Let $\alpha$, $\beta$ and $\gamma$ be real numbers such that the system of linear equations $$x + 2y + 3z = \alpha$$ $$4x + 5y + 6z = \beta$$ $$7x + 8y + 9z = \gamma - 1$$ is consistent. Let $|M|$ represent the determinant of the matrix $$M = \begin{pmatrix} \alpha & 1 & 0 \\ \beta & 2 & 1 \\ \gamma & 1 & 0 \end{pmatrix}.$$ Let $P$ be the plane containing all those $(\alpha, \beta, \gamma)$ for which the above system of linear equations is consistent, and $D$ be the square of the distance of the point $(0, 1, 0)$ from the plane $P$.
The value of $|M|$ is ____.

Let $\alpha$, $\beta$ and $\gamma$ be real numbers such that the system of linear equations $$x + 2y + 3z = \alpha$$ $$4x + 5y + 6z = \beta$$ $$7x + 8y + 9z = \gamma - 1$$ is consistent. Let $|M|$ represent the determinant of the matrix $$M = \begin{pmatrix} \alpha & 1 & 0 \\ \beta & 2 & 1 \\ \gamma & 1 & 0 \end{pmatrix}.$$ Let $P$ be the plane containing all those $(\alpha, \beta, \gamma)$ for which the above system of linear equations is consistent, and $D$ be the square of the distance of the point $(0, 1, 0)$ from the plane $P$.
The value of $D$ is ____.

Let $p , q , r$ be nonzero real numbers that are, respectively, the $10 ^ { \text {th } } , 100 ^ { \text {th } }$ and $1000 ^ { \text {th } }$ terms of a harmonic progression. Consider the system of linear equations
$$\begin{gathered} x + y + z = 1 \\ 10 x + 100 y + 1000 z = 0 \\ q r x + p r y + p q z = 0 \end{gathered}$$
List-I (I) If $\frac { q } { r } = 10$, then the system of linear equations has (II) If $\frac { p } { r } \neq 100$, then the system of linear equations has (III) If $\frac { p } { q } \neq 10$, then the system of linear equations has (IV) If $\frac { p } { q } = 10$, then the system of linear equations has
List-II (P) $x = 0 , \quad y = \frac { 10 } { 9 } , z = - \frac { 1 } { 9 }$ as a solution (Q) $x = \frac { 10 } { 9 } , \quad y = - \frac { 1 } { 9 } , z = 0$ as a solution (R) infinitely many solutions (S) no solution (T) at least one solution
The correct option is:
(A) (I) → (T); (II) → (R); (III) → (S); (IV) → (T)
(B) (I) → (Q); (II) → (S); (III) → (S); (IV) → (R)
(C) (I) → (Q); (II) → (R); (III) → (P); (IV) → (R)
(D) (I) → (T); (II) → (S); (III) → (P); (IV) → (T)

Let $\alpha , \beta$ and $\gamma$ be real numbers. Consider the following system of linear equations
$x + 2 y + z = 7$
$x + \alpha z = 11$
$2 x - 3 y + \beta z = \gamma$
Match each entry in List-I to the correct entries in List-II.
List-I
(P) If $\beta = \frac { 1 } { 2 } ( 7 \alpha - 3 )$ and $\gamma = 28$, then the system has
(Q) If $\beta = \frac { 1 } { 2 } ( 7 \alpha - 3 )$ and $\gamma \neq 28$, then the system has
(R) If $\beta \neq \frac { 1 } { 2 } ( 7 \alpha - 3 )$ where $\alpha = 1$ and $\gamma \neq 28$, then the system has
(S) If $\beta \neq \frac { 1 } { 2 } ( 7 \alpha - 3 )$ where $\alpha = 1$ and $\gamma = 28$, then the system has
List-II
(1) a unique solution
(2) no solution
(3) infinitely many solutions
(4) $x = 11 , y = - 2$ and $z = 0$ as a solution
(5) $x = - 15 , y = 4$ and $z = 0$ as a solution
The correct option is:
(A) $( P ) \rightarrow ( 3 ) \quad ( Q ) \rightarrow ( 2 ) \quad ( R ) \rightarrow ( 1 ) \quad ( S ) \rightarrow ( 4 )$
(B) $( P ) \rightarrow ( 3 ) \quad ( Q ) \rightarrow ( 2 ) \quad ( R ) \rightarrow ( 5 ) \quad ( S ) \rightarrow ( 4 )$
(C) $( P ) \rightarrow ( 2 ) \quad ( Q ) \rightarrow ( 1 ) \quad ( R ) \rightarrow ( 4 ) \quad ( S ) \rightarrow ( 5 )$
(D) $( P ) \rightarrow ( 2 ) \quad ( Q ) \rightarrow ( 1 ) \quad ( R ) \rightarrow ( 1 ) \quad ( S ) \rightarrow ( 3 )$

