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
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