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  • Let the system of linear equations <img alt="x+2 y+z=2, \alpha x+3 y-z=\alpha,-\alpha x+y+2 z=-\alpha" src="https://entrancecorner.oncodecogs.com/gif.latex?x&plus;2%20y&plus;z%3D2%2C%20%5Ca

Let the system of linear equations x+2 y+z=2, \alpha x+3 y-z=\alpha,-\alpha x+y+2 z=-\alpha be inconsistent. Then \alpha is equal to :
 

Option: 1

\frac{5}{2}


Option: 2

-\frac{5}{2}


Option: 3

\frac{7}{2}


Option: 4

-\frac{7}{2}


Answers (1)

best_answer

\mathrm{\Delta=\left|\begin{array}{ccc} 1 & 2 & 1 \\ \alpha & 3 & -1 \\ -\alpha & 1 & 2 \end{array}\right|}\\

\mathrm{=0 \Rightarrow 1 \times 7-2 \times(\alpha)+1 \times 4 \alpha=0 }\\

\mathrm{\quad \Rightarrow \alpha=-7 / 2}

\mathrm{\Delta_{x}=\left|\begin{array}{ccc} 2 & 2 & 1 \\ \alpha & 3 & -1 \\ -\alpha & 1 & 2 \end{array}\right|}

\mathrm{=2 \times 7-2 \times \alpha+1 \times 4 \alpha}

\mathrm{at \: \alpha=-7 / 2},

\mathrm{\Delta_{x}=14+2 \times \frac{7}{2}-4 \times \frac{7}{2}=7 \neq 0}

So the system of equations is inconsistent for  \mathrm{\alpha =\frac{-7}{2}}

Hence the correct answer is option 4

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