The point of intersection of the lines $\left( a ^ { 3 } + 3 \right) x + a y + a - 3 = 0$ and $\left( a ^ { 5 } + 2 \right) x + ( a + 2 ) y + 2 a + 3 = 0$ (a real) lies on the $y$-axis for
(1) no value of $a$
(2) more than two values of $a$
(3) exactly one value of $a$
(4) exactly two values of $a$

If the system of equations $$\begin{aligned} & x + y + z = 6 \\ & x + 2 y + 3 z = 10 \\ & x + 2 y + \lambda z = 0 \end{aligned}$$ has a unique solution, then $\lambda$ is not equal to
(1) 1
(2) 0
(3) 2
(4) 3

The number of values of $k$, for which the system of equations: $(k+1)x + 8y = 4k$ $kx + (k+3)y = 3k - 1$ has no solution, is:
(1) 2
(2) 3
(3) Infinite
(4) 1

If $a , b , c$ are non-zero real numbers and if the system of equations $$( a - 1 ) x = y + z$$ $$( b - 1 ) y = x + z$$ $$( c - 1 ) z = x + y$$ has a non-trivial solution, then $ab + bc + ca$ equals:
(1) $-1$
(2) $a + b + c$
(3) $abc$
(4) 1

If $S$ is the set of distinct values of $b$ for which the following system of linear equations
$$\begin{aligned} & x + y + z = 1 \\ & x + ay + z = 1 \\ & ax + by + z = 0 \end{aligned}$$
has no solution, then $S$ is:
(1) An empty set
(2) An infinite set
(3) A finite set containing two or more elements
(4) A singleton

If the system of linear equations $x + k y + 3 z = 0$ $3 x + k y - 2 z = 0$ $2 x + 4 y - 3 z = 0$ has a non-zero solution $( x , y , z )$, then $\frac { x z } { y ^ { 2 } }$ is equal to:
(1) 30
(2) - 10
(3) 10
(4) - 30

Let $S$ be the set of all real values of $k$ for which the system of linear equations $x + y + z = 2$ $2 x + y - z = 3$ $3 x + 2 y + k z = 4$ has a unique solution. Then, $S$ is :
(1) equal to $R - \{ 0 \}$
(2) an empty set
(3) equal to $R$
(4) equal to $\{ 0 \}$

Let $S$ be the set of all real values of $k$ for which the system of linear equations
$$\begin{aligned} & x + y + z = 2 \\ & 2 x + y - z = 3 \\ & 3 x + 2 y + k z = 4 \end{aligned}$$
has a unique solution. Then $S$ is
(1) an empty set
(2) equal to $\mathrm { R } - \{ 0 \}$
(3) equal to $\{ 0 \}$
(4) equal to $R$

If the system of linear equations $x - 4y + 7z = g$; $3y - 5z = h$; $-2x + 5y - 9z = k$ is consistent, then:
(1) $g + h + 2k = 0$
(2) $g + 2h + k = 0$
(3) $2g + h + k = 0$
(4) $g + h + k = 0$

If the system of equations $x + y + z = 5 , x + 2 y + 3 z = 9 , x + 3 y + \alpha z = \beta$ has infinitely many solutions, then $\beta - \alpha$ equals
(1) 8
(2) 21
(3) 5
(4) 18

