સમીકરણની સંહતિ begin{array}{l}α x + y + z = α - 1x + α y + z = α

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3 and 4 .Determinants and Matrices

hard

સમીકરણની સંહતિ $\begin{array}{l}\alpha x + y + z = \alpha - 1\\x + \alpha y + z = \alpha - 1\\x + y + \alpha z = \alpha - 1\end{array}$ નો ઉકેલ ખાલીગણ હોય તો $\alpha $ કિમત મેળવો.

$-2 $ નથી

$1$

$-2$

$-2 $ અથવા $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

$x$ ની . . . કિમત માટે $\left| {\,\begin{array}{*{20}{c}}{ – x}&1&0\\1&{ – x}&1\\0&1&{ – x}\end{array}\,} \right| = 0$ મળે.

easy

ધારોકે $\alpha \beta \gamma=45 ; \alpha, \beta, \gamma \in \mathbb{R}$. જો કોઈ $x, y, z \in \mathbb{R} x y z \neq 0$

માટે $x(\alpha, 1,2)+y(1, \beta, 2)+z(2,3, \gamma)=(0,0,0)$ હોય, તો $6 \alpha+4 \beta+\gamma=$…………..

hard

(JEE MAIN-2024)

$\left| {\,\begin{array}{*{20}{c}}1&1&1\\1&{1 – x}&1\\1&1&{1 + y}\end{array}\,} \right|$ = . . .

medium

જો $\omega = – \frac{1}{2} + i\frac{{\sqrt 3 }}{2}$. તો $\left| {\,\begin{array}{*{20}{c}}1&1&1\\1&{ – 1 – {\omega ^2}}&{{\omega ^2}}\\1&{{\omega ^2}}&{{\omega ^4}}\end{array}\,} \right|= . . . $

easy

(IIT-2002)

જો $\left| {\,\begin{array}{*{20}{c}}{{x^2} + x}&{x + 1}&{x – 2}\\{2{x^2} + 3x – 1}&{3x}&{3x – 3}\\{{x^2} + 2x + 3}&{2x – 1}&{2x – 1}\end{array}\,} \right| = Ax – 12$, તો $A$ મેળવો.

medium

(JEE MAIN-2015)