JEE MAIN - Mathematics (2005 - No. 16)
The system of equations
$$\matrix{ {\alpha \,x + y + z = \alpha - 1} \cr {x + \alpha y + z = \alpha - 1} \cr {x + y + \alpha \,z = \alpha - 1} \cr } $$
has no solutions, if $$\alpha $$ is :
$$-2$$
either $$-2$$ or $$1$$
not $$-2$$
$$1$$
Explanation
$$ax + y + z = \alpha - 1$$
$$x + \alpha \,y + z = \alpha - 1;$$
$$x + y + z\alpha = \alpha - 1$$
$$\Delta = \left| {\matrix{ \alpha & 1 & 1 \cr 1 & \alpha & 1 \cr 1 & 1 & \alpha \cr } } \right|$$
$$ = \alpha \left( {{\alpha ^2} - 1} \right) - 1\left( {\alpha - 1} \right) + 1\left( {1 - \alpha } \right)$$
$$ = \alpha \left( {\alpha - 1} \right)\left( {\alpha + 1} \right) - 1\left( {\alpha - 1} \right) - 1\left( {\alpha - 1} \right)$$
For infinite solutions, $$\Delta = 0$$
$$ \Rightarrow \left( {\alpha - 1} \right)\left[ {{\alpha ^2} + \alpha - 1 - 1} \right] = 0$$
$$ \Rightarrow \left( {\alpha - 1} \right)\left[ {{\alpha ^2} + \alpha - 2} \right] = 0$$
$$ \Rightarrow \alpha = - 2,1;$$
But $$\alpha \ne 1.\,\,\,$$ $$\therefore$$ $$\,\,\alpha = - 2$$
$$x + \alpha \,y + z = \alpha - 1;$$
$$x + y + z\alpha = \alpha - 1$$
$$\Delta = \left| {\matrix{ \alpha & 1 & 1 \cr 1 & \alpha & 1 \cr 1 & 1 & \alpha \cr } } \right|$$
$$ = \alpha \left( {{\alpha ^2} - 1} \right) - 1\left( {\alpha - 1} \right) + 1\left( {1 - \alpha } \right)$$
$$ = \alpha \left( {\alpha - 1} \right)\left( {\alpha + 1} \right) - 1\left( {\alpha - 1} \right) - 1\left( {\alpha - 1} \right)$$
For infinite solutions, $$\Delta = 0$$
$$ \Rightarrow \left( {\alpha - 1} \right)\left[ {{\alpha ^2} + \alpha - 1 - 1} \right] = 0$$
$$ \Rightarrow \left( {\alpha - 1} \right)\left[ {{\alpha ^2} + \alpha - 2} \right] = 0$$
$$ \Rightarrow \alpha = - 2,1;$$
But $$\alpha \ne 1.\,\,\,$$ $$\therefore$$ $$\,\,\alpha = - 2$$
Comments (0)
