1.

The range of real number `alpha` for which the equation `z + alpha|z-1| + 2i = 0` has a solution is :

Answer» `Z+alpha|Z-1|+2i=0`
`Z+2i=-alpha|Z-1|`
`alpha=-(Z+2i)/|Z-1|`
`alpha` is real
`|z-1|` is real
`Z=x+iy`
`-iy*2i=0`
`y=-2`
`alpha=-x/|(x-1)-2i|`
`|(x-1)-2i|^2=(-x/alpha)^2`
`(x-1)^2+4=x^2/alpha^2`
`x^2-2x+4=x^2/alpha^2`
`x^2(1-1/alpha^2)-2x+5=0`
`D>=0`
`b^2-4ac>=0`
`4-4(1-1/alpha^2)5>=0`
`alpha^2<=5/4`
`alpha^2-5/4<=0`
`(alpha-sqrt5/2)(alpha+sqrt5/2)<=0`
`alpha in [-sqrt5/2,sqrt5/2]`
option a is correct.


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