Existance of unique solution of the system of equations 2x + y + z = β , 10x - y + α z

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3 and 4 .Determinants and Matrices

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The existance of the unique solution of the system of equations$2x + y + z = \beta $ , $10x - y + \alpha z = 10$ and $4x+ 3y-z =6$ depends on

A

Both $\alpha $ and $\beta $

B

Neither $\beta $ nor $\alpha $

C

$\beta $ only

D

$\alpha $ only

Solution

For unique solution $\left|\begin{array}{ccc}{2} & {1} & {1} \\ {10} & {-1} & {\alpha} \\ {4} & {3} & {-1}\end{array}\right| \neq 0$

$\Rightarrow 2[1-3 \alpha]-1[-10-4 \alpha]+1[30+4] \neq 0$

$\Rightarrow \quad 46-2 \alpha \neq 0$

$\Rightarrow \quad \alpha \neq 23$

$\therefore \quad$ depend on only $\alpha$

Standard 12

Mathematics

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