Contains two elements.

If the system of linear equations $$\begin{aligned} & 2x + 2ay + az = 0 \\ & 2x + 3by + bz = 0 \\ & 2x + 4cy + cz = 0 \end{aligned}$$ where $a, b, c \in R$ are non-zero and distinct; has a non-zero solution, then
(1) $\frac { 1 } { a } , \frac { 1 } { b } , \frac { 1 } { c }$ are in $A.P$.
(2) $a, b, c$ are in $G.P$.
(3) $a + b + c = 0$
(4) $a, b, c$ are in $A.P$.

The system of linear equations $\lambda x + 2y + 2z = 5$ $2\lambda x + 3y + 5z = 8$ $4x + \lambda y + 6z = 10$ has
(1) no solution when $\lambda = 8$
(2) a unique solution when $\lambda = -8$
(3) no solution when $\lambda = 2$
(4) infinitely many solutions when $\lambda = 2$

The following system of linear equations $7 x + 6 y - 2 z = 0$ $3 x + 4 y + 2 z = 0$ $x - 2 y - 6 z = 0$, has
(1) infinitely many solutions, ( $x , y , z$ ) satisfying $y = 2z$
(2) no solution
(3) infinitely many solutions, $( x , y , z )$ satisfying $x = 2z$
(4) only the trivial solution

Let $\lambda \in \mathrm { R }$. The system of linear equations $2 x _ { 1 } - 4 x _ { 2 } + \lambda x _ { 3 } = 1$ $x _ { 1 } - 6 x _ { 2 } + x _ { 3 } = 2$ $\lambda x _ { 1 } - 10 x _ { 2 } + 4 x _ { 3 } = 3$ is inconsistent for :
(1) exactly one positive value of $\lambda$
(2) exactly one negative value of $\lambda$
(3) every value of $\lambda$
(4) exactly two values of $\lambda$

The number of $\theta \in (0,4\pi)$ for which the system of linear equations $3(\sin 3\theta) x - y + z = 2$ $3(\cos 2\theta) x + 4y + 3z = 3$ $6x + 7y + 7z = 9$ has no solution is
(1) 6
(2) 7
(3) 8
(4) 9