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Question

Find a complex number z satisfying the equation z+2|z+1|+i=0

A
2i
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B
2i
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C
2i
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D
None of these
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Solution

The correct option is B 2i
Let z=x+iy. Then,

z+2|z+1|+i=0

x+iy+2|(x+1)+iy|+i=0

x+2(x+1)2+y2+(y+1)i=0

x+2(x+1)2+2y2+(y+1)i=0

x+2(x+1)2+2=0 and y=1

2(x+1)2+2=x and y=1

2(x+1)2+2=x2 and y=1

x2+4x+4=0 and y=1

(x+2)2=0 and y=1

x=2 and y=1

Hence, z=x+iy=2i

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