Question

Let ${\alpha }_1,{\alpha }_2$ are two values of $\alpha$ for which the system $2\alpha x+y=5,x-6y=\alpha$ and $x+y=2$ is consistent, then $|2({\alpha }_1+{\alpha }_2)|$ is:

• A. $21$
• B. $23$
• C. $25$
• D. $27$

Answer 1

The correct option is B $23$

$2\alpha x+y=5;x-6y=\alpha$

$x+y=2$

Now $x=2-y$

$\therefore 2-y-6y=\alpha$

$\therefore 2-7y=\alpha$

Now $2\alpha (2-y)+y=5$

$\therefore 4\alpha -2\alpha y+y=5$

$\therefore y=\frac{2-\alpha }{7}$

$\therefore 4\alpha -2\alpha \left(\frac{2-\alpha }{7}\right)+\frac{2-\alpha }{7}=5$

$\therefore 28\alpha -4\alpha +2{\alpha }^2+2-\alpha =35$

$\therefore 2{\alpha }^2-23\alpha -33=0$

$\therefore 2({\alpha }_1+{\alpha }_2)=2\left(\frac{23}{\alpha }\right)=23$