The value of $k$  for which one of the roots of $x^2-x+3k=0$  is double of one of the roots of $x^2-x+k=0$  is

Let $\alpha$  be a root of $x^2-x+k=0$ ,

then $2\alpha$  is a root of $x^2-x+3k=0$ .

Then the equations are ${\alpha }^2-\alpha +k=0$  ….. ( 1 )

and $4{\alpha }^2-2\alpha +3k=0$   ….. ( 2 )

solve ( 1 ) & ( 2 )

we get,

$\Rightarrow$  $\frac{{\alpha }^2}{-3k+2k}=\frac{\alpha }{4k-3k}=\frac{1}{-2+4}$

$\Rightarrow$  ${\alpha }^2=-\frac{k}{2}$  and $\alpha =\frac{k}{2}$

Now, ${\alpha }^2={\left(\alpha \right)}^2\Rightarrow -\frac{k}{2}={\left(\frac{k}{2}\right)}^2$

$\Rightarrow$ $k^2+2k=0\Rightarrow k=-2$ (k not equals to 0)