For α, β ∈ R , suppose system of linear equations x-y+z=5 ; 2 x+2 y+α z=8 ; 3 x-y+4 z=β has in

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3 and 4 .Determinants and Matrices

medium

For $\alpha, \beta \in R$, suppose the system of linear equations $x-y+z=5$ ; $ 2 x+2 y+\alpha z=8 $ ; $3 x-y+4 z=\beta$ has infinitely many solutions. Then $\alpha$ and $\beta$ are the roots of

$x ^2-10 x +16=0$

$x^2+18 x+56=0$

$x^2-18 x+56=0$

$x^2+14 x+24=0$

(JEE MAIN-2023)

Solution

$\begin{array}{l}\left|\begin{array}{ccc}1 & -1 & 1 \\ 2 & 2 & \alpha \\ 3 & -1 & 4\end{array}\right|=0 ; 8+\alpha-2(-4+1)+3(-\alpha-2)=0 \\ 8+\alpha+6-3 \alpha-6=0 \\ \alpha=4\end{array}$

Standard 12

Mathematics

If $\omega $ be a complex cube root of unity, then $\left| {\,\begin{array}{*{20}{c}}1&\omega &{ – {\omega ^2}/2}\\1&1&1\\1&{ – 1}&0\end{array}\,} \right| = $

medium

The cubic $\left| {\begin{array}{*{20}{c}}
0&{a – x}&{b – x} \\
{ – a – x}&0&{c – x} \\
{ – b – x}&{ – c – x}&0
\end{array}} \right| = 0$ has a reperated root in $x$ then,

normal

If $A = \left[ {\begin{array}{*{20}{c}}
1&{\sin \,\theta }&1\\
{ – \,\sin \,\theta }&1&{\sin \,\theta }\\
{ – 1}&{ – \,\sin \,\theta }&1
\end{array}} \right];$ then for all $\theta \, \in \,\left( {\frac{{3\pi }}{4},\frac{{5\pi }}{4}} \right),$ det $(A)$ lies in the interval

hard

(JEE MAIN-2019)

The value of $\lambda $ for which the system of equations $2x – y – z = 12,$ $x – 2y + z = – 4,$ $x + y + \lambda z = 4$ has no solution is

medium

(IIT-2004)

If $A\, = \,\left[ \begin{gathered}
1\ \ \ \,1\ \ \ \,2\ \ \ \hfill \\
0\ \ \ \,2\ \ \ \,1\ \ \ \hfill \\
1\ \ \ \,0\ \ \ \,2\ \ \ \hfill \\
\end{gathered} \right]$ and $A^3 = (aA-I) (bA-I)$,where $a, b$ are integers and $I$ is a $3 × 3$ unit matrix then value of $(a + b)$ is equal to

normal

For $\alpha, \beta \in R$, suppose the system of linear equations $x-y+z=5$ ; $ 2 x+2 y+\alpha z=8 $ ; $3 x-y+4 z=\beta$ has infinitely many solutions. Then $\alpha$ and $\beta$ are the roots of

Solution

Similar Questions

If $\omega $ be a complex cube root of unity, then $\left| {\,\begin{array}{*{20}{c}}1&\omega &{ – {\omega ^2}/2}\\1&1&1\\1&{ – 1}&0\end{array}\,} \right| = $

The cubic $\left| {\begin{array}{*{20}{c}} 0&{a – x}&{b – x} \\ { – a – x}&0&{c – x} \\ { – b – x}&{ – c – x}&0 \end{array}} \right| = 0$ has a reperated root in $x$ then,

If $A = \left[ {\begin{array}{*{20}{c}} 1&{\sin \,\theta }&1\\ { – \,\sin \,\theta }&1&{\sin \,\theta }\\ { – 1}&{ – \,\sin \,\theta }&1 \end{array}} \right];$ then for all $\theta \, \in \,\left( {\frac{{3\pi }}{4},\frac{{5\pi }}{4}} \right),$ det $(A)$ lies in the interval

The value of $\lambda $ for which the system of equations $2x – y – z = 12,$ $x – 2y + z = – 4,$ $x + y + \lambda z = 4$ has no solution is

If $A\, = \,\left[ \begin{gathered} 1\ \ \ \,1\ \ \ \,2\ \ \ \hfill \\ 0\ \ \ \,2\ \ \ \,1\ \ \ \hfill \\ 1\ \ \ \,0\ \ \ \,2\ \ \ \hfill \\ \end{gathered} \right]$ and $A^3 = (aA-I) (bA-I)$,where $a, b$ are integers and $I$ is a $3 × 3$ unit matrix then value of $(a + b)$ is equal to

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The cubic $\left| {\begin{array}{*{20}{c}}
0&{a – x}&{b – x} \\
{ – a – x}&0&{c – x} \\
{ – b – x}&{ – c – x}&0
\end{array}} \right| = 0$ has a reperated root in $x$ then,

If $A = \left[ {\begin{array}{*{20}{c}}
1&{\sin \,\theta }&1\\
{ – \,\sin \,\theta }&1&{\sin \,\theta }\\
{ – 1}&{ – \,\sin \,\theta }&1
\end{array}} \right];$ then for all $\theta \, \in \,\left( {\frac{{3\pi }}{4},\frac{{5\pi }}{4}} \right),$ det $(A)$ lies in the interval

If $A\, = \,\left[ \begin{gathered}
1\ \ \ \,1\ \ \ \,2\ \ \ \hfill \\
0\ \ \ \,2\ \ \ \,1\ \ \ \hfill \\
1\ \ \ \,0\ \ \ \,2\ \ \ \hfill \\
\end{gathered} \right]$ and $A^3 = (aA-I) (bA-I)$,where $a, b$ are integers and $I$ is a $3 × 3$ unit matrix then value of $(a + b)$ is equal to