If the system of linear equations $$\begin{aligned} & x - 2 y + k z = 1 \\ & 2 x + y + z = 2 \\ & 3 x - y - k z = 3 \end{aligned}$$ has a solution $( x , y , z )$, $z \neq 0$, then $( x , y )$ lies on the straight line whose equation is:
(1) $4 x - 3 y - 4 = 0$
(2) $3 x - 4 y - 4 = 0$
(3) $3 x - 4 y - 1 = 0$
(4) $4 x - 3 y - 1 = 0$

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 $\lambda = 2$

Let $S$ be the set of all $\lambda \in R$ for which the system of linear equations $$2x - y + 2z = 2$$ $$x - 2y + \lambda z = -4$$ $$x + \lambda y + z = 4$$ has no solution. Then the set $S$
(1) Contains more than two elements
(2) Is an empty set
(3) Is a singleton
(4) Contains exactly two elements

If the system of equations $x + y + z = 2$ $2 x + 4 y - z = 6$ $3 x + 2 y + \lambda z = \mu$ has infinitely many solutions, then:
(1) $\lambda + 2 \mu = 14$
(2) $2 \lambda - \mu = 5$
(3) $\lambda - 2 \mu = - 5$
(4) $2 \lambda + \mu = 14$

The values of $\lambda$ and $\mu$ for which the system of linear equations $x + y + z = 2 , x + 2 y + 3 z = 5$, $x + 3 y + \lambda z = \mu$ has infinitely many solutions, are respectively
(1) 6 and 8
(2) 5 and 7
(3) 5 and 8
(4) 4 and 9

For the system of linear equations: $$x - 2 y = 1 , x - y + k z = - 2 , k y + 4 z = 6 , k \in R$$ Consider the following statements:
(A) The system has unique solution if $k \neq 2 , k \neq - 2$.
(B) The system has unique solution if $k = - 2$.
(C) The system has unique solution if $k = 2$.
(D) The system has no-solution if $k = 2$.
(E) The system has infinitely many solutions if $k = - 2$.

Consider the following system of equations: $$\begin{aligned} & x + 2 y - 3 z = a \\ & 2 x + 6 y - 11 z = b \\ & x - 2 y + 7 z = c \end{aligned}$$ where $a , b$ and $c$ are real constants. Then the system of equations :
(1) has a unique solution when $5 a = 2 b + c$
(2) has no solution for all $a , b$ and $c$
(3) has infinite number of solutions when $5 a = 2 b + c$
(4) has a unique solution for all $a , b$ and $c$

The number of integral values of $m$ so that the abscissa of point of intersection of lines $3 x + 4 y = 9$ and $y = m x + 1$ is also an integer, is:
(1) 1
(2) 2
(3) 3
(4) 0

The system of linear equations $3 x - 2 y - k z = 10$ $2 x - 4 y - 2 z = 6$ $x + 2 y - z = 5 m$ is inconsistent if:
(1) $k = 3 , \quad m \neq \frac { 4 } { 5 }$
(2) $k = 3 , \quad m = \frac { 4 } { 5 }$
(3) $k \neq 3 , \quad m \in R$
(4) $k \neq 3 , \quad m \neq \frac { 4 } { 5 }$

Let $[ \lambda ]$ be the greatest integer less than or equal to $\lambda$. The set of all values of $\lambda$ for which the system of linear equations $x + y + z = 4,3 x + 2 y + 5 z = 3,9 x + 4 y + ( 28 + [ \lambda ] ) z = [ \lambda ]$ has a solution is: (1) $R$ (2) $( - \infty , - 9 ) \cup [ - 8 , \infty )$ (3) $( - \infty , - 9 ) \cup ( - 9 , \infty )$ (4) $[ - 9 , - 8 )$

The following system of linear equations $2 x + 3 y + 2 z = 9$ $3 x + 2 y + 2 z = 9$ $x - y + 4 z = 8$
(1) has infinitely many solutions
(2) has a unique solution
(3) has a solution ( $\alpha , \beta , \gamma$ ) satisfying $\alpha + \beta ^ { 2 } + \gamma ^ { 3 } = 12$
(4) does not have any solution