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

For real numbers $\alpha$ and $\beta$, consider the following system of linear equations:

$x+y-z=2, x+2 y+\alpha z=1,2 x-y+z=\beta$. If the system has infinite solutions, then $\alpha+\beta$ is equal to $…..$

medium

(JEE MAIN-2021)

If $x = cy + bz,\,\,y = az + cx,\,\,z = bx + ay$ (where $x, y, z $ are not all zero) have a solution other than $x = 0$, $y = 0$, $z = 0$ then $a, b$ and $ c $ are connected by the relation

medium

(IIT-1978)

Find values of ${x},$ if $\left|\begin{array}{ll}2 & 3 \\ 4 & 5\end{array}\right|=\left|\begin{array}{ll}x & 3 \\ 2 x & 5\end{array}\right|$

easy

The number of values of $\theta \in (0,\pi)$ for which the system of linear equations
$x + 3y + 7z = 0$
$-x + 4y + 7z = 0$
$(sin\,3\theta )x + (cos\,2\theta )y + 2z = 0$ has a non-trivial solution, is

hard

(JEE MAIN-2019)

$\left| {\,\begin{array}{*{20}{c}}1&a&{{a^2} – bc}\\1&b&{{b^2} – ac}\\1&c&{{c^2} – ab}\end{array}\,} \right| = $

medium

(IIT-1988)

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

For real numbers $\alpha$ and $\beta$, consider the following system of linear equations: $x+y-z=2, x+2 y+\alpha z=1,2 x-y+z=\beta$. If the system has infinite solutions, then $\alpha+\beta$ is equal to $…..$

If $x = cy + bz,\,\,y = az + cx,\,\,z = bx + ay$ (where $x, y, z $ are not all zero) have a solution other than $x = 0$, $y = 0$, $z = 0$ then $a, b$ and $ c $ are connected by the relation

Find values of ${x},$ if $\left|\begin{array}{ll}2 & 3 \\ 4 & 5\end{array}\right|=\left|\begin{array}{ll}x & 3 \\ 2 x & 5\end{array}\right|$

The number of values of $\theta \in (0,\pi)$ for which the system of linear equations $x + 3y + 7z = 0$ $-x + 4y + 7z = 0$ $(sin\,3\theta )x + (cos\,2\theta )y + 2z = 0$ has a non-trivial solution, is

$\left| {\,\begin{array}{*{20}{c}}1&a&{{a^2} – bc}\\1&b&{{b^2} – ac}\\1&c&{{c^2} – ab}\end{array}\,} \right| = $

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For real numbers $\alpha$ and $\beta$, consider the following system of linear equations:

$x+y-z=2, x+2 y+\alpha z=1,2 x-y+z=\beta$. If the system has infinite solutions, then $\alpha+\beta$ is equal to $…..$

The number of values of $\theta \in (0,\pi)$ for which the system of linear equations
$x + 3y + 7z = 0$
$-x + 4y + 7z = 0$
$(sin\,3\theta )x + (cos\,2\theta )y + 2z = 0$ has a non-trivial solution, is