System of equations begin{array}{l}α x + y + z = α - 1x + α y + z = α - 1x + y + α z = α

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3 and 4 .Determinants and Matrices

hard

The system of equations $\begin{array}{l}\alpha x + y + z = \alpha - 1\\x + \alpha y + z = \alpha - 1\\x + y + \alpha z = \alpha - 1\end{array}$ has no solution, if $\alpha $ is

Not $ -2$

$1$

$-2$

Either $ -2$ or $ 1$

(AIEEE-2005)

Solution

(c) For no solution or infinitely many solutions $\left| {\,\begin{array}{*{20}{c}}\alpha &1&1\\1&\alpha &1\\1&1&\alpha \end{array}\,} \right| = 0 \Rightarrow \alpha = 1,\alpha = – 2$.

But for $\alpha = 1$, clearly there are infinitely many solutions and when we put $\alpha = – 2$ in given system of equations and adding them together $L.H.S$ $ \ne $ $R.H.S$. i.e., No solution.

Standard 12

Mathematics

The determinant $\left| {\,\begin{array}{*{20}{c}}a&b&{a – b}\\b&c&{b – c}\\2&1&0\end{array}\,} \right|$ is equal to zero if $a,b,c$ are in

easy

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 – $I$ List – $II$

($P$) If $\beta=\frac{1}{2}(7 \alpha-3)$ and $\gamma=28$, then the system has ($1$) a unique solution

($Q$) If $\beta=\frac{1}{2}(7 \alpha-3)$ and $\gamma \neq 28$, then the system has ($2$) no solution

($R$) If $\beta \neq \frac{1}{2}(7 \alpha-3)$ where $\alpha=1$ and $\gamma \neq 28$,

then the system has
($3$) infinitely many solutions

($S$) If $\beta \neq \frac{1}{2}(7 \alpha-3)$ where $\alpha=1$ and $\gamma=28$, then the system has ($4$) $x=11, y=-2$ and $z=0$ as a solution

($5$) $x=-15, y=4$ and $z=0$ as a solution

Then the system has

medium

(IIT-2023)

The value of the determinant $\left| {\begin{array}{*{20}{c}}{{a^2}}&a&1\\{\cos \,(nx)}&{\cos \,(n\, + \,1)\,x}&{\cos \,(n\, + \,2)\,x}\\{\sin \,(nx)}&{\sin \,(n\, + \,1)\,x}&{\sin \,(n\, + \,2)\,x}\end{array}} \right|$ is independent of :

normal

If the system of equations

$ 2 x+7 y+\lambda z=3 $

$ 3 x+2 y+5 z=4 $

$ x+\mu y+32 z=-1$

has infinitely many solutions, then $(\lambda-\mu)$ is equal to $\qquad$

hard

(JEE MAIN-2024)

If $a$, $b$, $c$, $d$, $e$, $f$ are in $G.P$., then the value of $\left| {\begin{array}{*{20}{c}}
{{a^2}}&{{d^2}}&x \\
{{b^2}}&{{e^2}}&y \\
{{c^2}}&{{f^2}}&z
\end{array}} \right|$ depends on

normal

List – $I$	List – $II$
($P$) If $\beta=\frac{1}{2}(7 \alpha-3)$ and $\gamma=28$, then the system has	($1$) a unique solution
($Q$) If $\beta=\frac{1}{2}(7 \alpha-3)$ and $\gamma \neq 28$, then the system has	($2$) no solution
($R$) If $\beta \neq \frac{1}{2}(7 \alpha-3)$ where $\alpha=1$ and $\gamma \neq 28$, then the system has	($3$) infinitely many solutions
($S$) If $\beta \neq \frac{1}{2}(7 \alpha-3)$ where $\alpha=1$ and $\gamma=28$, then the system has	($4$) $x=11, y=-2$ and $z=0$ as a solution
	($5$) $x=-15, y=4$ and $z=0$ as a solution

The system of equations $\begin{array}{l}\alpha x + y + z = \alpha - 1\\x + \alpha y + z = \alpha - 1\\x + y + \alpha z = \alpha - 1\end{array}$ has no solution, if $\alpha $ is

Solution

Similar Questions

The determinant $\left| {\,\begin{array}{*{20}{c}}a&b&{a – b}\\b&c&{b – c}\\2&1&0\end{array}\,} \right|$ is equal to zero if $a,b,c$ are in

The value of the determinant $\left| {\begin{array}{*{20}{c}}{{a^2}}&a&1\\{\cos \,(nx)}&{\cos \,(n\, + \,1)\,x}&{\cos \,(n\, + \,2)\,x}\\{\sin \,(nx)}&{\sin \,(n\, + \,1)\,x}&{\sin \,(n\, + \,2)\,x}\end{array}} \right|$ is independent of :

If the system of equations $ 2 x+7 y+\lambda z=3 $ $ 3 x+2 y+5 z=4 $ $ x+\mu y+32 z=-1$ has infinitely many solutions, then $(\lambda-\mu)$ is equal to $\qquad$

If $a$, $b$, $c$, $d$, $e$, $f$ are in $G.P$., then the value of $\left| {\begin{array}{*{20}{c}} {{a^2}}&{{d^2}}&x \\ {{b^2}}&{{e^2}}&y \\ {{c^2}}&{{f^2}}&z \end{array}} \right|$ depends on

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If the system of equations

$ 2 x+7 y+\lambda z=3 $

$ 3 x+2 y+5 z=4 $

$ x+\mu y+32 z=-1$

has infinitely many solutions, then $(\lambda-\mu)$ is equal to $\qquad$

If $a$, $b$, $c$, $d$, $e$, $f$ are in $G.P$., then the value of $\left| {\begin{array}{*{20}{c}}
{{a^2}}&{{d^2}}&x \\
{{b^2}}&{{e^2}}&y \\
{{c^2}}&{{f^2}}&z
\end{array}} \right|$ depends on