જો a ne b ne c, તો સમીકરણ left| {,begin{array | vedclass

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Question

(a) Obviously, on putting $x = 0$, we observe that the determinant becomes

${\Delta _{x = 0}} = \left| {\,\begin{array}{*{20}{c}}0&{ - a}&{

Vedclass · Accepted Answer

$x = 0$

Vedclass · Answer

(a) Obviously, on putting $x = 0$, we observe that the determinant becomes

${\Delta _{x = 0}} = \left| {\,\begin{array}{*{20}{c}}0&{ - a}&{ - b}\a&0&{ - c}\b&c&0\end{array}\,} ight| = a(bc) - b(ac) = 0$

$	herefore $ $x = 0$ is a root of the given equation.

Aliter : Expanding $\Delta $, we get

$\Delta \equiv - (x - a)\,[ - (x + b)(x - c)] + (x - b)\,[(x + a)\,(x + c)] = 0$

$ \Rightarrow $ $2{x^3} - (2\,\Sigma ab)x = 0$

$ \Rightarrow $ Either $x = 0$ or ${x^2} = \sum {ab} $ (i.e., $x = \pm \sum {ab} )$

Again $x = 0$ satisfies the given equation.

જો $a \ne b \ne c,$ તો સમીકરણ $\left| {\,\begin{array}{*{20}{c}}0&{x - a}&{x - b}\\{x + a}&0&{x - c}\\{x + b}&{x + c}&0\end{array}\,} \right| = 0$ નું સમાધાન કરે તેવી $x$ ની કિમત મેળવો.

Similar Questions

$\left| {\,\begin{array}{*{20}{c}}1&{\cos (\beta - \alpha )}&{\cos (\gamma - \alpha )}\\{\cos (\alpha - \beta )}&1&{\cos (\gamma - \beta )}\\{\cos (\alpha - \gamma )}&{\cos (\beta - \gamma )}&1\end{array}} \right|$ = . . .

જો $\omega $ એ એકનું કાલ્પનિક ઘનમૂળ હોય તો $\Delta = \left| {\begin{array}{*{20}{c}}1&{2\omega }\\\omega &{{\omega ^2}}\end{array}} \right|$, તો ${\Delta ^2}$ = . . .

સુરેખ સમીકરણ સંહતિ $3 x-2 y+z=b$ ; $5 x-8 y+9 z=3$ ; $2 x+y+a z=-1$ ને એક પણ ઉકેલ ન મળે તો,તે માટેની ક્રમયુક્ત જોડ $(a,b)$એ$\dots\dots\dots$ છે.

$\left| {\,\begin{array}{*{20}{c}}{bc}&{bc' + b'c}&{b'c'}\\{ca}&{ca' + c'a}&{c'a'}\\{ab}&{ab' + a'b}&{a'b'}\end{array}\,} \right|$ = . . .

જો $x + y - z = 0,\,3x - \alpha y - 3z = 0,\,\,x - 3y + z = 0$ ને શૂન્યતર ઉકેલ હોય, તો $\alpha$ ની કિમત મેળવો.