Let the system of linear equations $$x + 2y + z = 2$$, $$\alpha x + 3y - z = \alpha $$, $$ - \alpha x + y + 2z = - \alpha $$ be inconsistent. Then $$\alpha$$ is equal to :
JEE · Math · previous-year question
- A.$${5 \over 2}$$
- B.$$-$$$${5 \over 2}$$
- C.$${7 \over 2}$$
- D.$$-$$$${7 \over 2}$$correct
Answer
D. $$-$$$${7 \over 2}$$
Explanation
$$x + 2y + z = 2$$ $$\alpha x + 3y - z = \alpha $$ $$ - \alpha x + y + 2z = - \alpha $$ $$\Delta = \left| {\begin{matrix} 1 & 2 & 1 \\ \alpha & 3 & { - 1} \\ { - \alpha } & 1 & 2 \\ \end{matrix} } \right| = 1(6 + 1) - 2(2\alpha - \alpha ) + 1(\alpha + 3\alpha )$$ $$ = 7 + 2\alpha $$ $$\Delta = 0 \Rightarrow \alpha = - {7 \over 2}$$ $${\Delta _1} = \left| {\begin{matrix} 2 & 2 & 1 \\ \alpha & 3 & { - 1} \\ { - \alpha } & 1 & 2 \\ \end{matrix} } \right| = 14 + 2\alpha \ne 0$$ for $$\alpha = - {7 \over 2}$$ $$\therefore$$ For no solution $$\alpha = - {7 \over 2}$$
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